CA3086361A1 - Fast and partition-resilient blockchains - Google Patents

Abstract

In a transaction system in which transactions are organized in blocks, a new block Br of valid transactions is constructed, relative to a sequence of prior blocks B0,..., Br-1, by having an entity determine a quantity Q from the prior blocks, having the entity use a secret key in order to compute a string S uniquely associated to Q and the entity, having the entity compute from S a quantity T that is one of: S itself, a function of S, and/or a hash value of S, having the entity determine whether T possesses a given property, and, if T possesses the given property, having the entity digitally sign a hash value H of B r and make available S, B r and a digitally signed version of H, wherein, B r may be proposed in different steps of the round r and may be re-proposed multiple times during round r, and an entity may verify a hash value H of a block B independent of whether the entity has received B or not.

Description

Fast and Partition-Resilient Blockchains CROSS REFERENCE TO RELATED APPLICATIONS
This application claims priority to U.S. Prov. App. No. 62/607,558, filed December 19, 2017, and entitled FASTER BYZANTINE AGREEMENT IN PROPAGATION NETWORKS
WITH > 2/3 HONEST MAJORITY, and to U.S. Prov. App. No. 62/632,944, filed February 20, 2018, and entitled ALGORAND, and to U.S. Prov. App. No. 62/643,331, filed March 15, 2018, and entitled INCENTIVES AND TRANSACTION FEES IN ALGORAND, and to U.S. Prov. App. No. 62/777,410, filed December 10, 2018, and entitled VIRTUAL
BLOCKCHAIN PROTOCOLS FOR FAIR ELECTRONIC EXCHANGE, and to U.S. Prov.
App. No. 62/778,482, filed December 12, 2018, and entitled VIRTUAL BLOCKCHAIN
PROTOCOLS FOR FAIR ELECTRONIC EXCHANGE, which are all incorporated by reference herein.
TECHNICAL FIELD
This application relates to the field of electronic transactions and more particularly to the field of distributed public ledgers, securing the contents of sequence of transaction blocks, and the verification of electronic payments.
BACKGROUND OF THE INVENTION
A blockchain consists of an augmentable sequence of blocks: B1, B2,..., wherein each block consists of a number of transactions, the hash of the previous block, and other data e.g., the index of the block, time information, etc. Useful properties of a blockchain are that (P1) there is a unique block corresponding to each index 1, 2, ... , (P2) every user in the system eventually learns the content of every block, (P3) no one can alter the content or the order of the blocks, and (P4) any valid transactio will eventually enter a block in the chain.
Users can digitally sign messages, and thus each user possesses at least one public key and a corresponding secret key. In a blockchain, in general, one knows the public keys but not necessarily the user who owns it. Accordingly, we may identify a public key with its owner.
Several blockchain systems require a block to be certified by the digital signatures of sufficiently many users in the system. In some systems such certifying users belong to a fixed set of users. In some other systems they belong to a dynamically changing set.
The latter is preferrable, because an adversary would have a harder time to corrupt a dynamically changing set, particularly if the set is not only dynamic, but unpredictable as well.
A particularly effective way of selecting a set of users in a verifiable but unpredictable way is the cryptographic sortition technique is described in published PCT
patent application PCT/U52017/031037, which is incorporated by reference herein. Here, a user i belongs to a set of users empowered to act in some step s during the production of block number r based on the result of a computation that i performs via a secret key of his, using inputs s and r, and possibly other inputs and other data (e.g., the fact that the user has joined the system at least k blocks before block r, for some given integer k). For instance, i's computation may involve i's digital signature, szr's, of such inputs, hashing szr's, and checking whether the hash is less than a given threshold t. (Indeed, like any other string, a hashed value can be interpreted in some standard way as a number.) In this case, CT tr's = Str's is defined to be the credential of i for step s about block r. Such credential proves to anyone that i is indeed entitled to produce a (preferably signed) message mzr's, his voting message for step s in round r, in the process aimed at producing block r. In fact, i's digital signatures can be checked by anyone, and anyone can hash a given value, and then check whether the result is indeed less than (or equal to) a given number.
A blockchain works by propagating messages (e.g., blocks, transactions, voting messages, digital signatures, etc). Typically, but not exclusively, messages are propagated by gossiping them in a peer-to-peer fashion, or via relays. Several blockchain systems require the propagation network to guarantee the delivery of messages propagated by every honest user to other honest users within a bounded delay. Some further require the users to have (almost) aligned system clocks, so that the users propagate messages in a synchronized way _________ e.g., the users enter step 2 in the generation of block 100 at time 11:20:00am EST, the voting messages for this step are delivered by time 11:20:05am EST, and the users enter step 3 of block 100 then. A less demanding and thus preferable requirement, as imposed by Algorand, is that the users' clocks have (almost) the same speed, but the actual times shown on the clocks can be arbitrarily far from each other. A user starts his own step s in the generation of block r based on the messages he has received from the propagation network, and ends it based on messages received and how long his own clock has advanced since he started this step.
When the propagation network satisfies this requirement, Algorand ensures that the adversary cannot prevent the blockchain from functioning properly (including achieving properties P1 _______________________________________________________________ P4). However, this relies on the adversary not attacking the propagation network itself. Such attacks include any effort the adversary may take in order to violate the bounded delay of message delivery for a sufficiently large amount of users __________ e.g., by partitioning the users into two groups of equal size and controlling the message delivery channels between them, so that a message propagated by a user from group 1 may have indefinite delay before reaching any user in group 2.
It is thus desirable to weaken this requirement and provide blockchains and electronic money systems that do not suffer from the inefficiencies and insecurities of known decentralized approaches.
SUMMARY OF THE INVENTION.
According to the system described herein, an entity manages a transaction system in which transactions are organized in a sequence of blocks that are certified by digital signatures of a sufficient number of verifiers by the entity proposing a hash of a block B' that includes new valid transactions relative to a sequence of certified blocks B Br-1- if no rth block Br has been certified and by the entity proposing a hash of the block Br if the rth block Br has been verified by a sufficient number of other entities. A block may be certified by the entity only in response to confirming transactions for the block and confirming that the block was constructed and propagated by an entity entitled to construct and propagate the block. The entity may propose a hash value by digitally signing the hash value to provide a digitally-signed version of the hash value and the entity may propagate the digitally-signed version of the hash value to a network that includes other entities. If no rth block Br has been certified, the entity may also digitally sign and propagate the block B'. The entity may determine a quantity Q from the prior blocks and may use a secret key in order to compute a string S uniquely associated with Q
and computes from S a quantity T that is S itself, a function of S, and/or a hash value of S and the entity may determine whether to propose a hash value by determining whether T possesses a given property. S may be a signature of Q under a secret key of the entity, T may be a hash of S and T may possess the given property if T is less than a given threshold.
The entity may be

2 part of a network of entities and a particular one of the entities may construct and propagates the block Br . The rth block Br may be determined to be certified by the entity if the entity receives an indication that at least a predetermined number of the entities individually certify a hash value corresponding to the rth block Br . In response to the entity receiving the indication that a predetermined number of the entities individually certified the rth block Br , the entity may increment r to begin adding additional blocks to the sequence of blocks.
The particular one of the entities may be individually chosen by a predetermined number of the entities to be a leader. The rth block Br may be determined to be certifiable by the entity if the entity receives an indication that at least a predetermined number of the entities individually verify receiving an indication that the particular one of the entities has provided a hash value corresponding to the rth block Br to each of the predetermined number of the entities.
According further to the system described herein, an entity manages a transaction system in which transactions are organized in a sequence of certified blocks by the entity receiving a hash value of a block Br from an other entity that generated the block based on new valid transactions relative to a sequence of certified blocks B , . . . , Br-1,the entity certifying the block Br in response to a sufficient number of other entities having indicated receipt of the hash value of the block Br from the other entity and the hash value being valid for the block Br , the entity generating a new block B' based on new valid transactions relative to a sequence of certified blocks B ,... , Br-1- in response to an insufficient number of the other entities indicating receipt of the hash value of the block Br from the other entity, B' being different from Br , and the entity incrementing r to begin adding additional blocks to the sequence of blocks in response to the entity receiving the indication that a predetermined number of the entities individually certified the rth block Br or a predetermined number of the entities individually certified the new block B'. The blocks may be certified by digital signatures. New blocks may be proposed by different ones of the entities until receiving the indication that a predetermined number of the entities individually certified a previously proposed block. The entity may provides an indication that a new block should be generated in response to the hash value not being valid for the block Br . The entity may generate a new block B' based on new valid transactions relative to a sequence of certified blocks B ,.. . , Br-1- in response to a sufficient number of the other entities providing an indication that a new block should be generated.
The entity may provide an indication that the hash value of the block Br should be propagated in response to a sufficient number of the other entities having indicated receipt of the hash value of the block Br from the other entity and the hash value being valid for the block Br .
According further to the system described herein, an entity verifies a proposed hash value of a new block Br of transactions relative to a given a sequence of blocks, B ,..
. , Br-1, without access to the new block Br in a transaction system in which transactions are organized in blocks and blocks are certified by a set of digital signatures by having the entity determine a quantity Q from the prior blocks, having the entity compute a digital signature S of Q, having the entity compute from S a quantity T that is S itself, a function of S, and/or hash value of S, having the entity determine whether T possesses a given property, andfif T possesses the given property, having the entity verify the proposed hash value of the new block Br independent of confirming whether the proposed hash value corresponds to the new block Br . The entity may propagate the proposed hash value of the new block Br prior to receiving the new block Br .
According furtheer to the system described herein, in a transaction system in which transactions are organized in blocks, a new block Br of valid transactions is constructed, relative to a sequence of prior blocks B , BI-, ... , Br-1, by having an entity determine a quantity Q from the prior blocks, having the entity use a secret key in order to compute a string S uniquely associated to Q, having the entity compute from S a quantity T that is S
itself, a function of S, and/or a hash value of S, having the entity determine whether T possesses a given property

3 and, if T possesses the given property, having the the entity compute a hash value H of Br , digitally sign H and make available to others S, Br and a digitally signed version of H. The secret key may be a secret signing key corresponding to a public key of the entity and S is a digital signature of Q by the entity. T may be a binary expansion of a number and satisfies the given property if T is less than a given number p. S may be made available by making S
deducible from Br . Each user may have a balance in the transaction system and p may vary for each user according to the balance of each user.
According further to the system described herein, selecting a subset of users in a blockchain system to verify a data string m relative to a sequence of prior blocks B ,...
, Br-1-, includes causing at least some of the users to determine a quantity Q from the prior blocks, causing at least some of the users to compute a digital signature S of Q and other information, causing at least some of the users to determine a hash value of the digital signature, causing at least some of the users to compare the hash value to a pre-determined threshold, and causing the subset of the users to digitally sign m together with other information and make available to others S
and a digitally signed version of m in response to the hash value being below a pre-determined threshold for each of the subset of users. The digital signature may be credentialed if the hash value is below a pre-determined threshold. Each user may have a balance in the transaction system and the pre-determined threshold may vary for each user according to the balance of each user. The pre-determined threshold for each user may be chosen to cause the subset of the users to contain a minimum number of the users. The data string m may be a hash value of a new block Br . The data string m may be verified by at least a given number of credentialed signatures of m.
According further to the system described herein, selecting a subset of users in a blockchain system to certify a new block Br relative to a sequence of prior blocks B ,...
, Br-1-, includes causing at least some of the users to determine a quantity Q from the prior blocks, causing at least some of the users to compute a digital signature S of Q and other information, causing at least some of the users to determine a hash value of the digital signature, causing at least some of the users to compare the hash value to a pre-determined threshold, causing the subset of the users to determine Br is valid relative to B ,. .. , Br-1- in response to the hash value being below a pre-determined threshold for each of the subset of users and causing the subset of the users to digitally sign a hash value H of Br together with other information and make available to others S and a digitally signed version of H. A particular one of the users may digitally sign the new block Br only if the particular one of the users verifies information provided in the new block Br . Each user may have a balance in the transaction system and the pre-determined threshold may vary for each user according to the balance of each user. The pre-determined threshold for each user may be chosen to cause the subset of the users to contain a minimum number of the users. The block Br may be certified by at least a given number of credentialed signatures of H from users who have determined Br is valid relative to B ,...
, Br-1-.
According further to the system described herein, computer software, provided in a non-transitory computer-readable medium, includes executable code that implements any of the steps described herein.
The present invention dispenses with the requirement on the propagation network's message delivery delay for the security of certifying new blocks. A new block is first prepared (e.g., proposed, propagated, and/or agreed upon by at least some users) and then it is certified.
A user who has receive a newly constructed block, a hash value of a new block, and/or credentialed signatures within a desired time interval proceeds to verify and/or certify the new block. However, we wish to certify a new block even when messages propagated in the network may be indefinitely delayed. The certification of a block B guarantees that certain valuable properties apply to the block. A typical main property is to enable a user, even a

4 user who has not participated to or observed the preparation of a block B, to ascertain that B
has been added to the blockchain, or even that B is the rth block in the blockchain. Another valuable property (often referred to as finalization) guarantees that B will not disappear from the blockchain, due to a soft fork, even in the presence of a partition of the communication network on which the blockchain protocol is executed. A partition of the network may cause the users to be separated into multiple groups, with messages propagated from one group not reaching users in other groups. A partition may be resolved after an indefinite amount of time, after which the network again guarantees message delivery after bounded delays.
Assume that a block B has been prepared, in any fashion and in any number of steps.
Realizing that a block has been properly prepared requires time and effort, and the verification of various pieces of evidence. A certificate of B consists of a given number of users' digital signatures with valid credentials. Such a certificate of B vouches that the users who have produced such signatures have participated to or observed the preparation of B. At least, it vouches that, if one of the digital signatures of the certificate has been produced by an honest user, then that user has checked that B has been properly prepared.
In the system described herein, while a user is collecting evidences for one block B being properly prepared as the rth block in the blockchain relative to prior blocks B ,. .. , Br-1-, the user may construct and propose a new block B' as the rth block in the blockchain relative to B , ... , Br-1- if he has evidence that a certificate of B has not been generated in the system. A
user proposes B' by determining a quantity Q from prior blocks, computing a string S uniquely associated to Q using a secret key, computing from S a quantity T that is S
itself, a function of S, and/or a hash value of S, determining whether T possesses a given property and, if T
possesses the given property, computing a hash value H' of B', digitally signing H' and making available to others S, B' and a digitally signed version of H'. The secret key may be a secret signing key corresponding to a public key of the entity and S is a digital signature of Q by the entity. T may be a binary expansion of a number and satisfies the given property if T is less than a given number p. S may be made available by making S deducible from B'.
Each user may have a balance in the transaction system and p may vary for each user according to the balance of each user.
In the system described herein, while a user is collecting evidences for one block B being properly prepared as the rth block in the blockchain relative to prior blocks B ,. .. , Br-1-, the user may re-propose B as the rth block in the blockchain if he has evidence that a certificate of B may have been generated in the system but have not been made available to him. A user may re-propose B without having received B itself but rather having received a given number of users' digital signatures with valid credentials verifying a hash value of B.
A user may re-propose B by determining a quantity Q from prior blocks, computing a string S uniquely associated to Q using a secret key, computing from S a quantity T that is S itself, a function of S, and/or a hash value of S, determining whether T possesses a given property and, if T
possesses the given property, digitally signing a hash value H of B and making available to others S and a digitally signed version of H. The secret key may be a secret signing key corresponding to a public key of the entity and S is a digital signature of Q by the entity. T
may be a binary expansion of a number and satisfies the given property if T is less than a given number p. S
may be made available by making S deducible from B'. Each user may have a balance in the transaction system and p may vary for each user according to the balance of each user.
Proposing new blocks and re-proposing existing blocks may happen indefinite amount of times during the generation of a rth block of the blockchain and may be carried out by different users. A block B may have more than one certificates generated from different steps. However, during the generation of a rth block of the blockchain, one and only one block will have a certificate and thus be considered by the users as the rth block of the blockchain.

5 The efficiency of the system described herein derives from the following facts. First, a user i may verify and/or re-propose a hash value of a block B before the user may receive B itself. Second, a new block B' may be proposed as the rth block before all users may have collected evidence that a previously proposed block B as the rth block does not have a certificate generated in the system. Indeed, B' may be proposed as soon as one user has collected such evidence. Third, a previously proposed block B as the rth block of the blockchain may be re-proposed before all users may have collected evidence that a certificate for B may have been generated in the system but not made available to all. Indeed, B may be re-proposed as soon as one user has collected such an evidence.
An evidence may consists of a set of credentialed signatures properly formed to verify a data string m. Evidences for different purposes may consist of different numbers of signatures. The security of the system described herein derives from proper choices of pre-determined thresholds that users compare the hash values of their signatures to when verifying different data strings, and from proper choices of numbers of signatures sufficient to form evidences for different purposes. For instance, let p be the maximum percentage of malicious users in the system.
Typically, malicious users are in a minority _______________________________ e.g., p < 1/3. Then the pre-determined threshold t and the sufficient number n of signatures forming a certificate for a block may be chosen so that, with sufficiently high probability, (a) for any possible block value B, there are n or more credentialed signatures of honest users to form a certificate for B and (b) in any certificate of B, more than 2/3 of credentialed signatures belongs to honest users.
The system described herein is agnostic about whether ephemeral keys are used in the blockchain: when users propose new blocks, re-propose existing blocks, or verify data strings, the users may use long-term secret keys to generate credentialed signatures, where the keys may be used repeatedly during the life time of the system, or the users may use ephemeral secret keys where one key is used only once, or the users may use combinations of long-term keys and ephemeral keys.
As part of the system described herein, a user i may not only have a credentialed signature for participating in the generation of a block, but also a credentialed signature with a weight (essentially a credentialed signature associated with a number of votes).
Indeed, the weight of i's credential signature for participating in the generation of a block may depend on how much money i has in the system. Indeed, rather than having a single pre-determined threshold t for all users for participating in block generation, each user i may have his own threshold tz that is higher the higher i's amount of money is. And a user i may have different thresholds for participating in block generation in different ways ________________________ e.g., proposing a new block, re-proposing a block, or verifying a data string m of a specific format. For simplicity, but without limitation intended, we shall continue to describe our system treating a user i with a weight-n credentialed signature as n users, each having a (weight-1) credentialed signature.
Below, after quickly recalling the traditional Algorand system, we provide an example of the preferred embodiment, without any limitation intended, based on Algorand.
BRIEF DESCRIPTION OF THE DRAWINGS.
Embodiments of the system described herein are explained in more details in accordance with the figures of the drawings, which are briefly described as follows.
FIG. 1 is a schematic representation of a network and computing stations according to an embodiment of the system described herein.
FIG. 2 is a schematic and conceptual summary of the first step of Algorand system, where a new block of transactions is proposed.
FIG. 3 is a schematic and conceptual summary of the agreement and certification of a new block in the Algorand system.

6 FIG. 4 is a schematic diagram illustrating a Merkle tree and an authenticating path for a value contained in one of its nodes.
FIG. 5 is a schematic diagram illustrating the 8 Merkle trees corresponding to the first 8 blocks constructed in a blocktree.
FIG. 6 is a schematic representation of a partitioned network and computing stations according to an embodiment of the system described herein.
DETAILED DESCRIPTION OF VARIOUS EMBODIMENTS.
The system described herein provides a mechanisms for distributing transaction verification and propagation so that no entity is solely responsible for performing calculations to verify and/or propagate transaction information. Instead, each of the participating entities shares in the calculations that are performed to propagate transaction in a verifiable and reliable manner.
Referring to FIG. 1, a diagram 20 shows a plurality of computing workstations 22a-22c connected to a data network 24, such as the Internet. The workstations 22a-22c communicate with each other via the network 24 to provide distributed transaction propagation and verification, as described in more detail elsewhere herein. The system may accommodate any number of workstations capable of providing the functionality described herein, provided that the workstations 22a-22c are capable of communicating with each other.
Each of the workstations 22a-22c may independently perform processing to propagate transactions to all of the other workstations in the system and to verify transactions, as described in more detail elsewhere herein.
FIG. 2 diagrammatically and conceptually summarizes the first step of a round r in the Algorand system, where each of a few selected users proposes his own candidate for the rth block. Specifically, the step begins with the users in the system, a, ... , z, individually undergo the secret cryptographic sortition process, which decides which users are selected to propose a block, and where each selected user secretly computes a credential proving that he is entitled to produce a block. Only users b, d, and h are selected to propose a block, and their respectively r 1 r computed credentials are o-b' ,a1, and o-hr'l. Each selected user i assembles his own proposed block, B"' , ephemerally signs it (i.e., digitally signs it with an ephemeral key, as explained later on), and propagates to the network together with his own credential. The leader of the round is the selected user whose credential has the smallest hash. The figure indicates the leader to be user d. Thus his proposed block, B, is the one to be given as input to the Binary agreement protocol.
FIG. 3 diagrammatically and conceptually summarizes Algorand's process for reaching agreement and certifying a proposed block as the official rth block, Br .
Since the first step of Algorand consists of proposing a new block, this process starts with the second step. This step actually coincides with the first step of Algorand's preferred Byzantine agreement protocol, BA*. Each step of this protocol is executed by a different "committee" of players, randomly selected by secret cryptographic sortition (not shown in this figure).
Accordingly, the users selected to perform each step may be totally different. The number of Steps of BA* may vary.
Fig. 3 depicts an execution of BA* involving 7 steps: from Algorand's 2 through Algorand's step 8. The users selected to perform step 2 are a, e, and q. Each user i E
{a, e, q} propagates to the network his credential, CT tr that proves that i is indeed entitled to send a message in step 2 of round r of Algorand, and his message proper of this step, mzr's, ephemerally signed.
Steps 3-7 are not shown. In the last step 8, the figure shows that the corresponding selected users, b, f, and x, having reached agreement on Br as the official block of round r, propagate their own ephemeral signatures of block Br (together, these signatures certify Br ) and their own credentials, proving that they are entitled to act in Step 8.
FIG. 4 schematically illustrates a Merkle tree and one of its autenticating path. Specifically,

7 Fig. 4.A illustrates a full Merkle tree of depth 3. Each node x, where x is denoted by a binary string of length < 3, stores a value vx. If x has length < 2, then vx = H(vxo, vi). For the Merkle tree of Fig. 4.a, Fig. 4.B illustrates the authenticating path of the value v010.
FIG. 5 schematically illustrates the 8 Merkle trees, corresponding to the first 8 blocks constructed in a blocktree, constructed within a full binary tree of depth 3.
In Fig. 5.i, nodes marked by an integer belong to Merkle tree T. Contents of nodes marked by i (respectively, by i) are temporary (respectively, permanent).
The description herein focuses on transactions that are payments and on describing the system herein as a money platform. Those skilled in the art will realize that the system described herein can handle all kinds of transactions as well.
The system described herein has a very flexible design and can be implemented in various, but related, ways. We illustrate its flexibility by detailing two possible embodiments of its general design. From them, those skilled in the art can appreciate how to derive all kinds of other implementations as well.
To facilitate understanding the invention, and allow to internal cross reference of its various parts, we organize its presentation in numbered and titled sections. The first sections are common to both of the detailed embodiments.
1 Introduction Money is becoming increasingly virtual. It has been estimated that about 80%
of United States dollars today only exist as ledger entries. Other financial instruments are following suit.
In an ideal world, in which we could count on a universally trusted central entity, immune to all possible cyber attacks, money and other financial transactions could be solely electronic.
Unfortunately, we do not live in such a world. Accordingly, decentralized cryptocurrencies, such as Bitcoin, and "smart contract" systems, such as Ethereum, have been proposed. At the heart of these systems is a shared ledger that reliably records a sequence of transactions, as varied as payments and contracts, in a tamperproof way. The technology of choice to guarantee such tamperproofness is the blockchain. Blockchains are behind applications such as cryptocurrencies, financial applications, and the Internet of Things. Several techniques to manage blockchain-based ledgers have been proposed: proof of work, proof of stake, practical Byzantine fault-tolerance, or some combination.
Currently, however, ledgers can be inefficient to manage. For example, Bitcoin's proof-of-work approach requires a vast amount of computation, is wasteful and scales poorly. In addition, it de facto concentrates power in very few hands.
We therefore wish to put forward a new method to implement a public ledger that offers the convenience and efficiency of a centralized system run by a trusted and inviolable authority, without the inefficiencies and weaknesses of current decentralized implementations. We call our approach Algorand, because we use algorithmic randomness to select, based on the ledger constructed so far, a set of verifiers who are in charge of constructing the next block of valid transactions. Naturally, we ensure that such selections are provably immune from manipulations and unpredictable until the last minute, but also that they ultimately are universally clear.
Algorand's approach is quite democratic, in the sense that neither in principle nor de facto it creates different classes of users (as "miners" and "ordinary users" in Bitcoin). In Algorand "all power resides with the set of all users".
One notable property of Algorand is that its transaction history may fork only with very small probability (e.g., one in a trillion, that is, or even 10-18). Algorand can also address some legal and political concerns.

8 The Algorand approach applies to blockchains and, more generally, to any method of generating a tamperproof sequence of blocks. We actually put forward a new method alternative to, and more efficient than, blockchains that may be of independent interest.
1.1 Bitcoin's Assumption and Technical Problems Bitcoin is a very ingenious system and has inspired a great amount of subsequent research.
Yet, it is also problematic. Let us summarize its underlying assumption and technical problems which are actually shared by essentially all cryptocurrencies that, like Bitcoin, are based on proof-of-work.
For this summary, it suffices to recall that, in Bitcoin, a user may own multiple public keys of a digital signature scheme, that money is associated with public keys, and that a payment is a digital signature that transfers some amount of money from one public key to another.
Essentially, Bitcoin organizes all processed payments in a chain of blocks, B1, B2,..., each consisting of multiple payments, such that, all payments of B1, taken in any order, followed by those of B2, in any order, etc., constitute a sequence of valid payments. Each block is generated, on average, every 10 minutes.
This sequence of blocks is a chain, because it is structured so as to ensure that any change, even in a single block, percolates into all subsequent blocks, making it easier to spot any alteration of the payment history. (As we shall see, this is achieved by including in each block a cryptographic hash of the previous one.) Such block structure is referred to as a blockchain.
Assumption: Honest Majority of Computational Power Bitcoin assumes that no malicious entity (nor a coalition of coordinated malicious entities) controls the majority of the computational power devoted to block generation. Such an entity, in fact, would be able to modify the blockchain, and thus re-write the payment history, as it pleases. In particular, it could make a payment 0, obtain the benefits paid for, and then "erase" any trace of 0.
Technical Problem 1: Computational Waste Bitcoin's proof-of-work approach to block generation requires an extraordinary amount of computation. Currently, with just a few hundred thousands public keys in the system, the top 500 most powerful supercomputers can only muster a mere 12.8% percent of the total computational power required from the Bitcoin players. This amount of computation would greatly increase, should significantly more users join the system.
Technical Problem 2: Concentration of Power Today, due to the exorbitant amount of computation required, a user, trying to generate a new block using an ordinary desktop (let alone a cell phone), expects to lose money. Indeed, for computing a new block with an ordinary computer, the expected cost of the necessary electricity to power the computation exceeds the expected reward. Only using pools of specially built computers (that do nothing other than "mine new blocks"), one might expect to make a profit by generating new blocks. Accordingly, today there are, de facto, two disjoint classes of users: ordinary users, who only make payments, and specialized mining pools, that only search for new blocks.
It should therefore not be a surprise that, as of recently, the total computing power for block generation lies within just five pools. In such conditions, the assumption that a majority of the computational power is honest becomes less credible.
Technical Problem 3: Ambiguity In Bitcoin, the blockchain is not necessarily unique.
Indeed its latest portion often forks: the blockchain may be say B1, , Bk, Bik+2, according to one user, and B1, , Bk, N+1, N+2, N+3 according another user. Only after

9 several blocks have been added to the chain, can one be reasonably sure that the first k 3 blocks will be the same for all users. Thus, one cannot rely right away on the payments contained in the last block of the chain. It is more prudent to wait and see whether the block becomes sufficiently deep in the blockchain and thus sufficiently stable.
Separately, law-enforcement and monetary-policy concerns have also been raised about Bitcoin.' 1.2 Algorand, in a Nutshell Setting Algorand works in a very tough setting. Briefly, (a) Permissionless and Permissioned Environments. Algorand works efficiently and securely even in a totally permissionless environment, where arbitrarily many users are allowed to join the system at any time, without any vetting or permission of any kind. Of course, Algorand works even better in a permissioned environment.
(b) Very Adversarial Environments. Algorand withstands a very powerful Adversary, who can (1) instantaneously corrupt any user he wants, at any time he wants, provided that, in a permissionless environment, 2/3 of the money in the system belongs to honest user. (In a permissioned environment, irrespective of money, it suffices that 2/3 of the users are honest.) (2) totally control and perfectly coordinate all corrupted users; and (3) schedule the delivery of all messages, provided that each message m sent by a honest user reaches reaches all (or sufficiently many of) the honest users within a time Arn, which solely depends on the size of m.
Main Properties Despite the presence of our powerful adversary, in Algorand = The amount of computation required is minimal. Essentially, no matter how many users are present in the system, each of fifteen hundred users must perform at most a few seconds of computation.
= A new block is generated quickly and will de facto never leave the blockchain. That is, Algorand's blockchain may fork only with negligible probability (i.e., less than one in a trillion or 10-18). Thus, users can relay on the payments contained in a new block as soon as the block appears.
= All power resides with the users themselves. Algorand is a truy distributed system. In particular, there are no exogenous entities (as the "miners" in Bitcoin), who can control which transactions are recognized.
'The (pseudo) anonymity offered by Bitcoin payments may be misused for money laundering and/or the financing of criminal individuals or terrorist organizations. Traditional banknotes or gold bars, that in principle offer perfect anonymity, should pose the same challenge, but the physicality of these currencies substantially slows down money transfers, so as to permit some degree of monitoring by law-enforcement agencies.
The ability to "print money" is one of the very basic powers of a nation state. In principle, therefore, the massive adoption of an independently floating currency may curtail this power.
Currently, however, Bitcoin is far from being a threat to governmental monetary policies, and, due to its scalability problems, may never be.

Algorand's Techniques.
1. A NEW AND FAST BYZANTINE AGREEMENT PROTOCOL. Algorand generates a new block via an inventive cryptographic, message-passing, binary Byzantine agreement (BA) protocol, BA*. Protocol BA* not only satisfies some additional properties (that we shall soon discuss), but is also very fast. Roughly said, its binary-input version consists of a 3-step loop, in which a player i sends a single message m, to all other players. Executed in a complete and synchronous network, with more than 2/3 of the players being honest, with probability >
1/3, after each loop the protocol ends in agreement. (We stress that protocol BA* satisfies the original definition of Byzantine agreement, without any weakenings.) Algorand leverages this binary BA protocol to reach agreement, in our different commu-nication model, on each new block. The agreed upon block is then certified, via a prescribed number of digital signature of the proper verifiers, and propagated through the network.
2. SECRET CRYPTOGRAPHIC SORTITION. Although very fast, protocol BA* would benefit from further speed when played by millions of users. Accordingly, Algorand chooses the players of BA* to be a much smaller subset of the set of all users. To avoid a different kind of concentration-of-power problem, each new block Br will be constructed and agreed upon, via a new execution of B A* , by a separate set of selected verifiers, SVr. In principle, selecting such a set might be as hard as selecting Br directly. We traverse this potential problem by a novel approach that we term secret cryptographic sortition. Sortition is the practice of selecting officials at random from a large set of eligible individuals.
(Sortition was practiced across centuries: for instance, by the republics of Athens, Florence, and Venice. In modern judicial systems, random selection is often used to choose juries. Random sampling has also been advocated for elections.) In a decentralized system, of course, choosing the random coins necessary to randomly select the members of each verifier set SVr is problematic. We thus resort to cryptography in order to select each verifier set, from the population of all users, in a way that is guaranteed to be automatic (i.e., requiring no message exchange) and random.
In a similar fashion we select a user, the leader, in charge of proposing the new block Br , and the verifier set SVr, in charge to reach agreement on the block proposed by the leader. The inventive system leverages some information, Qr-1, that is deducible from the content of the previous block and is non-manipulatable even in the presence of a very strong adversary.
3. THE QUANTITY (SEED) Q. . We use the the last block Br ¨1 in the blockchain in order to automatically determine the next verifier set and leader in charge of constructing the new block Br . The challenge with this approach is that, by just choosing a slightly different payment in the previous round, our powerful Adversary gains a tremendous control over the next leader.
Even if he only controlled only 1/1000 of the players/money in the system, he could ensure that all leaders are malicious. (See the Intuition Section 4.1.) This challenge is central to all proof-of-stake approaches, and, to the best of our knowledge, it has not, up to now, been satisfactorily solved.
To meet this challenge, we purposely construct, and continually update, a separate and carefully defined quantity, Qr , which provably is, not only unpredictable, but also not influentiable, by our powerful Adversary. We may refer to Qr as the rth seed, as it is from Qr that Algorand selects, via secret cryptographic sortition, all the users that will play a special role in the generation of the rth block. The seed Qr will be deducible from the block Br ¨1 .
4. SECRET CREDENTIALS. Randomly and unambiguously using the current last block, Br ¨1 , in order to choose the verifier set and the leader in charge of constructing the new block, Br , is not enough. Since Br ¨1 must be known before generating Br , the last non-influentiable quantity Qr-1 deducible from Br-1 must be known too. Accordingly, so are the verifiers and the leader in charge to compute the block Br . Thus, our powerful Adversary might immediately corrupt all of them, before they engage in any discussion about Br , so as to get full control over the block they certify.
To prevent this problem, leaders (and actually verifiers too) secretly learn of their role, but can compute a proper credential, capable of proving to everyone that indeed have that role.
When a user privately realizes that he is the leader for the next block, first he secretly assembles his own proposed new block, and then disseminates it (so that can be certified) together with his own credential. This way, though the Adversary will immediately realize who the leader of the next block is, and although he can corrupt him right away, it will be too late for the Adversary to influence the choice of a new block. Indeed, he cannot "call back" the leader's message no more than a powerful government can put back into the bottle a message virally spread by WikiLeaks.
As we shall see, we cannot guarantee leader uniqueness, nor that everyone is sure who the leader is, including the leader himself! But, in Algorand, unambiguous progress will be guaranteed.
5. PLAYER REPLACEABILITY. After he proposes a new block, the leader might as well "die"
(or be corrupted by the Adversary), because his job is done. But, for the verifiers in SVr, things are less simple. Indeed, being in charge of certifying the new block Br with sufficiently many signatures, they must first run Byzantine agreement on the block proposed by the leader. The problem is that, no matter how efficient it is, BA* requires multiple steps and the honesty of > 2/3 of its players. This is a problem, because, for efficiency reasons, the player set of BA*
consists the small set SVr randomly selected among the set of all users. Thus, our powerful Adversary, although unable to corrupt 1/3 of all the users, can certainly corrupt all members of SW!
Fortunately we'll prove that protocol B A* , executed by propagating messages in a peer-to-peer fashion, is player-replaceable. This novel requirement means that the protocol correctly and efficiently reaches consensus even if each of its step is executed by a totally new, and randomly and independently selected, set of players. Thus, with millions of users, each small set of players associated to a step of BA* most probably has empty intersection with the next set.
In addition, the sets of players of different steps of BA* will probably have totally different cardinalities. Furthermore, the members of each set do not know who the next set of players will be, and do not secretly pass any internal state.
The replaceable-player property is actually crucial to defeat the dynamic and very powerful Adversary we envisage. We believe that replaceable-player protocols will prove crucial in lots of contexts and applications. In particular, they will be crucial to execute securely small sub-protocols embedded in a larger universe of players with a dynamic adversary, who, being able to corrupt even a small fraction of the total players, has no difficulty in corrupting all the players in the smaller sub-protocol.
An Additional Property/Technique: Lazy Honesty A honest user follows his prescribed instructions, which include being online and run the protocol. Since, Algorand has only modest computation and communication requirement, being online and running the protocol "in the background" is not a major sacrifice. Of course, a few "absences" among honest players, as those due to sudden loss of connectivity or the need of rebooting, are automatically tolerated (because we can always consider such few players to be temporarily malicious).
Let us point out, however, that Algorand can be simply adapted so as to work in a new model, in which honest users to be offline most of the time. Our new model can be informally introduced as follows.
Lazy Honesty. Roughly speaking, a user i is lazy-but-honest if (1) he follows all his prescribed instructions, when he is asked to participate to the protocol, and (2) he is asked to participate to the protocol only rarely, and with a suitable advance notice.
With such a relaxed notion of honesty, we may be even more confident that honest people will be at hand when we need them, and Algorand guarantee that, when this is the case, The system operates securely even if, at a given point in time, the majority of the participating players are malicious.
2 Preliminaries 2.1 Cryptographic Primitives Ideal Hashing. We shall rely on an efficiently computable cryptographic hash function, H, that maps arbitrarily long strings to binary strings of fixed length.
Following a long tradition, we model H as a random oracle, essentially a function mapping each possible string s to a randomly and independently selected (and then fixed) binary string, H(s), of the chosen length.
In our described embodiments, H has 256-bit long outputs. Indeed, such length is short enough to make the system efficient and long enough to make the system secure.
For instance, we want H to be collision-resilient. That is, it should be hard to find two different strings x and y such that H(x) = H(y). When H is a random oracle with 256-bit long outputs, finding any such pair of strings is indeed difficult. (Trying at random, and relying on the birthday paradox, would require 2256/2 = 2128 trials.) Digital Signing. Digital signatures allow users to to authenticate information to each other without sharing any sharing any secret keys. A digital signature scheme consists of three fast algorithms: a probabilistic key generator G, a signing algorithm S, and a verification algorithm V.
Given a security parameter k, a sufficiently high integer, a user i uses G to produce a pair of k-bit keys (i.e., strings): a "public" key pk, and a matching "secret" signing key ski. Crucially, a public key does not "betray" its corresponding secret key. That is, even given knowledge of pkz, no one other than i is able to compute sk, in less than astronomical time.
User i uses sk, to digitally sign messages. For each possible message (binary string) m, first hashes m and then runs algorithm S on inputs H(m) and sk, so as to produce the k-bit string sigpk,(m) S(H(m), ski) .2 The binary string sigpk,(m) is referred to as i's digital signature of m (relative to pk,), and can be more simply denoted by sigz(m), when the public key pk, is clear from context.
Everyone knowing pk, can use it to verify the digital signatures produced by i. Specifically, on inputs (a) the public key pk, of a player i, (b) a message m, and (c) a string s, that is, i's alleged digital signature of the message m, the verification algorithm V
outputs either YES or NO.
The properties we require from a digital signature scheme are:
1. Legitimate signatures are always verified: If s = sigz(m), then V(pk,,m,$) = YES; and 2Since H is collision-resilient it is practically impossible that, by signing m one "accidentally signs" a different message a.

2. Digital signatures are hard to forge: Without knowledge of sk, the time to find a string s such that V(pk,,m,$) =Y ES, for a message m never signed by i, is astronomically long.
(Following strong security requirements, this is true even if one can obtain the signature of any other message.) Accordingly, to prevent anyone else from signing messages on his behalf, a player i must keep his signing key sk, secret (hence the term "secret key"), and to enable anyone to verify the messages he does sign, i has an interest in publicizing his key pk, (hence the term "public key").
Signatures with Message Retrievability In general, a message m is not retrievable from its signature sigz(m). In order to virtually deal with digital signatures that satisfy the conceptually convenient "message retrievability" property (i.e., to guarantee that the signer and the message are easily computable from a signature, we define S/Gpki (m) = (i, m, sigpk,(m)) and S IG,(m) = (i, m, sigz(m)), if pk, is clear.
Unique Digital Signing. We also consider digital signature schemes (G, 5, V) satisfying the following additional property.
3. Uniqueness. It is hard to find strings pk', m, s, and s' such that s s' and V (pk' , m, s) = V (pk' ,m, s') = 1.
(Note that the uniqueness property holds also for strings pk' that are not legitimately generated public keys. In particular, however, the uniqueness property implies that, if one used the specified key generator G to compute a public key pk together with a matching secret key sk, and thus knew sk, it would be essentially impossible also for him to find two different digital signatures of a same message relative to pk.) Remarks = FROM UNIQUE SIGNATURES TO VERIFIABLE RANDOM FUNCTIONS. Relative to a digital signature scheme with the uniqueness property, the mapping m H(sigz(m)) associates to each possible string m, a unique, randomly selected, 256-bit string, and the correctness of this mapping can be proved given the signature sigz(m).
That is, ideal hashing and digital signature scheme satisfying the uniqueness property essentially provide an elementary implementation of a verifiable random function (VRF).
A VRF is a special kind of digital signature. We may write VRF (m) to indicate such a special signature of i of a message m. In addition to satisfy the uniqueness property, verifiable random functions produce outputs that are guaranteed to be sufficiently random.
That is, V RF,(m) is essentially random, and unpredictable until it is produced. By contrast, SIC(m) need not be sufficiently random. For instance, user i may choose his public key so that SIC(m) always is a k-bit string that is (lexicographically) small (i.e., whose first few bits could always be Os). Note, however, that, since H is an ideal hash function, H(S/Gz(m)) will always be a random 256-bit string. In our preferred embodiments we make extensive use of hashing digital signatures satisfying the uniqueness property precisely to be able to associate to each message m and each user i a unique random number. Should one implement Algorand with VRFs, one can replace H(S/Gz(m)) with VRF,(m). In particular, a user i need not first to compute SIG,(m), then H(SIG,(m)) (in order, ______ say to compare H(S/Gz(m)) with a number p). He might directly compute V RF,(m). In sum, it should be understood that H(S/Gz(m)) can be interpreted as VRF,(m), or as a sufficiently random number, easily computed by player i, but unpredictable to anyone else, unambiguously associated to i and m.
= THREE DIFFERENT NEEDS FOR DIGITAL SIGNATURES. In Algorand, a user i relies on digital signatures for (1) Authenticating i's own payments. In this application, keys can be "long-term" (i.e., used to sign many messages over a long period of time) and come from a ordinary signature scheme.
(2) Generating credentials proving that i is entitled to act at some step s of a round r. Here, keys can be long-term, but must come from a scheme satisfying the uniqueness property.
(3) Authenticating the message i sends in each step in which he acts. Here, keys must be ephemeral (i.e., destroyed after their first use), but can come from an ordinary signature scheme.
= A SMALL-COST SIMPLIFICATION. For simplicity, we envision each user i to have a single long-term key. Accordingly, such a key must come from a signature scheme with the uniqueness property. Such simplicity has a small computational cost. Typically, in fact, unique digital signatures are slightly more expensive to produce and verify than ordinary signatures.
2.2 The Idealized Public Ledger Algorand tries to mimic the following payment system, based on an idealized public ledger.
1. The Initial Status. Money is associated with individual public keys (privately generated and owned by users). Letting pki, , pk 3 be the initial public keys and al, , a3 their respective initial amounts of money units, then the initial status is So = (pki, ai), , (pk, a3) , which is assumed to be common knowledge in the system.
2. Payments. Let pk be a public key currently having a > 0 money units, pk' another public key, and a' a non-negative number no greater than a. Then, a (valid) payment is a digital signature, relative to pk, specifying the transfer of a' monetary units from pk to pk', together with some additional information. In symbols, = S Gpk(pk, pk' , , I, H(1)), where / represents any additional information deemed useful but not sensitive (e.g., time information and a payment identifier), and I any additional information deemed sensitive (e.g., the reason for the payment, possibly the identities of the owners of pk and the pk', and so on).
We refer to pk (or its owner) as the payer, to each pk' (or its owner) as a payee, and to a' as the amount of the payment 0.
Free Joining Via Payments. Note that users may join the system whenever they want by generating their own public/secret key pairs. Accordingly, the public key pk' that appears in the payment above may be a newly generated public key that had never "owned" any money before.

3. The Magic Ledger. In the Idealized System, all payments are valid and appear in a tamper-proof list L of sets of payments "posted on the sky" for everyone to see:
L = PAY1, PAY2, , Each block PAY r+1 consists of the set of all payments made since the appearance of block PAY. In the ideal system, a new block appears after a fixed (or finite) amount of time.
Discussion.
= More General Payments and Unspent Transaction Output. More generally, if a public key pk owns an amount a, then a valid payment of pk may transfer the amounts a, a'2, respectively to the keys pei, , so long as E3 ,31 < a .
In Bitcoin and similar systems, the money owned by a public key pk is segregated into separate amounts, and a payment made by pk must transfer such a segregated amount a in its entirety. If pk wishes to transfer only a fraction a' < a of a to another key, then it must also transfer the balance, the unspent transaction output, to another key, possibly pk itself.
Algorand also works with keys having segregated amounts. However, in order to focus on the novel aspects of Algorand, it is conceptually simpler to stick to our simpler forms of payments and keys having a single amount associated to them.
= Current Status. The Idealized Scheme does not directly provide information about the current status of the system (i.e., about how many money units each public key has). This information is deducible from the Magic Ledger.
In the ideal system, an active user continually stores and updates the latest status information, or he would otherwise have to reconstruct it, either from scratch, or from the last time he computed it. (Yet, we later on show how to augment Algorand so as to enable its users to reconstruct the current status in an efficient manner.) = Security and "Privacy". Digital signatures guarantee that no one can forge a payment of another user. In a payment 0, the public keys and the amount are not hidden, but the sensitive information I is. Indeed, only H(/) appears in 0, and since H is an ideal hash function, H(/) is a random 256-bit value, and thus there is no way to figure out what I was better than by simply guessing it. Yet, to prove what I was (e.g., to prove the reason for the payment) the payer may just reveal I. The correctness of the revealed I
can be verified by computing H(/) and comparing the resulting value with the last item of 0.
In fact, since H is collision resilient, it is hard to find a second value I' such that H(/) = H(//).
2.3 Basic Notions and Notations Keys, Users, and Owners Unless otherwise specified, each public key ("key" for short) is long-term and relative to a digital signature scheme with the uniqueness property. A public key i joins the system when another public key j already in the system makes a payment to i.
For color, we personify keys. We refer to a key i as a "he", say that i is honest, that i sends and receives messages, etc. User is a synonym for key. When we want to distinguish a key from the person to whom it belongs, we respectively use the term "digital key" and "owner".
Permissionless and Permissioned Systems. A system is permissionless, if a digital key is free to join at any time and an owner can own multiple digital keys; and its permissioned, otherwise.

Unique Representation Each object in Algorand has a unique representation. In particular, each set {(x, y, z,...) : x E X,y E Y, z E Z, ...} is ordered in a pre-specified manner: e.g., first lexicographically in x, then in y, etc.
Same-Speed Clocks There is no global clock: rather, each user has his own clock. User clocks need not be synchronized in any way. We assume, however, that they all have the same speed.
For instance, when it is 12pm according to the clock of a user i, it may be 2:30pm according to the clock of another user j, but when it will be 12:01 according to i's clock, it will 2:31 according to j's clock. That is, "one minute is the same (sufficiently, essentially the same) for every user".
Rounds Algorand is organized in logical units, r = 0, 1,..., called rounds.
We consistently use superscripts to indicate rounds. To indicate that a non-numerical quantity Q (e.g., a string, a public key, a set, a digital signature, etc.) refers to a round r, we simply write Q. . Only when Q is a genuine number (as opposed to a binary string interpretable as a number), do we write Q(r), so that the symbol r could not be interpreted as the exponent of Q.
At (the start of a) round r > 0, the set of all public keys is PK' , and the system status is Sr = (i, 4r), .) :i E
where a(zr) is the amount of money available to the public key i. Note that PK
is deducible from Sr, and that Sr may also specify other components for each public key i.
For round 0, PK is the set of initial public keys, and S is the initial status. Both PK and S are assumed to be common knowledge in the system. For simplicity, at the start of round r, so are PK1, , PK' and S1, , Sr .
In a round r, the system status transitions from Sr to Sr+1: symbolically, Round r: Sr Sr+1.
Payments In Algorand, the users continually make payments (and disseminate them in the way described in subsection 2.7). A payment of a user i E PK' has the same format and semantics as in the Ideal System. Namely, = SIG,(i, i', a, I, H(I)) .
Payment is individually valid at a round r (is a round-r payment, for short) if (1) its amount a is less than or equal to a,r), and (2) it does not appear in any official payset PAY'' for r' < r. (As explained below, the second condition means that has not already become effective.
A set of round-r payments of i is collectively valid if the sum of their amounts is at most a (7') .
Paysets A round-r payset P is a set of round-r payments such that, for each user i, the payments of i in P (possibly none) are collectively valid. The set of all round-r paysets is PAY(r). A round-r payset P is maximal if no superset of P is a round-r payset.
We actually suggest that a payment also specifies a round p, = SIG,(p,i,i' , a, I, H(I)) , and cannot be valid at any round outside [p, p k], for some fixed non-negative integer k.3 3This simplifies checking whether 0 has become "effective" (i.e., it simplifies determining whether some payset PAY' contains 0. When k = 0, if 0 = SIG, (r, , a, I, H(I)) , and 0 cl PAY', then i must re-submit 0.

Official Paysets For every round r, Algorand publicly selects (in a manner described later on) a single (possibly empty) payset, PAY', the round's official payset.
(Essentially, PAY' represents the round-r payments that have "actually" happened.) As in the Ideal System (and Bitcoin), (1) the only way for a new user j to enter the system is to be the recipient of a payment belonging to the official payset PAY of a given round r;
and (2) PAY' determines the status of the next round, Sr-hi, from that of the current round, Sr. Symbolically, PAY' : Sr S" 1.
Specifically, 1. the set of public keys of round r 1, PK" 1, consists of the union of PKr and the set of all payee keys that appear, for the first time, in the payments of PAY'; and 2. the amount of money a( r+1) that a user i owns in round r +1 is the sum of az(r) i.e., the amount of money i owned in the previous round (0 if i 0 PK') ________________ and the sum of amounts paid to i according to the payments of PAYr. .
In sum, as in the Ideal System, each status Sr-hi is deducible from the previous payment history:
PAY , ... ,PAYr.
2.4 Blocks and Proven Blocks In Algorando, the block Br corresponding to a round r specifies: r itself; the set of payments of round r, PAYr; a quantity SIGer(Qr-1), to be explained, and the hash of the previous block, H(Br-1). Thus, starting from some fixed block B , we have a traditional blockchain:
B1 = (1, PAY', Sicei (Q ), H(B )), B2 = (2, PAY2,Sice2(Q1), H(B1))= = = =
In Algorand, the authenticity of a block is actually vouched by a separate piece of information, a "block certificate" C E RTr , which turns Br into a proven block, Br . The Magic Ledger, therefore, is implemented by the sequence of the proven blocks, B2, ...
Discussion As we shall see, CERTr consists of a set of digital signatures for H(Br), those of a majority of the members of SY", together with a proof that each of those members indeed belongs to SY". We could, of course, include the certificates CERTr in the blocks themselves, but find it conceptually cleaner to keep it separate.) In Bitcoin each block must satisfy a special property, that is, must "contain a solution of a crypto puzzle", which makes block generation computationally intensive and forks both inevitable and not rare. By contrast, Algorand's blockchain has two main advantages: it is generated with minimal computation, and it will not fork with overwhelmingly high probability.
Each block Bz is safely final as soon as it enters the blockchain.
2.5 Acceptable Failure Probability To analyze the security of Algorand we specify the probability, F, with which we are willing to accept that something goes wrong (e.g., that a verifier set SY" does not have an honest majority). As in the case of the output length of the cryptographic hash function H, also F is a parameter. But, as in that case, we find it useful to set F to a concrete value, so as to get a more intuitive grasp of the fact that it is indeed possible, in Algorand, to enjoy simultaneously sufficient security and sufficient efficiency. To emphasize that F is parameter that can be set as desired, in the first and second embodiments we respectively set F = 10-12 and F = 10-18 .
Discussion Note that 10-12 is actually less than one in a trillion, and we believe that such a choice of F is adequate in our application. Let us emphasize that 10-12 is not the probability with which the Adversary can forge the payments of an honest user. All payments are digitally signed, and thus, if the proper digital signatures are used, the probability of forging a payment is far lower than 10-12, and is, in fact, essentially 0. The bad event that we are willing to tolerate with probability F is that Algorand's blockchain forks. Notice that, with our setting of F and one-minute long rounds, a fork is expected to occur in Algorand's blockchain as infrequently as (roughly) once in 1.9 million years. By contrast, in Bitcoin, a forks occurs quite often.
A more demanding person may set F to a lower value. To this end, in our second embodiment we consider setting F to 10-18. Note that, assuming that a block is generated every second, 1018 is the estimated number of seconds taken by the Universe so far: from the Big Bang to present time. Thus, with F = 10-18, if a block is generated in a second, one should expect for the age of the Universe to see a fork.
2.6 The Adversarial Model Algorand is designed to be secure in a very adversarial model. Let us explain.
Honest and Malicious Users A user is honest if he follows all his protocol instructions, and is perfectly capable of sending and receiving messages. A user is malicious (i.e., Byzantine, in the parlance of distributed computing) if he can deviate arbitrarily from his prescribed instructions.
The Adversary The Adversary is an efficient (technically polynomial-time) algorithm, personified for color, who can immediately make malicious any user he wants, at any time he wants (subject only to an upperbound to the number of the users he can corrupt).
The Adversary totally controls and perfectly coordinates all malicious users.
He takes all actions on their behalf, including receiving and sending all their messages, and can let them deviate from their prescribed instructions in arbitrary ways. Or he can simply isolate a corrupted user sending and receiving messages. Let us clarify that no one else automatically learns that a user i is malicious, although i's maliciousness may transpire by the actions the Adversary has him take.
This powerful adversary however, = Does not have unbounded computational power and cannot successfully forge the digital signature of an honest user, except with negligible probability; and = Cannot interfere in any way with the messages exchanges among honest users.
Furthermore, his ability to attack honest users is bounded by one of the following assumption.
Honesty Majority of Money We consider a continuum of Honest Majority of Money (HMM) assumptions: namely, for each non-negative integer k and real h> 1/2, HHMk > h: the honest users in every round r owned a fraction greater than h of all money in the system at round r ¨ k.

Discussion. Assuming that all malicious users perfectly coordinate their actions (as if controlled by a single entity, the Adversary) is a rather pessimistic hypothesis. Perfect coordination among too many individuals is difficult to achieve. Perhaps coordination only occurs within separate groups of malicious players. But, since one cannot be sure about the level of coordination malicious users may enjoy, we'd better be safe than sorry.
Assuming that the Adversary can secretly, dynamically, and immediately corrupt users is also pessimistic. After all, realistically, taking full control of a user's operations should take some time.
The assumption HMMk > h implies, for instance, that, if a round (on average) is implemented in one minute, then, the majority of the money at a given round will remain in honest hands for at least two hours, if k = 120, and at least one week, if k = 10, 000.
Note that the HMM assumptions and the previous Honest Majority of Computing Power assumptions are related in the sense that, since computing power can be bought with money, if malicious users own most of the money, then they can obtain most of the computing power.
2.7 The Communication Model We envisage message propagation __ i.e., "peer-to-peer gossip"4 ____________ to be the only means of communication, and assume that every propagated message reaches almost all honest users in a timely fashion. We essentially assume that each message m propagated by honest user reaches, within a given amount of time that depends on the length of m, all honest users. (It actually suffices that m reaches a sufficiently high percentage of the honest users.) The BA Protocol BA* in a Traditional Setting As already emphasized, Byzantine agreement is a key ingredient of Algorand.
Indeed, it is through the use of such a BA protocol that Algorand is unaffected by forks.
However, to be secure against our powerful Adversary, Algorand must rely on a BA protocol that satisfies the new player-replaceability constraint. In addition, for Algorand to be efficient, such a BA
protocol must be very efficient.
BA protocols were first defined for an idealized communication model, synchronous complete networks (SC networks). Such a model allows for a simpler design and analysis of BA protocols.
Accordingly, in this section, we introduce a new BA protocol, BA*, for SC
networks and ignoring the issue of player replaceability altogether. The protocol BA* is a contribution of separate value. Indeed, it is the most efficient cryptographic BA protocol for SC
networks known so far.
To use it within our Algorand protocol, we modify BA* a bit, so as to account for our different communication model and context.
We start by recalling the model in which BA* operates and the notion of a Byzantine agreement.
3.1 Synchronous Complete Networks and Matching Adver-saries In a SC network, there is a common clock, ticking at each integral times r =
1, 2, ...
4Essentially, as in Bitcoin, when a user propagates a message m, every active user i receiving m for the first time, randomly and independently selects a suitably small number of active users, his "neighbors", to whom he forwards m, possibly until he receives an acknowledgement from them. The propagation of m terminates when no user receives m for the first time.

At each even time click r, each player i instantaneously and simultaneously sends a single message m'z',3 (possibly the empty message) to each player j, including himself. Each m'' is correctly received at time click r 1 by player j, together with the identity of the sender i.
Again, in a communication protocol, a player is honest if he follows all his prescribed instructions, and malicious otherwise. All malicious players are totally controlled and perfectly coordinated by the Adversary, who, in particular, immediately receives all messages addressed to malicious players, and chooses the messages they send.
The Adversary can immediately make malicious any honest user he wants at any odd time click he wants, subject only to a possible upperbound t to the number of malicious players.
That is, the Adversary "cannot interfere with the messages already sent by an honest user i", which will be delivered as usual.
The Adversary also has the additional ability to see instantaneously, at each even round, the messages that the currently honest players send, and instantaneously use this information to choose the messages the malicious players send at the same time tick.
3.2 The Notion of a Byzantine Agreement The notion of Byzantine agreement might have been first introduced for the binary case, that is, when every initial value consists of a bit. However, it was quickly extended to arbitrary initial values. By a BA protocol, we mean an arbitrary-value one.
Definition 3.1. In a synchronous network, let P be a n-player protocol, whose player set is common knowledge among the players, t a positive integer such that n> 2t 1. We say that P
is an arbitrary-value (respectively, binary) (n, t)-Byzantine agreement protocol with soundness a E (0, 1) if, for every set of values V not containing the special symbol I
(respectively, for V = {0,1}), in an execution in which at most t of the players are malicious and in which every player i starts with an initial value vz E V, every honest player j halts with probability 1, outputting a value out, E V U {I} so as to satisfy, with probability at least a, the following two conditions:
1. Agreement: There exists out E V U {I} such that out, = out for all honest players i.
2. Consistency: if, for some value v E V, vz = v for all players i, then out =
V.
We refer to out as P's output, and to each out, as player i's output.
3.3 The BA Notation In our BA protocols, a player is required to count how many players sent him a given message in a given step. Accordingly, for each possible value v that might be sent, #:(v) (or just #z(v) when s is clear) is the number of players j from which i has received v in step s.
Recalling that a player i receives exactly one message from each player j, if the number of players is n, then, for all i and s, > #:(v) = n.
3.4 The New Binary BA Protocol BBA*
In this section we present a new binary BA protocol, BBA*, which relies on the honesty of more than two thirds of the players and is very fast: no matter what the malicious players might do, each execution of its main loop not only is trivially executed, but brings the players into agreement with probability 1/3.
In BBA*, each player has his own public key of a digital signature scheme satisfying the unique-signature property. Since this protocol is intended to be run on synchronous complete network, there is no need for a player i to sign each of his messages.
Digital signatures are used to generate a sufficiently common random bit in Step 3. (In Algorand, digital signatures are used to authenticate all other messages as well.) The protocol requires a minimal set-up: a common random string r, independent of the players' keys. (In Algorand, r is actually replaced by the quantity Qr.) Protocol BBA* is a 3-step loop, where the players repeatedly exchange Boolean values, and different players may exit this loop at different times. A player i exits this loop by propagating, at some step, either a special value 0* or a special value 1*, thereby instructing all players to "pretend" they respectively receive 0 and 1 from i in all future steps.
(Alternatively said:
assume that the last message received by a player j from another player i was a bit b. Then, in any step in which he does not receive any message from i, j acts as if i sent him the bit b.) The protocol uses a counter y, representing how many times its 3-step loop has been executed. At the start of BBA*, y = 0. (One may think of 'y as a global counter, but it is actually increased by each individual player every time that the loop is executed.) There are n > 3t+1, where t is the maximum possible number of malicious players. A binary string x is identified with the integer whose binary representation (with possible leadings Os) is x; and lsb(x) denotes the least significant bit of x.
PROTOCOL BBA*
(COMMUNICATION) STEP 1. [Coin-Fixed-To-0 Step] Each player i sends bz.
1.1 If (0) > 2t +1, then i sets b, = 0, sends 0*, outputs out, = 0, and HALTS.
1.2 If #(1) > 2t +1, then, then i sets I), = 1.
1.3 Else, i sets I), = 0.
(COMMUNICATION) STEP 2. [Coin-Fixed-To-1 Step] Each player i sends bz.
2.1 If Yq(1) > 2t +1, then i sets b, =1, sends 1*, outputs out, =1, and HALTS.

2.2 If Yq(0)> 2t +1, then i set b, =0.
2.3 Else, i sets b, =1.
(COMMUNICATION) STEP 3. [Coin-Genuinely-Flipped Step] Each player i sends b, and SIG,(r,ry).
3.1 If e(0) > 2t 1, then i sets b, = 0.
3.2 If e(1)> 2t +1, then i sets I), = 1.
3.3 Else, letting S, = fj E N who have sent i a proper message in this step 3 1, i sets I), = c lsb(minics, H(SIG,(r,-)/))); increases -y, by I; and returns to Step I.
Theorem 3.1. Whenever n > 3t +1, BBA* is a binary (n,t)-BA protocol with soundness I.
A proof of Theorem 3.1 can be found in https://people.csail.mit.edu/silvio/
SelectedScien-tificPapers/DistributedComputation/BYZANTINEAGREEMENTMADETRIVIAL. 15 pdf.

3.5 Graded Consensus and the Protocol GC
Let us recall, for arbitrary values, a notion of consensus much weaker than Byzantine agreement.
Definition 3.2. Let P be a protocol in which the set of all players is common knowledge, and each player i privately knows an arbitrary initial value v.
We say that P is an (n, t)-graded consensus protocol if, in every execution with n players, at most t of which are malicious, every honest player i halts outputting a value-grade pair (vi, 9,2), where g, E {0,1, 2}, so as to satisfy the following three conditions:
I. For all honest players i and j, gi ¨ 931 < 1.
2. For all honest players i and j, gt, 9,1 > 0 v3 = v3.
3. If v'1 = = = = = vn = v for some value v, then v, = v and g, = 2 for all honest players i.
The following two-step protocol GC is a graded consensus protocol in the literature. To match the steps of protocol Algorand'i of section 4.1, we respectively name 2 and 3 the steps of GC. (Indeed, the first step of A/gorand'i is concerned with something else:
namely, proposing a new block.) PROTOCOL GC
STEP 2. Each player i sends v't to all players.
STEP 3. Each player i sends to all players the string x if and only if #(x) >
2f 1.
OUTPUT DETERMINATION. Each player i outputs the pair (vi, g,) computed as follows:
= If, for some x, #(x) > 2f 1, then v, = x and g, = 2.
= If, for some x, (x) > t 1, then v, = x and g, = 1.
= Else, v, = I and g, = 0.
Since protocol GC is a protocol in the literature, it is known that the following theorem holds.
Theorem 3.2. If n> 3f 1, then GC is a (n, t)-graded broadcast protocol.
3.6 The Protocol BA*
We now describe the arbitrary-value BA protocol BA* via the binary BA protocol BBA* and the graded-consensus protocol GC. Below, the initial value of each player i is v.
PROTOCOL BA*
STEPS 1 AND 2. Each player i executes GC, on input et, so as to compute a pair (v,, g,).
STEP 3, ... Each player i executes BBA* _______________________________ with initial input 0, if g, = 2, and I otherwise so as to compute the bit out,.
OUTPUT DETERMINATION. Each player i outputs v, if out, = 0, and I otherwise.
Theorem 3.3. Whenever n > 3t 1, BA* is a (n,t)-BA protocol with soundness I.

Proof. We first prove Consistency, and then Agreement.
PROOF OF CONSISTENCY. Assume that, for some value v E V, viz = v. Then, by property 3 of graded consensus, after the execution of GC, all honest players output (v, 2).
Accordingly, 0 is the initial bit of all honest players in the end of the execution of BBA*.
Thus, by the Agreement property of binary Byzantine agreement, at the end of the execution of B A* , out, = 0 for all honest players. This implies that the output of each honest player i in BA* is v, = v. El PROOF OF AGREEMENT. Since BBA* is a binary BA protocol, either (A) out, = 1 for all honest player i, or (B) out, = 0 for all honest player i.
In case A, all honest players output I in BA*, and thus Agreement holds.
Consider now case B. In this case, in the execution of BBA*, the initial bit of at least one honest player i is 0.
(Indeed, if initial bit of all honest players were 1, then, by the Consistency property of BBA*, we would have outi = 1 for all honest j.) Accordingly, after the execution of GC, i outputs the pair (v, 2) for some value v. Thus, by property 1 of graded consensus, gj > 0 for all honest players j. Accordingly, by property 2 of graded consensus, tt? = v for all honest players j. This implies that, at the end of BA*, every honest player j outputs v. Thus, Agreement holds also in case B. El Since both Consistency and Agreement hold, BA* is an arbitrary-value BA
protocol. =
Protocol BA* works also in gossiping networks, and in fact satisfies the player replaceability property that is crucial for Algorand to be secure in the envisaged very adversarial model.
The Player Replaceability of BBA* and BA* Let us now provide some intuition of why the protocols BA* and BBA* can be adapted to be executed in a network where communication is via peer-to-peer gossiping, satisfy player replaceability. For concreteness, assume that the network has 10M users and that each step x of BBA* (or BA*) is executed by a committee of

10,000 players, who have been randomly selected via secret cryptographic sortition, and thus have credentials proving of being entitled to send messages in step x. Assume that each message sent in a given step specifies the step number, is digitally signed by its sender, and includes the credential proving that its sender is entitled to speak in that step.
First of all, if the percentage h of honest players is sufficiently larger than 2/3 (e.g., 75%), then, with overwhelming probability, the committee selected at each step has the required 2/3 honest majority.
In addition, the fact that the 10,000-strong randomly selected committee changes at each step does not impede the correct working of either BBA* or BA*. Indeed, in either protocol, a player i in step s only reacts to the multiplicity with which, in Step s ¨
1, he has received a given message m. Since we are in a gossiping network, all messages sent in Step s ¨ 1 will (immediately, for the purpose of this intuition) reach all users, including those selected to play in step s. Furthermore because all messages sent in step s ¨ 1 specify the step number and include the credential that the sender was indeed authorized to speak in step s ¨ 1. Accordingly, whether he happened to have been selected also in step s ¨ 1 or not, a user i selected to play in step s is perfectly capable of correctly counting the multiplicity with which he has received a correct step s ¨ 1 message. It does not at all matter whether he has been playing all steps so far or not. All users are in "in the same boat" and thus can be replaced easily by other users.

11 PCT/US2018/066481 4 Two Embodiments of Algorand As discussed, at a very high level, a round of Algorand ideally proceeds as follows. First, a randomly selected user, the leader, proposes and circulates a new block. (This process includes initially selecting a few potential leaders and then ensuring that, at least a good fraction of the time, a single common leader emerges.) Second, a randomly selected committee of users is selected, and reaches Byzantine agreement on the block proposed by the leader.
(This process includes that each step of the BA protocol is run by a separately selected committee.) The agreed upon block is then digitally signed by a given threshold (TH) of committee members.
These digital signatures are propagated so that everyone is assured of which is the new block.
(This includes circulating the credential of the signers, and authenticating just the hash of the new block, ensuring that everyone is guaranteed to learn the block, once its hash is made clear.) In the next two sections, we present two embodiments of the basic Algorand design, A/gorand'i and Algorand, that respectively work under a proper majority-of-honest-users assumption. In Section 7 we show how to adopts these embodiments to work under a honest-majority-of-money assumption.
Algorand'i only envisages that > 2/3 of the committee members are honest. In addition, in Algorand, the number of steps for reaching Byzantine agreement is capped at a suitably high number, so that agreement is guaranteed to be reached with overwhelming probability within a fixed number of steps (but potentially requiring longer time than the steps of Algorand). In the remote case in which agreement is not yet reached by the last step, the committee agrees on the empty block, which is always valid.
Algorand12 envisages that the number of honest members in a committee is always greater than or equal to a fixed threshold tH (which guarantees that, with overwhelming probability, at least 2/3 of the committee members are honest). In addition, Algorand12 allows Byzantine agreement to be reached in an arbitrary number of steps (but potentially in a shorter time than Algorand).
Those skilled in the art will realize that many variants of these basic embodiments can be derived. In particular, it is easy, given Algorand, to modify A/gorand'i so as to enable to reach Byzantine agreement in an arbitrary number of steps.
Both embodiments share the following common core, notations, notions, and parameters.
4.1 A Common Core Objectives Ideally, for each round r, Algorand should satisfy the following properties:
1. Perfect Correctness. All honest users agree on the same block Br .
2. Completeness I. With probability 1, the block Br has been chosen by a honest user.
(Indeed a malicious user may always choose a block whose payset contains the payments of just his "friends".) Of course, guaranteeing perfect correctness alone is trivial: everyone always chooses the official payset PAY to be empty. But in this case, the system would have completeness 0.
Unfortunately, guaranteeing both perfect correctness and completeness 1 is not easy in the presence of malicious users. Algorand thus adopts a more realistic objective.
Informally, letting h denote the percentage of users who are honest, h> 2/3, the goal of Algorand is Guaranteeing, with overwhelming probability, perfect correctness and completeness close to h.
Privileging correctness over completeness seems a reasonable choice: payments not processed in one round can be processed in the next, but one should avoid forks, if possible.

Led Byzantine Agreement Disregarding excessive time and communication for a moment, perfect Correctness could be guaranteed as follows. At the start of round r, each user i proposes his own candidate block B. . Then, all users reach Byzantine agreement on just one of the candidate blocks. As per our introduction, the BA protocol employed requires a 2/3 honest majority and is player replaceable. Each of its step can be executed by a small and randomly selected set of verifiers, who do not share any inner variables.
Unfortunately, this approach does not quite work. This is so, because the candidate blocks proposed by the honest users are most likely totally different from each other. Indeed, each honest user sees different payments. Thus, although the sets of payments seen by different honest users may have a lot of overlap, it is unlikely that all honest users will construct a propose the same block. Accordingly, the consistency agreement of the BA
protocol is never binding, only the agreement one is, and thus agreement may always been reached on I rather than on a good block.
Algorand' avoids this problem as follows. First, a leader for round r, fr, is selected. Then, fr propagates his own candidate block, B. Finally, the users reach agreement on the block they actually receive from fr. Because, whenever fr is honest, Perfect Correctness and Completeness 1 both hold, Algorand' ensures that fr is honest with probability close to h.
Leader Selection In Algorand's, the rth block is of the form Br = (r, PAYr, H(Br-1).
As already mentioned in the introduction, the quantity Qr-1- is carefully constructed so as to be essentially non-manipulatable by our very powerful Adversary. (Later on in this section, we shall provide some intuition about why this is the case.) At the start of a round r, all users know the blockchain so far, B , , Br-1, from which they deduce the set of users of every prior round: that is, PK1, , PKr-1. A potential leader of round r is a user i such that .H (SIG, (r, 1, Qr ¨1)) < p .
Let us explain. Note that, since the quantity Qr-1- is deducible from block Br-1, because of the message retrievability property of the underlying digital signature scheme. Furthermore, the underlying signature scheme satisfies the uniqueness property. Thus, SIG, (r,1,Qr-1) is a binary string uniquely associated to i and r. Accordingly, since H is a random oracle, H (SIG, (r, 1, Qr-1)) is a random 256-bit long string uniquely associated to i and r. The symbol "." in front of H (SIG, (r, 1, Qr-1)) is the decimal (in our case, binary) point, so that r, .H (SIG, (r, 1, Qr-1)) is the binary expansion of a random 256-bit number between 0 and 1 uniquely associated to i and r. Thus the probability that r, is less than or equal to p is essentially p.
The probability p is chosen so that, with overwhelming (i.e., 1 ¨ F) probability, at least one potential verifier is honest. (If fact, p is chosen to be the smallest such probability.) Note that, since i is the only one capable of computing his own signatures, he alone can determine whether he is a potential verifier of round 1. However, by revealing his own credential, SIG (r, 1, Qr-1), i can prove to anyone to be a potential verifier of round r.
The leader r is defined to be the potential leader whose hashed credential is smaller that the hashed credential of all other potential leader j: that is, H(oTer,'s) <
H(os).
Note that, since a malicious fr may not reveal his credential, the correct leader of round r may never be known, and that, barring improbable ties, r is indeed the only leader of round r.
Let us finally bring up a last but important detail: a user i can be a potential leader (and thus the leader) of a round r only if he belonged to the system for at least k rounds. This guarantees the non-manipulatability of Qr and all future Q-quantities. In fact, one of the potential leaders will actually determine Q.
.
Verifier Selection Each step s > 1 of round r is executed by a small set of verifiers, SV".
Again, each verifier i E SVr's is randomly selected among the users already in the system k rounds before r, and again via the special quantity Qr-1-. Specifically, i E
Picr¨k is a verifier in SV", if .H (SIG, (r, s,Qr-1)) < .
Once more, only i knows whether he belongs to SV",but, if this is the case, he could prove it by exhibiting his credential az" = H(SIGt (r,s,Qr-1)). A verifier i E SVr's sends a message, mtr's, in step s of round r, and this message includes his credential CT tr's , so as to enable the verifiers f the nest step to recognize that mtr's is a legitimate step-s message.
The probability p' is chosen so as to ensure that, in SV", letting #good be the number of honest users and #bad the number of malicious users, with overwhelming probability the following two conditions hold.
For embodiment Algoran4:
(1) #good > 2 = #bad and (2) #good 4 = #bad < 2n, where n is the expected cardinality of SV".
For embodiment Algorand12:
(1) #good > tH and (2) #good +2#bad < 2tH, where tH is a specified threshold.
These conditions imply that, with sufficiently high probability, (a) in the last step of the BA
protocol, there will be at least given number of honest players to digitally sign the new block Br (b) only one block per round may have the necessary number of signatures, and (c) the used BA protocol has (at each step) the required 2/3 honest majority.
Clarifying Block Generation If the round-r leader fr is honest, then the corresponding block is of the form Br = (r, PAY, sic (Qr ¨1) H (Br-1)) , where the payset PAY r is maximal. (recall that all paysets are, by definition, collectively valid.) Else (i.e., if fr is malicious), Br has one of the following two possible forms:
Br = (r, PAY, SIGt (Qr ¨1) H (Br ¨1)) and Br = BrE: (r,0, Q-1 H (Br ¨1)) .
In the first form, PAY r is a (non-necessarily maximal) payset and it may be PAY r = 0; and i is a potential leader of round r. (However, i may not be the leader fr . This may indeed happen if if fr keeps secret his credential and does not reveal himself.) The second form arises when, in the round-r execution of the BA protocol, all honest players output the default value, which is the empty block BrE: in our application.
(By definition, the possible outputs of a BA protocol include a default value, generically denoted by I. See section 3.2.) Note that, although the paysets are empty in both cases, Br = (Qr H (Br-1)) and BrE: are syntactically different blocks and arise in two different situations: respectively, "all went smoothly enough in the execution of the BA protocol", and "something went wrong in the BA protocol, and the default value was output".
Let us now intuitively describe how the generation of block Br proceeds in round r of Algorand/ . In the first step, each eligible player, that is, each player i E
PKr¨k , checks whether he is a potential leader. If this is the case, then i is asked, using of all the payments he has seen so far, and the current blockchain, B , , Br-1, to secretly prepare a maximal payment set, PAY:, and secretly assembles his candidate block, Br = (r, PAY:, SIG, (Qr-1-) ,H (Br-1)).
That,is, not only does he include in B, as its second component, the just prepared payset, but also, as its third component, his own signature of Qr-1, the third component of the last block, Br-1. Finally, he propagates his round-r-step-1 message, mrt'l, which includes (a) his candidate block B'z' , (b) his proper signature of his candidate block (i.e., his signature of the hash of Br , and (c) his own credential 0-tr'l, proving that he is indeed a potential verifier of round r.
(Note that, until an honest i produces his message mtr'1, the Adversary has no clue that i is a potential verifier. Should he wish to corrupt honest potential leaders, the Adversary might as well corrupt random honest players. However, once he sees mrt'l, since it contains i's credential, the Adversary knows and could corrupt i, but cannot prevent n4", which is virally propagated, from reaching all users in the system.) In the second step, each selected verifier j E SVr'2 tries to identify the leader of the round.
Specifically, j takes the step-1 credentials, , o-tr'1, contained in the proper step-1 message rntr'l he has received; hashes all of them, that is, computes H (o-zr ;1) , ,H (o-tr'1); finds the credential, o-re'l, whose hash is lexicographically minimum; and considers fri to be the leader of round r.
Recall that each considered credential is a digital signature of Qr-1, that SIG, (r, 1, Qr-1) is uniquely determined by i and Qr-1, that H is random oracle, and thus that each H(S/G3 (r, 1, Qr-1) is a random 256-bit long string unique to each potential leader i of round r.
From this we can conclude that, if the 256-bit string Qr-1 were itself randomly and independently selected, than so would be the hashed credentials of all potential leaders of round r. In fact, all potential leaders are well defined, and so are their credentials (whether actually computed or not). Further, the set of potential leaders of round r is a random subset of the users of round r ¨ k, and an honest potential leader i always properly constructs and propagates his message m13', which contains i's credential. Thus, since the percentage of honest users is h, no matter what the malicious potential leaders might do (e.g., reveal or conceal their own credentials), the minimum hashed potential-leader credential belongs to a honest user, who is necessarily identified by everyone to be the leader fr of the round r.
Accordingly, if the 256-bit string Qr-1 were itself randomly and independently selected, with probability exactly h (a) the leader fr is honest and (b) 3 = fr for all honest step-2 verifiers j.
In reality, the hashed credential are, yes, randomly selected, but depend on Qr-1, which is not randomly and independently selected. A careful analysis, however, guarantees that Qr-1 is sufficiently non-manipulatable to guarantee that the leader of a round is honest with probability h' sufficiently close to h: namely, h'> h2 (1+ h ¨ h2). For instance, if h =
80%, then h'> .7424.
Having identified the leader of the round (which they correctly do when the leader fr is honest), the task of the step-2 verifiers is to start executing BA* using as initial values what they believe to be the block of the leader. Actually, in order to minimize the amount of communication required, a verifier j E SVr'2 does not use, as his input value tt; to the Byzantine protocol, the block B3 that he has actually received from .3 (the user j believes to be the leader), but the the leader, but the hash of that block, that is, tt; = H(B3). Thus, upon termination of the BA protocol, the verifiers of the last step do not compute the desired round-r block Br , but compute (authenticate and propagate) H(Br). Accordingly, since H(Br) is digitally signed by sufficiently many verifiers of the last step of the BA protocol, the users in the system will realize that H(Br) is the hash of the new block. However, they must also retrieve (or wait for, since the execution is quite asynchronous) the block Br itself, which the protocol ensures that is indeed available, no matter what the Adversary might do.
Asynchrony and Timing A/gorand'i and Algoranc112 have a significant degree of asynchrony.
This is so because the Adversary has large latitude in scheduling the delivery of the messages being propagated. In addition, whether the total number of steps in a round is capped or not, there is the variance contribute by the number of steps actually taken.
As soon as he learns the certificates of B ,. , Br-1, a user i computes Qr-1 and starts working on round r, checking whether he is a potential leader, or a verifier in some step s of round r.
Assuming that i must act at step s, in light of the discussed asynchrony, i relies on various strategies to ensure that he has sufficient information before he acts.
For instance, he might wait to receive at least a given number of messages from the verifiers of the previous step (as in Alg oranciii), or wait for a sufficient time to ensure that he receives the messages of sufficiently many verifiers of the previous step (as in Algoranc112).
The Seed Cr and the Look-Back Parameter k Recall that, ideally, the quantities Qr should random and independent, although it will suffice for them to be sufficiently non-manipulatable by the Adversary.
At a first glance, we could choose Qr-1 to coincide with H (PAYr-1). An elementary analysis reveals, however, that malicious users may take advantage of this selection mechanism.5 Some additional effort shows that myriads of other alternatives, based on traditional block quantities are easily exploitable by the Adversary to ensure that malicious leaders are very frequent. We instead specifically and inductively define our brand new quantity Qr so as to be able to prove that it is non-manipulatable by the Adversary. Namely, Qr H(S/aer (V-1), r), if Br is not the empty block, and Qr H (Qr-1 , r) otherwise.
The intuition of why this construction of Qr works is as follows. Assume for a moment that Qr-1 is truly randomly and independently selected. Then, will so be Qr? When fr is honest the answer is (roughly speaking) yes. This is so because H(S/Ger(=),r) {0, 1}256 {0, 1}256 is a random function. When fr is malicious, however, Qr is no longer univocally defined from Qr-1 and fr . There are at least two separate values for Qr . One continues to be Qr H (S aer(Qr-1) , r), and the other is H(Qr-1, r). Let us first argue that, while the second 5We are at the start of round r ¨ 1. Thus, Qr-2 = PAYr-2 is publicly known, and the Adversary privately knows who are the potential leaders he controls. Assume that the Adversary controls 10% of the users, and that, with very high probability, a malicious user w is the potential leader of round r ¨1. That is, assume that H (SIGõ, (r ¨ 2, 1, Qr-2)) is so small that it is highly improbable an honest potential leader will actually be the leader of round r ¨ 1. (Recall that, since we choose potential leaders via a secret cryptographic sortition mechanism, the Adversary does not know who the honest potential leaders are.) The Adversary, therefore, is in the enviable position of choosing the payset PAY' he wants, and have it become the official payset of round r ¨ 1. However, he can do more. He can also ensure that, with high probability, (*) one of his malicious users will be the leader also of round r, so that he can freely select what PAY r will be. (And so on. At least for a long while, that is, as long as these high-probability events really occur.) To guarantee (*), the Adversary acts as follows. Let PAY' be the payset the Adversary prefers for round r ¨ 1.
Then, he computes H(PAY') and checks whether, for some already malicious player z, H(PAY')) is particularly small, that is, small enough that with very high probability z will be the leader of round r. If this is the case, then he instructs w to choose his candidate block to be B,r-1 = (r ¨ 1, PAY', H (Br-2). Else, he has two other malicious users x and y to keep on generating a new payment 0', from one to the other, until, for some malicious user z (or even for some fixed user z) H (SIG. (PAY' U {0})) is particularly small too. This experiment will stop quite quickly.
And when it does the Adversary asks w to propose the candidate block Bzr-1 =
(r ¨ 1, PAY' U {0}, H (Br-2)).

choice is somewhat arbitrary, a second choice is absolutely mandatory. The reason for this is that a malicious fr can always cause totally different candidate blocks to be received by the honest verifiers of the second step.6 Once this is the case, it is easy to ensure that the block ultimately agreed upon via the BA protocol of round r will be the default one, and thus will not contain anyone's digital signature of Qr-1. But the system must continue, and for this, it needs a leader for round r. If this leader is automatically and openly selected, then the Adversary will trivially corrupt him. If it is selected by the previous Qr-1 via the same process, than fr will again be the leader in round r +1. We specifically propose to use the same secret cryptographic sortition mechanism, but applied to a new Q-quantity: namely, H(Qr-1, r). By having this quantity to be the output of H guarantees that the output is random, and by including r as the second input of H, while all other uses of H have either a single input or at least three inputs, "guarantees" that such a Qr is independently selected. Again, our specific choice of alternative Qr does not matter, what matter is that fr has two choice for Qr, and thus he can double his chances to have another malicious user as the next leader.
The options for Qr may even be more numerous for the Adversary who controls a malicious tr. For instance, let x, y, and z be three malicious potential leaders of round r such that H (o-xr4) <H (ar4) <H (0-zr'1) and H (o-zr'1) is particulary small. That is, so small that there is a good chance that H (o-zr'1) is smaller of the hashed credential of every honest potential leader. Then, by asking x to hide his credential, the Adversary has a good chance of having y become the leader of round r ¨ 1.
This implies that he has another option for Qr: namely, H (SIGy (Qr-1) r).
Similarly, the Adversary may ask both x and y of withholding their credentials, so as to have z become the leader of round r ¨ 1 and gaining another option for Qr: namely, H (51-Gz (Qr-1) r).
Of course, however, each of these and other options has a non-zero chance to fail, because the Adversary cannot predict the hash of the digital signatures of the honest potential users.
A careful, Markov-chain-like analysis shows that, no matter what options the Adversary chooses to make at round r ¨ 1, as long as he cannot inject new users in the system, he cannot decrease the probability of an honest user to be the leader of round r 40 much below h.
This is the reason for which we demand that the potential leaders of round r are users already existing in round r ¨ k. It is a way to ensure that, at round r ¨ k, the Adversary cannot alter by much the probability that an honest user become the leader of round r. In fact, no matter what users he may add to the system in rounds r ¨ k through r, they are ineligible to become potential leaders (and a fortiori the leader) of round r. Thus the look-back parameter k ultimately is a security parameter. (Although, as we shall see in section ??, it can also be a kind of "convenience parameter" as well.) Ephemeral Keys Although the execution of our protocol cannot generate a fork, except with negligible probability, the Adversary could generate a fork, at the rth block, after the legitimate block r has been generated.
Roughly, once Br has been generated, the Adversary has learned who the verifiers of each step of round r are. Thus, he could therefore corrupt all of them and oblige them to certify a new block Br . Since this fake block might be propagated only after the legitimate one, users that have been paying attention would not be fooled.7 Nonetheless, B would be syntactically correct and we want to prevent from being manufactured.
6For instance, to keep it simple (but extreme), "when the time of the second step is about to expire", r could directly email a different candidate block B, to each user i. This way, whoever the step-2 verifiers might be, they will have received totally different blocks.
7Consider corrupting the news anchor of a major TV network, and producing and broadcasting today a We do so by means of a new rule. Essentially, the members of the verifier set SW"' of a step s of round r use ephemeral public keys pktr's to digitally sign their messages. These keys are single-use-only and their corresponding secret keys skr's are destroyed once used. This way, if a verifier is corrupted later on, the Adversary cannot force him to sign anything else he did not originally sign.
Naturally, we must ensure that it is impossible for the Adversary to compute a new key and convince an honest user that it is the right ephemeral key of verifier i E
SW"' to use in step s.
4.2 Common Summary of Notations, Notions and Parameters Notations = r > 0: the current round number.
= s> 1: the current step number in round r.
= Br: the block generated in round r.
= PKr: the set of public keys by the end of round r ¨ 1 and at the beginning of round r.
= Sr: the system status by the end of round r ¨ 1 and at the beginning of round r.8 = PAYr: the payset contained in B.
= fr: round-r leader. fr chooses the payset PAY of round r (and determines the next Qr).
= Qr: the seed of round r, a quantity (i.e., binary string) that is generated at the end of round r and is used to choose verifiers for round r 1. Qr is independent of the paysets in the blocks and cannot be manipulated by fr = SVr's: the set of verifiers chosen for step s of round r.
= SVr: the set of verifiers chosen for round r, SVr = Us>iSVr's.
= MSVr,s and HSVr's: respectively, the set of malicious verifiers and the set of honest verifiers in SW's. MSVr's U HSVr's = SVr's and MSVr's n HSVr's = 0.
= ni E Z+ and n E Z+: respectively, the expected numbers of potential leaders in each SVr'1, and the expected numbers of verifiers in each SW's, for s > 1.
Notice that n1 << n, since we need at least one honest honest member in SVr,i, but at least a majority of honest members in each SW"' for s > 1.
= h E (0,1): a constant greater than 2/3. h is the honesty ratio in the system. That is, the fraction of honest users or honest money, depending on the assumption used, in each PK' is at least h.
= H: a cryptographic hash function, modelled as a random oracle.
= I: A special string of the same length as the output of H.
= F E (0,1): the parameter specifying the allowed error probability. A
probability < F is considered "negligible", and a probability > 1 ¨ F is considered "overwhelming".
= Ph E (0,1): the probability that the leader of a round r, fr , is honest.
Ideally ph = h. With the existence of the Adversary, the value of ph will be determined in the analysis.
newsreel showing secretary Clinton winning the last presidential election.
Most of us would recognize it as a hoax. But someone getting out of a coma might be fooled.
81n a system that is not synchronous, the notion of "the end of round r ¨1"
and "the beginning of round r" need to be carefully defined. Mathematically, PK' and S' are computed from the initial status S and the blocks B', ...,B'.

= k E Z+: the look-back parameter. That is, round r ¨ k is where the verifiers for round r are chosen from ____ namely, SVr c pKr¨k.a = /31 E (0,1): for the first step of round r, a user in round r ¨ k is chosen to be in SVr,1 with probability pi = p E (0, 1): for each step s > 1 of round r, a user in round r ¨ k is chosen to be in SVr's with probability p ________ = CERTr: the certificate for B. It is a set of tH signatures of H(Br) from proper verifiers in round r.
= Br (Br ,CERTr) is a proven block.
A user i knows Br if he possesses (and successfully verifies) both parts of the proven block.
Note that the CERTr seen by different users may be different.
= Tzr: the (local) time at which a user i knows B. In the Algorand protocol each user has his own clock. Different users' clocks need not be synchronized, but must have the same speed. Only for the purpose of the analysis, we consider a reference clock and measure the players' related times with respect to it.
= art's and t3tr's: respectively the (local) time a user i starts and ends his execution of Step s of round r.
= A and A: essentially, the upper-bounds to, respectively, the time needed to execute Step 1 and the time needed for any other step of the Algorand protocol.
Parameter A upper-bounds the time to propagate a single 1MB block.
Parameter A upperbounds the time to propagate one small message per verifier in a Step s > 1.
We assume that A <4,\.
Notions = Verifier selection.
For each round r and step s > 1, SV"
E PK" : .H(SIGt(r,s,Qr-1)) < pl.
Each user i E PK" privately computes his signature using his long-term key and decides whether i E SVr's or not. If i E SVr's, then SIG,(r,s,Qr-1) is i's (r, s)-credential, compactly denoted by o-tr's.
For the first step of round r, SVr,1 and atr'l are similarly defined, with p replaced by pl.
The verifiers in SVr,1 are potential leaders.
= Leader selection.
User i E SVr,1 is the leader of round r, denoted by fr , if H(a4) < H (ar'l) for all potential leaders j E SVr,i. Whenever the hashes of two players' credentials are compared, in the unlikely event of ties, the protocol always breaks ties lexicographically according to the (long-term public keys of the) potential leaders.
By definition, the hash value of player r's credential is also the smallest among all users in PK". Note that a potential leader cannot privately decide whether he is the leader or not, without seeing the other potential leaders' credentials.
Since the hash values are uniform at random, when SVr'i is non-empty, r always exists and is honest with probability at least h. The parameter n1 is large enough so as to ensure that each SVr'i is non-empty with overwhelming probability.
9Strict1y speaking, "r ¨ k" should be "max{0, r ¨ k}".

= Block structure.
A non-empty block is of the form Br = (r, PAY', S/Ge,(Qr-1), H(Br-1)), and an empty block is of the form BEr = (r, 0, Qr H (Br-1)).
Note that a non-empty block may still contain an empty payset PAY', if no payment occurs in this round or if the leader is malicious. However, a non-empty block implies that r 1 the identity of fr , his credential cte,' and SIG(Q1) have all been timely revealed. The protocol guarantees that, if the leader is honest, then the block will be non-empty with overwhelming probability.
= Seed Q.
.
If Br is non-empty, then Qr H (S IG.e,(Qr-1),r), otherwise Qr H(Qr-1,r).
Parameters = Relationships among various parameters.
________ The verifiers and potential leaders of round r are selected from the users in PK", where k is chosen so that the Adversary cannot predict Qr-1 back at round r ¨
k ¨ 1 with probability better than F: otherwise, he will be able to introduce malicious users for round r ¨ k, all of which will be potential leaders/verifiers in round r, succeeding in having a malicious leader or a malicious majority in SV" for some steps s desired by him.
________ For Step 1 of each round r, n1 is chosen so that with overwhelming probability, SVr'l 0.
= Example choices of important parameters.
________ The outputs of H are 256-bit long.
________ h = 80%, ni = 35.
A = 1 minute and A = 15 seconds.
= Initialization of the protocol.
The protocol starts at time 0 with r = 0. Since there does not exist "B-1" or "CERT-1", syntactically B-1 is a public parameter with its third component specifying Q-1, and all users know B-1 at time 0.
5 A lgorand'i In this section, we construct a version of Algorancli working under the following assumption.
HONEST MAJORITY OF USERS ASSUMPTION: More than 2/3 of the users in each PK are honest.
In Section 7, we show how to replace the above assumption with the desired Honest Majority of Money assumption.
5.1 Additional Notations and Parameters Notations = m E Z+: the maximum number of steps in the binary BA protocol, a multiple of 3.

= Lr <m/3: a random variable representing the number of Bernoulli trials needed to see a 1, when each trial is 1 with probability and there are at most m/3 trials. If all trials fail then Lr m/3. Lr will be used to upper-bound the time needed to generate block B.
= tH 23n +1: the number of signatures needed in the ending conditions of the protocol.
= C ERT': the certificate for Br . It is a set of tH signatures of H(Br) from proper verifiers in round r.
Parameters = Relationships among various parameters.
________ For each step s> 1 of round r, n is chosen so that, with overwhelming probability, IHSVr'sl> 21M SVr,s1 and IHSVr's1+ LIIMSVrl< 2n.
The closer to 1 the value of h is, the smaller n needs to be. In particular, we use (variants of) Chernoff bounds to ensure the desired conditions hold with overwhelming probability.
________ m is chosen such that Lr <m/3 with overwhelming probability.
= Example choices of important parameters.
F =10-12.
________ n c=--' 1500, k = 40 and m = 180.
5.2 Implementing Ephemeral Keys in Algorandi As already mentioned, we wish that a verifier i E SVr's digitally signs his message mrz's of step s in round r, relative to an ephemeral public key pk, using an ephemeral secrete key skr's that he promptly destroys after using. We thus need an efficient method to ensure that every user can verify that pk,r's is indeed the key to use to verify i's signature of mzr's. We do so by a (to the best of our knowledge) new use of identity-based signature schemes.
At a high level, in such a scheme, a central authority A generates a public master key, PMK, and a corresponding secret master key, SMK. Given the identity, U, of a player U, A computes, via SMK, a secret signature key sku relative to the public key U, and privately gives sku to U. (Indeed, in an identity-based digital signature scheme, the public key of a user U is U itself!) This way, if A destroys SMK after computing the secret keys of the users he wants to enable to produce digital signatures, and does not keep any computed secret key, then U is the only one who can digitally sign messages relative to the public key U. Thus, anyone who knows "U's name", automatically knows U's public key, and thus can verify U's signatures (possibly using also the public master key PMK).
In our application, the authority A is user i, and the set of all possible users U coincides with the round-step pair (r, s) in __ say S = {i} x {r' ,r' 106} X {1, ,m 3}, where r' is a given round, and m 3 the upperbound to the number of steps that may occur within a round. This way, pks (i, r, s), so that everyone seeing i's signature SIG", s(Mt") can, with overwhelming probability, immediately verify it for the first million rounds r following r'.
In other words, i first generates PMK and SMK. Then, he publicizes that PMK is i's master public key for any round r E , 106], and uses SMK to privately produce and store the secret key sktr's for each triple (i, r, s) E S. This done, he destroys SMK. If he determines that he is not part of SV", then i may leave sktr's alone (as the protocol does not require that he aunthenticates any message in Step s of round r). Else, i first uses skr's to digitally sign his , message m,rs , and then destroys sky.

Note that i can publicize his first public master key when he first enters the system. That is, the same payment that brings i into the system (at a round r' or at a round close to r'), may also specify, at i's request, that i's public master key for any round r E
[r', r' 106] is PMK _____ e.g., by including a pair of the form (PMK, [r', r' 106]).
Also note that, since m 3 is the maximum number of steps in a round, assuming that a round takes a minute, the stash of ephemeral keys so produced will last i for almost two years.
At the same time, these ephemeral secret keys will not take i too long to produce. Using an elliptic-curve based system with 32B keys, each secret key is computed in a few microseconds.
Thus, if m 3 = 180, then all 180M secret keys can be computed in less than one hour.
When the current round is getting close to r' 106, to handle the next million rounds, i generates a new (PMK', SMK') pair, and informs what his next stash of ephemeral keys is by for example _________________________________________________________________ having SIG,(PMK',[r' 106 1, r' 2 = 106 1]) enter a new block, either as a separate "transaction" or as some additional information that is part of a payment. By so doing, i informs everyone that he/she should use PMK' to verify i's ephemeral signatures in the next million rounds. And so on.
(Note that, following this basic approach, other ways for implementing ephemeral keys without using identity-based signatures are certainly possible. For instance, via Merkle trees.1 ) Other ways for implementing ephemeral keys are certainly possible __________ e.g., via Merkle trees.
5.3 Matching the Steps of Algorand' with those of BA*
As we said, a round in Algorand'i has at most m 3 steps.
STEP 1. In this step, each potential leader i computes and propagates his candidate block B, together with his own credential, o-zr'l.
Recall that this credential explicitly identifies i. This is so, because atr'l = SIGt(r,1,Qr-1).
Potential verifier i also propagates, as part of his message, his proper digital signature of H(Bfl. Not dealing with a payment or a credential, this signature of i is relative to his r ephemeral public key pk1; : that is, he propagates sigpk ,i(H(R;)).
Given our conventions, rather than propagating B'z' and sigpk i(H(Bn), he could have , propagated S/Gpk,i(H(Bn). However, in our analysis we need to have explicit access to sig ,i(H(Bn).
STEPS 2. In this step, each verifier i sets f'; to be the potential leader whose hashed credential is the smallest, and B'z' to be the block proposed by et'. Since, for the sake of efficiency, we wish to agree on H(Br), rather than directly on Br i propagates the message he would have propagated in the first step of BA* with initial value et = H(Bn. That is, he propagates v't, after ephemerally signing it, of course. (Namely, after signing it relative to the right ephemeral public key, which in this case is pktr'2.) Of course too, i also transmits his own credential.
Since the first step of BA* consists of the first step of the graded consensus protocol GC, Step 2 of A/gorand' corresponds to the first step of GC.
' In this method, i generates a public-secret key pair (pC's , skr's) for each round-step pair (r, s) in say r', , r' 1061 x {1, , m 3}. Then he orders these public keys in a canonical way, stores the jth public key in the jth leaf of a Merkle tree, and computes the root value R,, which he publicizes. When he wants to sign a message relative to key pks , i not only provides the actual signature, but also the authenticating path for plsr's relative to R,. Notice that this authenticating path also proves that pls's is stored in the jth leaf. Form this idea, the rest of the details can be easily filled.

STEPS 3. In this step, each verifier i E SVr'2 executes the second step of BA*. That is, he sends the same message he would have sent in the second step of GC. Again, i's message is ephemerally signed and accompanied by i's credential. (From now on, we shall omit saying that a verifier ephemerally signs his message and also propagates his credential.) STEP 4. In this step, every verifier i E SVr'4 computes the output of GC, (v,, g,), and ephemerally signs and sends the same message he would have sent in the third step of BA*, that is, in the first step of BBA*, with initial bit 0 if g, = 2, and 1 otherwise.
STEP s = 5, ... , m 2. Such a step, if ever reached, corresponds to step s ¨1 of BA*, and thus to step s ¨ 3 of BBA*.
Since our propagation model is sufficiently asynchronous, we must account for the possibility that, in the middle of such a step s, a verifier i E SVr's is reached by information proving him that block Br has already been chosen. In this case, i stops his own execution of round r of Algorand', and starts executing his round-(r 1) instructions.
Accordingly, the instructions of a verifier i E SVr's, in addition to the instructions corresponding to Step s ¨ 3 of BBA*, include checking whether the execution of BBA*
has halted in a prior Step s'. Since BBA* can only halt is a Coin-Fixed-to-0 Step or in a Coin-Fixed-to-1 step, the instructions distinguish whether A (Ending Condition 0): ¨ 2 0 mod 3, or B (Ending Condition 1): s' ¨ 2 1 mod 3.
In fact, in case A, the block Br is non-empty, and thus additional instructions are necessary to ensure that i properly reconstructs Br, together with its proper certificate CERTr. In case B, the block Br is empty, and thus i is instructed to set Br =
BrE: =
(r,0,Qr-1,H(Br-1)), and to compute CERTr.
If, during his execution of step s, i does not see any evidence that the block Br has already been generated, then he sends the same message he would have sent in step s¨ 3 of BBA*.
STEP Tri 3. If, during step m 3, i E SVr'm+3 sees that the block Br was already generated in a prior step s', then he proceeds just as explained above.
Else, rather then sending the same message he would have sent in step m of BBA*, i is instructed, based on the information in his possession, to compute Br and its corresponding certificate CERTr.
Recall, in fact, that we upperbound by m 3 the total number of steps of a round.
5.4 The Actual Protocol Recall that, in each step s of a round r, a verifier i E SVr's uses his long-term public-secret key pair to produce his credential, az"
SIG,(r,s,Qr-1), as well as SIG, (Qr-1) in case s = 1.
Verifier i uses his ephemeral secret key sktr's to sign his (r, s)-message mtr's. For simplicity, when r and s are clear, we write esig,(x) rather than sigpv,. (x) to denote i's proper ephemeral signature of a value x in step s of round r, and write ESIGi(x) instead of S/Gpij,s(x) to denote (i,x,esigz(x)).
Step 1: Block Proposal Instructions for every user i E PKr¨k: User i starts his own Step 1 of round r as soon as he knows Br-1.

= User i computes Qr-1 from the third component of Br-1 and checks whether i E SVr'1 or not.
= If i SW,1, then i stops his own execution of Step 1 right away.
= If i E SVr'1, that is, if i is a potential leader, then he collects the round-r payments that have been propagated to him so far and computes a maximal payset PAY: from them.
Next, he computes his "candidate block" _8'; = (r, PAY, SIG,(Qr-1), H (Br-1)).
Finally, he computes the message r 1 r Tnz' = (Br, esigz(H(R;)), o-zr '1 ), destroys his ephemeral secret key sk1z' , and then propagates r,1 .
Remark. In practice, to shorten the global execution of Step 1, it is important that the (r,1)-messages are selectively propagated. That is, for every user i in the system, for the first (r,1)-message that he ever receives and successfully verifies," player i propagates it as usual.
For all the other (r,1)-messages that player i receives and successfully verifies, he propagates it only if the hash value of the credential it contains is the smallest among the hash values of the credentials contained in all (r,1)-messages he has received and successfully verified so far.
Furthermore, as suggested by Georgios Vlachos, it is useful that each potential leader i also r 1 propagates his credential separately: those small messages travel faster than blocks, ensure timely propagation of the mir'l's where the contained credentials have small hash values, while make those with large hash values disappear quickly.
"That is, all the signatures are correct and both the block and its hash are valid although i does not check whether the included payset is maximal for its proposer or not.

Step 2: The First Step of the Graded Consensus Protocol GC
Instructions for every user i E PKr¨k: User i starts his own Step 2 of round r as soon as he knows Br-1.
= User i computes Qr-1 from the third component of Br-1 and checks whether i E SVr'2 or not.
= If i SVr'2 then i stops his own execution of Step 2 right away.
= If i E SVr'2, then after waiting an amount of time t2 A A, i acts as follows.
1. He finds the user such that H(OTer'1) < H (0-r:1) for all credentials ar:1 that are part of the successfully verified (r, 1)-messages he has received so far.12 r 1 2. If he has received from a valid message me' = (Be,7e esige(H (Bn r ), af'1),13 then i sets v = H (Br )* otherwise i sets v' I.
3. i computes the message mir'2 (E S G
ir'2),14 destroys his ephemeral secret key skir'2, and then propagates mir'2.
Step 3: The Second Step of GC
Instructions for every user i E PKr¨k: User i starts his own Step 3 of round r as soon as he knows Br-1 .
= User i computes Qr ¨1 from the third component of Br-1 and checks whether i E SVr'3 or not.
= If i SVr,3, then i stops his own execution of Step 3 right away.
= If i E SVr'3, then after waiting an amount of time t3 t2 2A = 3A A, i acts as follows.
r 1. If there exists a value v' I such that, among all the valid messages mi2 he has received, more than 2/3 of them are of the form (E S IG J(v), ar:2), without any contradiction,15 then he computes the message mir'3 (ESIGi(21),CTir'3). Otherwise, he computes mri''3 (ESiGi(i), 2. i destroys his ephemeral secret key skir'3, and then propagates mir'3.
Step 4: Output of GC and The First Step of BB A*
Instructions for every user i E PKr¨k: User i starts his own Step 4 of round r as soon as he knows Br-1.
12Essentially, user i privately decides that the leader of round r is user L.
13Again, player L's signatures and the hashes are all successfully verified, and PAY/ in BT is a valid payset for round r __ although i does not check whether PAY/ is maximal for f or not.
14The message mir'2 signals that player i considers v,' to be the hash of the next block, or considers the next block to be empty.
15That is, he has not received two valid messages containing ESIG,(v') and a different ESIG3(v") respectively, from a player j. Here and from here on, except in the Ending Conditions defined later, whenever an honest player wants messages of a given form, messages contradicting each other are never counted or considered valid.

= User i computes Qr-1 from the third component of Br-1 and checks whether i E SVr''I or not.
= If i SVr,4, then i his stops his own execution of Step 4 right away.
= If i E SVr'4, then after waiting an amount of time t4 t3 2A = 5A A, i acts as follows.
1. He computes vz and gz, the output of GC, as follows.
(a) If there exists a value v' I such that, among all the valid messages mr'3 he has received, more than 2/3 of them are of the form (ES/G3(e), ar'3), then he sets II and gt 2.
(b) Otherwise, if there exists a value v' I such that, among all the valid messages mr,3 he has received, more than 1/3 of them are of the form (ESIG3(0,0-ir'3), then he sets v3 v' and gt 1.16 (c) Else, he sets v3 H (Bn and gt 0.
2. He computes bz, the input of BRA*, as follows:
bt 0 if gt = 2, and bt 1 otherwise.
3. He computes the message Mrt'Ll = (ESiGt (10 , ESIG z(vz), rT:'4), destroys his ephemeral secret key sktr'4, and then propagates mir'4.
Step s, 5 < s < m 2, s ¨ 2 0 mod 3: A Coin-Fixed-To-0 Step of B B A*
Instructions for every user i E PKr¨k: User i starts his own Step s of round v as soon as he knows Br-1.
= User i computes Qr-1 from the third component of Br-1 and checks whether i E
SVr's.
= If i SVr,s, then i stops his own execution of Step s right away.
= If i E SVr's then he acts as follows.
¨ He waits until an amount of time ts ts_, 2A = (2s ¨ 3)A A has passed.
¨ Ending Condition 0: If, during such waiting and at any point of time, there exists a string v I and a step s' such that (a) 5 < s' < s, s' ¨ 2 0 mod 3 ________ that is, Step s' is a Coin-Fixed-To-0 step, (b) i has received at least tH =
1 valid messages mr.i's'-1 = (ES/G3(0), ES/G3(v), 0_ ir,s') ¨1. 17 , and (c) i has received a valid message mir'l = (B; , esig 3 (H (BD), 0- 3r4) with v = H (B; ), then, i stops his own execution of Step s (and in fact of round v) right away without propagating anything; sets Br = B;; and sets his own C E RTr to be the set of messages mr-1.2' of sub-step (b).18 16It can be proved that the v' in case (b), if exists, must be unique.
17Such a message from player j is counted even if player i has also received a message from j signing for 1.
Similar things for Ending Condition 1. As shown in the analysis, this is done to ensure that all honest users know B within time A from each other.
18User i now knows B' and his own round r finishes. He still helps propagating messages as a generic user, but does not initiate any propagation as a (r, s)-verifier. In particular, he has helped propagating all messages in his C E RT' , which is enough for our protocol. Note that he should also set b, 0 for the binary BA protocol, but b, is not needed in this case anyway. Similar things for all future instructions.

¨ Ending Condition I: If, during such waiting and at any point of time, there exists a step s' such that (a') 6 < < s, s' ¨ 2 1 mod 3 ______________________________________________ that is, Step s' is a Coin-Fixed-To-1 step, and (b') i has received at least tH valid messages mr.'s'-1 = (ES/G3(1), ES/G3(v3), as'l)19 , then, i stops his own execution of Step s (and in fact of round r) right away without propagating anything; sets Br = BEr; and sets his own CERT to be the set of messages mr,s'-1 of sub-step (b').
¨ Otherwise, at the end of the wait, user i does the following.
He sets vz to be the majority vote of the vi's in the second components of all the valid ,s-1 mr 's he has received.
He computes bz as follows.
If more than 2/3 of all the valid Mir's-1's he has received are of the form (ES/G3(0), ES/G3 (V3), air's-1), then he sets I), O.
Else, if more than 2/3 of all the valid mir's¨i's he has received are of the form (ES/G(1), ES/G3 (v3), ctri's-1), then he sets I), 1.
Else, he sets b = U.
He computes the message mrz's (ES/G,(1),), E S
, destroys his ephemeral secret key sks, and then propagates m.
Step s, 6 < s <m 2, s ¨ 2 1 mod 3: A Coin-Fixed-To-1 Step of B B A*
Instructions for every user i E PKr¨k: User i starts his own Step s of round r as soon as he knows = User i computes Qr-1 from the third component of Br-1 and checks whether i E SVr's or not.
= If i SV", then i stops his own execution of Step s right away.
= If i E SVr's then he does the follows.
¨ He waits until an amount of time ts (2s ¨ 3)A A has passed.
¨ Ending Condition 0: The same instructions as Coin-Fixed-To-0 steps.
¨ Ending Condition I: The same instructions as Coin-Fixed-To-0 steps.
¨ Otherwise, at the end of the wait, user i does the following.
He sets vz to be the majority vote of the vi 's in the second components of all the valid Mr,s-1s,3 he has received.
He computes I), as follows.
If more than 2/3 of all the valid mr's-1's he has received are of the form (ES/G3 (0), ES/G3 (v3), oTs-1), then he sets I), U.
Else, if more than 2/3 of all the valid mri's-1's he has received are of the form (ES/G3(1),ES/G3 (v3), a''), then he sets bz 1.
191n this case, it does not matter what the vi's are.

Else, he sets b = 1.
He computes the message mrz's (ES/G,(1),), E S Gz(Vz), Us), destroys his ephemeral secret key skir's, and then propagates ms.
Step s, 7 < s <m 2, s ¨ 2 2 mod 3: A Coin-Genuinely-Flipped Step of BBA*
Instructions for every user i E PKr¨k: User i starts his own Step s of round r as soon as he knows Br-1.
= User i computes Qr-1 from the third component of Br-1 and checks whether i E SV" or not.
= If i E SV", then i stops his own execution of Step s right away.
= If i E SV" then he does the follows.
¨ He waits until an amount of time ts (2s ¨ 3)A A has passed.
¨ Ending Condition 0: The same instructions as Coin-Fixed-To-0 steps.
¨ Ending Condition I: The same instructions as Coin-Fixed-To-0 steps.
¨ Otherwise, at the end of the wait, user i does the following.
He sets vz to be the majority vote of the vi's in the second components of all the valid Mr,s¨l's he has received.

He computes bt as follows.
If more than 2/3 of all the valid Mr's-1's he has received are of the form (ESIG3(0),ESIG3(V3), o-ir's-1), then he sets bt U.
Else, if more than 2/3 of all the valid mir's¨l's he has received are of the form (ES/G(1), ES/G3 (v3), ctir's-1), then he sets bt 1.
Else, let SVr's be the set of (r, s-1)-verifiers from whom he has received a valid r,s-1 r's-1 message m I), isb(miniESV"
. He sets _i H(o-i)).
, He computes the message mtr's (ES/Gz(bz), E SIGt(Vt),0-tr's), destroys his ephemeral secret key sky, and then propagates ms.
Step m 3: The Last Step of BBA* 20 Instructions for every user i E PKr¨k: User i starts his own Step m 3 of round r as soon as he knows Br-1.
= User i computes Qr-1 from the third component of Br-1 and checks whether i E ST7r,"2+3 or not.
= If i E ST7r,n2+3, then i stops his own execution of Step m 3 right away.
= If i E ST7r,m+3 then he does the follows.
20With overwhelming probability BBA* has ended before this step, and we specify this step for completeness.

¨ He waits until an amount of time tm+3 tm+2 2A = (2m 3)A A has passed.
¨ Ending Condition 0: The same instructions as Coin-Fixed-To-0 steps.
¨ Ending Condition I: The same instructions as Coin-Fixed-To-0 steps.
¨ Otherwise, at the end of the wait, user i does the following.
He sets out, 1 and Br B.
He computes the message Mir'rn+3 = (ESIG,(out,),ESIG,(H(Br)), 0-tr" 3), destroys his ephemeral secret key skir'm+3, and then propagates mir'ru+3 to certify B.21 Reconstruction of the Round-r Block by Non-Verifiers Instructions for every user i in the system: User i starts his own round v as soon as he knows Br-1, and waits for block information as follows.
¨ If, during such waiting and at any point of time, there exists a string v and a step s' such that (a) 5 < < m 3 with s' ¨ 2 0 mod 3, r ¨1 r ¨1 (b) i has received at least tH valid messages m; =
(ES/G3(0), ES/G3(v), o-3' ), and (c) i has received a valid message mir'l = esigi (H (B;)), air4) with v = H
(B;), then, i stops his own execution of round v right away; sets Br = B; ; and sets his own CERT to be the set of messages mr's'-1 of sub-step (b).
¨ If, during such waiting and at any point of time, there exists a step s' such that (a') 6<8' < m 3 with s' ¨ 2 1 mod 3, and r ¨1 r (b') i has received at least tH valid messages m3' = (ES/G3(1), ES/G3(v3), o- ¨13' ), then, i stops his own execution of round v right away; sets Br = BEr; and sets his own CERT' to be the set of messages mr's'-1 of sub-step (b').
¨ If, during such waiting and at any point of time, i has received at least tH
valid messages r,m+3 = (ESIG3(1),ESIG3(H(Ber)), air'711+3), then i stops his own execution of round v right away, sets Br = BEr, and sets his own CERT' to be the set of messages m3r'ru+3 for 1 and H(BEr).
6 Algorand'2 In this section, we construct a version of Algorand' working under the following assumption.
HONEST MAJORITY OF USERS ASSUMPTION: More than 2/3 of the users in each PKr are honest.
In Section 7, we show how to replace the above assumption with the desired Honest Majority of Money assumption.
21A certificate from Step m 3 does not have to include ES/G,(out,). We include it for uniformity only: the certificates now have a uniform format no matter in which step they are generated.

6.1 Additional Notations and Parameters for Algorand2 Notations = p, E Z+: a pragmatic upper-bound to the number of steps that, with overwhelming probability, will actually taken in one round. (As we shall see, parameter p, controls how many ephemeral keys a user prepares in advance for each round.) = _Lir: a random variable representing the number of Bernoulli trials needed to see a 1, when each trial is 1 with probability Lr will be used to upper-bound the time needed to generate block B.
= tH: a lower-bound for the number of honest verifiers in a step s> 1 of round r, such that with overwhelming probability (given n and p), there are > tH honest verifiers in SW,s.
Parameters = Relationships among various parameters.
________ For each step s> 1 of round r, n is chosen so that, with overwhelming probability, IHSW,s1 > tH and IHSV"1+21MSV"I <2tH.
Note that the two inequalities above together imply IHSV"1 > 21MSVrl: that is, there is a 2/3 honest majority among selected verifiers.
The closer to 1 the value of h is, the smaller n needs to be. In particular, we use (variants of) Chernoff bounds to ensure the desired conditions hold with overwhelming probability.
= Specific choices of important parameters.
F =10-18.
________ n ,^,f, 4000, tH ,^,f, 0.69n, k = 70.
6.2 Implementing Ephemeral Keys in Alg rand/2 Recall that a verifier i E SVr's digitally signs his message mrz's of step s in round r, relative to an ephemeral public key pk,r's, using an ephemeral secrete key skr's that he promptly destroys after using. When the number of possible steps that a round may take is capped by a given integer we have already seen how to practically handle ephemeral keys. For example, as we have explained in A/gorand'i (where p, = m 3), to handle all his possible ephemeral keys, from a round r' to a round r' 106, i generates a pair (PMK, SMK), where PMK
public master key of an identity based signature scheme, and SMK its corresponding secret master key. User i publicizes PMK and uses SMK to generate the secret key of each possible ephemeral public key (and destroys SMK after having done so). The set of i's ephemeral public keys for the relevant rounds is S = fil x {r', , r' 106} x {1, , i}. (As discussed, as the round r' 106 approaches, i "refreshes" his pair (PMK, SMK).) In practice, if p, is large enough, a round of Algoranc112 will not take more than p, steps. In principle, however, there is the remote possibility that, for some round r the number of steps actually taken will exceed When this happens, i would be unable to sign his message mzr's for any step s > ,a, because he has prepared in advance only p, secret keys for round r. Moreover, he could not prepare and publicize a new stash of ephemeral keys, as discussed before. In fact, to do so, he would need to insert a new public master key PMK' in a new block.
But, should round r take more and more steps, no new blocks would be generated.
However, solutions exist. For instance, i may use the last ephemeral key of round r, as follows. He generates another stash of key-pairs for round r _____________ e.g., by (1) generating another master key pair (PMK, SMK); (2) using this pair to generate another, say, 106 ephemeral ¨r, +1 ¨r, 106 keys, sk, , . . . , , corresponding to steps p, 1, ..., 106 of round r; (3) using sk,r'11 to digitally sign PMK (and any (r, ,u)-message if i E ST/T'A), relative to pk,r41; and (4) erasing SMK and sk,r41. Should i become a verifier in a step ,a+s with s E
{1,...,106}, then i digitally ¨r, signs his (r, p, s)-message mzr'11 s relative to his new key pk s, = (i, r, p s). Of course, to verify this signature of i, others need to be certain that this public key corresponds to i's new public master key PMK. Thus, in addition to this signature, i transmits his digital signature of PMK relative to pk,r41.
Of course, this approach can be repeated, as many times as necessary, should round r continue for more and more steps! The last ephemeral secret key is used to authenticate a new master public key, and thus another stash of ephemeral keys for round r. And so on.
6.3 The Actual Protocol Algorand'2 Recall again that, in each step s of a round r, a verifier i E SV" uses his long-term public-secret key pair to produce his credential, az"
SIGt(r, s, Q'), as well as SIG t (Qr-1) in case s = 1. Verifier i uses his ephemeral key pair, (pktr's, sktr's), to sign any other message m that may be required. For simplicity, we write esigt(m), rather than sigpv,s(m), to denote i's proper ephemeral signature of m in this step, and write ES/G (m) instead of S/Gpv,s(m) (i, m, esigt(m)).
Step 1: Block Proposal Instructions for every user i E PKr¨k: User i starts his own Step 1 of round r as soon as he has CERT'', which allows i to unambiguously compute H(B1) and Q'.
= User i uses Qr ¨1 to check whether i E SVr'l or not. If i SVr'1, he does nothing for Step 1.
= If i E SW,1, that is, if i is a potential leader, then he does the following.
(a) If i has seen B , , Br-1 himself (any B = B can be easily derived from its hash value in C E RT3 and is thus assumed "seen"), then he collects the round-r payments that have been propagated to him so far and computes a maximal payset PAY:
from them.
(b) If i hasn't seen all B , , Br-1 yet, then he sets PAY: =0.
(c) Next, i computes his "candidate block" .0; = (r, PAY,SIGt(Qr-1), H (Br-1)).
(c) Finally, i computes the message mzr'l = (Br, esigt(H(M), atr"), destroys his ephemeral secret key sktr'1, and then propagates two messages, mir'1 and (S/Gt(Qr-1), o1), separately but simultaneously.22 Selective Propagation To shorten the global execution of Step 1 and the whole round, it is important that the (r, 1)-messages are selectively propagated. That is, for every user j in the system, 22When i is the leader, S/G1(CI-1) allows others to compute Qr = H(SIG,(Qr 1),r)=

= For the first (r, 1)-message that he ever receives and successfully verifies,23 whether it contains a block or is just a credential and a signature of Qr-1, player j propagates it as usual.
= For all the other (r, 1)-messages that player j receives and successfully verifies, he propagates it only if the hash value of the credential it contains is the smallest among the hash values of the credentials contained in all (r, 1)-messages he has received and successfully verified so far.
r = However, if j receives two different messages of the form mi'1 from the same player i,24 he discards the second one no matter what the hash value of i's credential is.
Note that, under selective propagation it is useful that each potential leader i propagates his credential separately from m1:25 those small messages travel faster than blocks, ensure timely propagation of the mir'l's where the contained credentials have small hash values, while make those with large hash values disappear quickly.
Step 2: The First Step of the Graded Consensus Protocol GC
Instructions for every user i E PKr¨k: User i starts his own Step 2 of round r as soon as he has CE RTr-1.
= User i waits a maximum amount of time t2 = A A. While waiting, i acts as follows.
r 1. After waiting for time 2A, he finds the user such that H(o1-f' ) < H(o1) for all credentials Cir:1 that are part of the successfully verified (r, 1)-messages he has received so far.26 2. If he has received a block B', which matches the hash value H (Br ¨1) con-tained in C ERTr-1 ,27 and if he has received from a valid message mt.'1 =
(BT , esige(H (BD), <1,28 then i stops waiting and sets t/i (H (BD ,).
3. Otherwise, when time t2 runs out, i sets t/i I.
4. When the value of t/i has been set, i computes Qr ¨1 from CERT'l and checks whether i E .9Vr'2 or not.
5. If i E SVr'2 i computes the message m7'2 = (ES/Gi(Vii), ari'2),29 destroys his ephemeral r 2 r 2 secret key ski' , and then propagates . Otherwise, i stops without propagating anything.
23That is, all the signatures are correct and, if it is of the form mr,'1, both the block and its hash are valid although j does not check whether the included payset is maximal for i or not.
24Which means i is malicious.
25We thank Georgios Vlachos for suggesting this.
26Essentially, user i privately decides that the leader of round r is user f.
270f course, if CERTr-1 indicates that Br-1 = B:-1, then i has already "received" Br-1 the moment he has CERTr-1.
28Again, player L's signatures and the hashes are all successfully verified, and PAY/ in H; is a valid payset for round r ____________________________________________________________________ although i does not check whether PAW' is maximal for or not. If BT contains an empty payset, then there is actually no need for i to see Br-1 before verifying whether BT
is valid or not.
29The message m,,r'2 signals that player i considers the first component of v, to be the hash of the next block, or considers the next block to be empty.

Step 3: The Second Step of GC
Instructions for every user i E PKr¨k: User i starts his own Step 3 of round v as soon as he has CERT'l.
= User i waits a maximum amount of time t3 t2 2A = 3A A. While waiting, i acts as follows.
1. If there exists a value v such that he has received at least tH valid messages mir'2 of the form (ES/G3(v), cr3r'2), without any contradiction,30 then he stops waiting and sets VI = V.
2. Otherwise, when time t3 runs out, he sets v" = I.
3. When the value of v" has been set, i computes Qr-1 from CERT'l and checks whether i E SVr'3 or not.
4. If i E SVr'3, then i computes the message mr3 = (ESiGt(V/), a:'3), destroys his r r ephemeral secret key sk,3, , and then propagates m,3i . Otherwise, i stops without propagating anything.
Step 4: Output of GC and The First Step of BBA*
Instructions for every user i E PKr¨k: User i starts his own Step 4 of round v as soon as he finishes his own Step 3.
= User i waits a maximum amount of time 2A.31 While waiting, i acts as follows.
1. He computes v3 and gt, the output of GC, as follows.
(a) If there exists a value v" I such that he has received at least tH valid messages mr'3 = (ES/G3 (V/), ar'3), then he stops waiting and sets v3 vi and g, 2.
r r,3 (b) If he has received at least tH valid messages Tri3'3 = (ES/G3 (I), o-3 ), then he stops waiting and sets v3 I and g, 0.32 (c) Otherwise, when time 2A runs out, if there exists a value v"
I such that he has received at least [41 valid messages mr'3 = (ES/G3 (e) o-r'3) then he sets Vz= V/ and g, 1.33 (d) Else, when time 2A runs out, he sets v3 I and g, 0.
2. When the values v3 and g, have been set, i computes bz, the input of BBA*, as follows:
b, 0 if g, = 2, and b, 1 otherwise.
3. i computes Qr-1 from CERT'l and checks whether i E SVr'4 or not.
4. If i E SVr'4, he computes the message mir'4 (ES/G,(1),), ES/Gz(vz), atr'4), destroys r 4 r 4 his ephemeral secret key sk,' , and propagates mi' . Otherwise, i stops without propagating anything.
30That is, he has not received two valid messages containing ESIG,(v) and a different ESIG1(1)) respectively, from a player j. Here and from here on, except in the Ending Conditions defined later, whenever an honest player wants messages of a given form, messages contradicting each other are never counted or considered valid.
31Thus, the maximum total amount of time since i starts his Step 1 of round r could be Li t3 2A = 5A A.
32Whether Step (b) is in the protocol or not does not affect its correctness.
However, the presence of Step (b) allows Step 4 to end in less than 2A time if sufficiently many Step-3 verifiers have "signed I."
331t can be proved that the v' in this case, if exists, must be unique.

Step s, 5 < s < m 2, s ¨ 2 0 mod 3: A Coin-Fixed-To-0 Step of BBA*
Instructions for every user i E PKr¨k: User i starts his own Step s of round v as soon as he finishes his own Step s ¨1.
= User i waits a maximum amount of time 2A.34 While waiting, i acts as follows.
¨ Ending Condition 0: If at any point there exists a string v I and a step s' such that (a) 5 < s' < s, s' ¨ 2 0 mod 3 ________ that is, Step s' is a Coin-Fixed-To-0 step, (b) i has received at least tH valid messages r,s' ¨1 = (ESIGi(0), ESIGi(V), Or'-1),35 and 711./
(c) i has received a valid message (S/Gi(Qr-1), ar.'1) with j being the second component of v, then, i stops waiting and ends his own execution of Step s (and in fact of round v) right away without propagating anything as a (v, s)-verifier; sets H(B) to be the first r component of v; and sets his own CERT' to be the set of messages ms'-1 i' of step (b) together with (S/Gi(Qr-1), 1) .36 ¨ Ending Condition I: If at any point there exists a step s' such that (a') 6 < s' < s, s' ¨ 2 1 mod 3 _______ that is, Step s' is a Coin-Fixed-To-1 step, and (b') i has received at least tH valid messages mr's'-1 = (ES/G(1), ES/Gi(vi), r ,s' ¨1) 37 then, i stops waiting and ends his own execution of Step s (and in fact of round v) right away without propagating anything as a (v, s)-verifier; sets Br = BEr;
and sets his own CERT to be the set of messages mr's'-1 of sub-step (b').
¨ If at any point he has received at least tH valid mr.'s l's of the form (ES/G(1), ES/Gi(Vi), Ts-1), then he stops waiting and sets bi 1.
¨ If at any point he has received at least tH valid mr's l's of the form (ES/Gi(0), ESIGi(Vi), 0-3r.'s-1), but they do not agree on the same v, then he stops waiting and sets bi 0.
¨ Otherwise, when time 2A runs out, i sets bi 0.
¨ When the value bi has been set, i computes Qr from CERT'l and checks whether i E SVr's.
¨ If i E SW's, i computes the message 'mri's = (ES/Gi(bi), ES IGi(Vi), air's) with vi being the value he has computed in Step 4, destroys his ephemeral secret key sky, and then propagates mir's. Otherwise, i stops without propagating anything.
34Thus, the maximum total amount of time since i starts his Step 1 of round r could be ts ts_i 2A =
(2s ¨ 3)A A.
35Such a message from player j is counted even if player i has also received a message from j signing for 1.
Similar things for Ending Condition 1. As shown in the analysis, this is to ensure that all honest users know CERT' within time A from each other.
36User i now knows H(B) and his own round r finishes. He just needs to wait until the actually block B' is propagated to him, which may take some additional time. He still helps propagating messages as a generic user, but does not initiate any propagation as a (r, s)-verifier. In particular, he has helped propagating all messages in his CERT', which is enough for our protocol. Note that he should also set b, 0 for the binary BA protocol, but b, is not needed in this case anyway. Similar things for all future instructions.
37In this case, it does not matter what the vj's are.

Step s, 6 < s < m 2, s ¨ 2 1 mod 3: A Coin-Fixed-To-1 Step of BB A*
Instructions for every user i E PKr¨k: User i starts his own Step s of round r as soon as he finishes his own Step s ¨1.
= User i waits a maximum amount of time 2A. While waiting, i acts as follows.
¨ Ending Condition 0: The same instructions as in a Coin-Fixed-To-0 step.
¨ Ending Condition I: The same instructions as in a Coin-Fixed-To-0 step.
¨ If at any point he has received at least tH valid mr's l's of the form (ES/G3(0), ES/G3 (v3), ctri's 1 then he stops waiting and sets bt 0.38 ¨ Otherwise, when time 2A runs out, i sets bt 1.
¨ When the value I), has been set, i computes Qr from C ERT'l and checks whether i E SV".
¨ If i E SVr's, i computes the message mzr's = (ES/G,(1),), E S IG,(Vt) az") with vz being the value he has computed in Step 4, destroys his ephemeral secret key skr's, and then propagates Tris. Otherwise, i stops without propagating anything.
Step s, 7 < s <m 2, s ¨ 2 2 mod 3: A Coin-Genuinely-Flipped Step of BB A*
Instructions for every user i E PKr¨k: User i starts his own Step s of round r as soon as he finishes his own step s ¨ 1.
= User i waits a maximum amount of time 2A. While waiting, i acts as follows.
¨ Ending Condition 0: The same instructions as in a Coin-Fixed-To-0 step.
¨ Ending Condition I: The same instructions as in a Coin-Fixed-To-0 step.
¨ If at any point he has received at least tH valid mr's¨l's of the form (ES/G3(0), ES/G3 (v3), ar's-1 then he stops waiting and sets bt U.
¨ If at any point he has received at least tH valid mr's¨l's of the form (ES/G3(1), ES/G3 (v3), ar's-1 then he stops waiting and sets bt 1.
¨ Otherwise, when time 2A runs out, letting SVtr's 1 be the set of (r, s¨ 1)-verifiers from whom he has received a valid message Mr's-1 i sets b = lsb(min ESV r' s_i H
(ar's-1)).
, ¨ When the value bt has been set, i computes Qr from C ERT'l and checks whether i E SV".
¨ If i E SVr's i computes the message mzr's = (ES/Gz(bz), ESIGt(Vt) az") with vz being the value he has computed in Step 4, destroys his ephemeral secret key sk, and then propagates mir's. Otherwise, i stops without propagating anything.
Remark. In principle, as considered in subsection 6.2, the protocol may take arbitrarily many steps in some round. Should this happens, as discussed, a user i E SV" with s > p, has exhausted his stash of pre-generated ephemeral keys and has to authenticate his (r, s)-message mrt's by a "cascade" of ephemeral keys. Thus i's message becomes a bit longer and transmitting these longer messages will take a bit more time. Accordingly, after so many steps of a given round, the value of the parameter A will automatically increase slightly. (But it reverts to the original A once a new block is produced and a new round starts.) 38Note that receiving tH valid (r, s ¨ 1)-messages signing for 1 would mean Ending Condition 1.

Reconstruction of the Round-r Block by Non-Verifiers Instructions for every user i in the system: User i starts his own round r as soon as he has = i follows the instructions of each step of the protocol, participates the propagation of all messages, but does not initiate any propagation in a step if he is not a verifier in it.
= i ends his own round r by entering either Ending Condition 0 or Ending Condition 1 in some step, with the corresponding CERT'.
= From there on, he starts his round r +1 while waiting to receive the actual block Br (unless he has already received it), whose hash H(Br) has been pinned down by CERT'.
Again, if CERT indicates that Br = BE' , the i knows Br the moment he has CERT'.
7 Protocol Algorand' with Honest Majority of Money We now, finally, show how to replace the Honest Majority of Users assumption with the much more meaningful Honest Majority of Money assumption. The basic idea is (in a proof-of-stake flavor) "to select a user i E Picr¨k to belong to SV" with a weight (i.e., decision power) proportional to the amount of money owned by i."39 By our HMM assumption, we can choose whether that amount should be owned at round r ¨k or at (the start of) round r. Assuming that we do not mind continual participation, we opt for the latter choice. (To remove continual participation, we would have opted for the former choice. Better said, for the amount of money owned at round r ¨ k ¨ 2, 000.) There are many ways to implement this idea. The simplest way would be to have each key hold at most 1 unit of money and then select at random n users i from Picr¨k such that 1.
The Next Simplest Implementation The next simplest implementation may be to demand that each public key owns a maximum amount of money M, for some fixed M. The value M is small enough compared with the total amount of money in the system, such that the probability a key belongs to the verifier set of more than one step in __ say _______________________________________________ k rounds is negligible. Then, a key i E Pick, owning an amount of money ci,r) in round r, is chosen to belong to SW,' if (r) .H (SIG, (r, s, Qr-1)) < p az ______________________ .
And all proceeds as before.
A More Complex Implementation The last implementation "forced a rich participant in the system to own many keys".
An alternative implementation, described below, generalizes the notion of status and consider each user i to consist of K +1 copies (i, v), each of which is independently selected to be a verifier, and will own his own ephemeral key (pk, sk) in a step s of a round r. The value K depends on the amount of money a(zr) owned by i in round r.
Let us now see how such a system works in greater detail.
--39We should say PKk2,000 ' so as to replace continual participation. For simplicity, since one may wish to require continual participation anyway, we use PK'-k as before, so as to carry one less parameter.

Number of Copies Let n be the targeted expected cardinality of each verifier set, and let az(r) be the amount of money owned by a user i at round v. Let .Ar be the total amount of money owned by the users in PKr-k at round v, that is, Ar = >_: C4r) .
tEPKr-k If i is an user in PK", then i's copies are (i, 1), , (i, K 1), where [n = a(r) K=
Ar EXAMPLE. Let n = 1, 000, Ar = 109, and o,r) = 3.7 millions. Then, K=
[103 = (3.7 = 106)]
= [3.7] =3 .

Verifiers and Credentials Let i be a user in PK" with K 1 copies.
r For each v = 1, , K, copy (i, v) belongs to SVr's automatically. That is, i's credential is o- 's S Gt((i, v), r, s , Qr-1), but the corresponding condition becomes .H(o-tr'vs) < 1, which is always true.
For copy (i, K 1), for each Step s of round v, i checks whether .H (SIGt((i, K 1), v, S,Qr-1)) < aCr) - K .
- Ar If so, copy (i, K 1) belongs to SV". To prove it, i propagates the credential r,1 EXAMPLE. As in the previous example, let n = 1K, or) = 3.7M, K = 1B, and i has copies: (i, 1), , (i, 4). Then, the first 3 copies belong to SW,' automatically. For the 4th one, conceptually, Algorand' independently rolls a biased coin, whose probability of Heads is 0.7.
Copy (i, 4) is selected if and only if the coin toss is Heads.
(Of course, this biased coin flip is implemented by hashing, signing, and comparing as we have done all along in this application so as to enable i to prove his result.) Business as Usual Having explained how verifiers are selected and how their credentials are computed at each step of a round v, the execution of a round is similar to that already explained.
8 Algorand with the Assumption of No Network Partition Essentially, in Algorand, blocks are gnerated in rounds. In a round v, (1) A properly credentialed leader proposes a new block and then (2) Properly credentialed users run, over several steps, a proper Byzantine agreement (BA) protocol on the block proposed.
The preferred BA protocol is BA*. The block proposal step can be considered step 1, so that the steps of BA* are 2, 3,...
Only a proper user i, randomly selected among the users in the system, is entitled to send r,s a message mt in step s of round r. Algorand is very fast and secure because such a user i checks whether he is entitled to speak. If this is the case, user i actually obtains a proof, a credential. If it is his turn to speak in step s of round r, i propagates in the network both his credential, az" and his digitally signed message tri:'s. The credential proves to other users that they should take in consideration the message mtr's.
A necessary condition for user i to be entitled to speak in step s of round r is that he was already in the system a few rounds ago. Specifically, k rounds before round r, where k is a parameter termed the 'look-back' parameter. That is, to be eligible to speak in round r, i must belong to Pick, the set of all public keys/users already in the system at round r ¨ k. (Users can be identified with their public keys.) This condition is easy to verify in the sense that it is derivable from the blockchain.
The other condition is that H(S/Gt(r, s, Qr-1)) < p where p is a given probability that controls the expected number of verifiers in SW'', that is, the set of users entitled to speak in step s of round r. If this condition is satisfied, then i's credential is defined to be az" SIGt(r, s, Qr Of course, only i can figure out whether he belongs to ST7r,s. All other users, who lack knowledge of i's secret signing key, have no idea about it. However, if i E SV", then i can demonstrate that this is the case to anyone by propagating his credential az" given the blockchain so far.
Recall in fact that (1) Qr-1 is easily computable from the previous block, Br-1, although essentially unpredictable sufficiently many blocks before, and (2) anyone can verify i's digital signatures (relative to his long-term key in the system).
Recall that, in the versions of Algorand so far, new blocks are proposed only once in a round r, __ that is, in step 1. The BA protocol has the users reach consensus on one of them (or the empty block), and does not further propose new blocks or re-propose blocks that have already been proposed for round r. When the network is not partitioned and the upper-bounds for the time to propagate messages are met, the users reach consensus efficiently and securely.
9 Algorand Resilient Against Network Partition Let us describe a new embodiment of Algorand, Algorand2, that dispenses the assumption of no network partition. We present the new protocol under the Honest Majority of Users (HMU) assumption. Using the same approach as in Section 7, the HMU assumption can be replaced with the Honest Majority of Money (HMM) assumption.
9.1 Communication Model When the network is not partitioned, small messages take time A to propagate to all honest users, and blocks take time A to propagate to all honest users, as before.
When the network is partitioned into more than one groups of users, the adversary determines whether a message m propagated by a user from one group will be delivered to users in other groups, who in other groups will receive m, and when they will receive m. A network partition may be resolved at an indefinite time in the future and messages propagated during the partition are delivered to all users after the partition is resolved. For simplicity, but without limitation intended, we describe the new embodiment assuming messages propagated during a network partition are delivered to all users immediately after the partition is resolved. For example, if a network partition lasts from time t1 to time t2, and let M be the set of messages propagated during the partition, then all users receive messages in M by time t2. Those skilled in the art will realize that the system described herein can handle other situations where messages in M take certain amount of time to reach all users or are re-propagated by users who have received them.
9.2 Choices of Parameters The expected committee size n and the threshold tH are chosen according to the following conditions. Let PK be the set of users, HPK and MPK respectively the set of honest and malicious users. Let HPKi be an arbitrary subset of HPK with half the size.
When each user i E PK is selected independently and randomly with probability p = *, let HSVi and MSV respectively be the set of selected ones from HPKi and MPK. Then with overwhelming probability, 11 1MSVI<tH.
Moreover, let HSV be the set of selected ones from HPK. Then with overwhelming probability, IHSV1 > tH.
Note that the above two conditions imply IHSV1 > 21MSVI.
For example, when h = 80% and PK is large enough, we may choose n = 3, 500 and tH = 2, 625.
9.3 General Structure Rounds. The protocol generates one block every round. A round consists of periods 1, 2, ...
and a period consists of steps 1, 2, .... At any moment in time, each user i is working on exactly one round-period pair. In particular, we use r.p to refer to period p of round r.
In step 1 of period 1, users propose new blocks. In step 1 of following periods, users propose new blocks or re-propose blocks that have been proposed in earlier periods.
Committees. Each step s of period r.p has a committee chosen by cryptographic self-selection, denoted by SW,P,s. We use the same look-back parameter as in Section 4.1, denoted by k. For example, k = 70. A user i is eligible to be selected in round-r committees if i E PKr-k.
The committee for Step 1 of each period has expected size n1 (e.g., 35) and all other committees have expected size n. Committee members for Step 1 are referred to as potential leaders.
Note that for simplicity, but without limitation intended, we describe the new embodiment herein with the same expected committee size n for all steps other than step 1 of each period.
Those skilled in the art will realize that different committees may have different sizes and can appreciate how to derive all kinds of other implementations as well.
Keys. All credentials for cryptographic self-selection are signed with users' long-term keys for a digital signature scheme with unique signatures, so are the random seeds Qr specified in the blocks. All other messages are signed using ephemeral keys of corresponding steps. In general we will use SIG(m) to denote user i's signature for message m, without specifying the keys.

Note that for simplicity, but without limitation intended, we describe the new embodiment herein with ephemeral keys. Those skilled in the art can appreciate how to derive other implementations with message-credentialed blockchains, including using techniques introduced in existing versions of Algorand.
Definition 9.1. Credential: User i's credential o-ir'P's for a round r, period p and step s is S Gz(Qr , r, p, s).
A committee member for a step always propagates his corresponding credential together with his message for that step, and we will not explicitly mention the propagation of credentials.
Definition 9.2. Leader: The leader fr.p for period r.p is the user arg m-nd Esvr=P=1 H (CT tr 'P'1) =
When a user i identifies his own leader for period r.p, i sets tz,r..p to be the user arg mind E H (CT dr'P'1), where Si is the set of potential leaders from which i has received valid credentials.
Definition 9.3. Valid block: We call a block proposed during round r valid if and only if all its transactions are valid with respect to blocks B , , = = = , Br-1 and the seed Qr specified by it follows the rules of the protocol.
Voting Messages. The committee members generate three types of voting messages.
Definition 9.4. Cert-vote: User i's cert-vote for a value v for period r.p is the signature SIG,(v,"cert",r.p).
We say a user i cert-votes a value v for period r.p when he propagates SIGz(v,"cert",r.p).
Definition 9.5. Soft-vote: User i's soft-vote for a value v for period r.p is the signature SIGz(v,"soft",r.p).
We say a user i soft-votes a value v for period p when he propagates SIGz(v,"so ft" ,r.p).
Definition 9.6. Next-vote: User i's next-vote for a value v for period r.p and step s is the signature SIG,(v,"next",r.p.$).
We say a user i next-votes a value v when he propagates SIGz(v,"next",r.p.$).40 The values v that will be voted upon are either values in the range of the hash function H
or a special symbol I of the same length but outside the range of H.41-Timers. Each user i keeps a timer clock, which he resets to 0 every time he starts a new period. As long as i remains in the same period, clock keeps counting. The users' individual timers do not need to be synchronized or almost synchronized. The only requirement is that they have the same speed.
40In each period, the soft-votes are generated only in one step, and so are the cert-votes. Thus they do not need to specify the corresponding step. The next-votes may be generated in multiple steps and need to specify the step numbers.
41The block structure in this manuscript is the same as in existing versions of Algorand, with the header of a block B, Header(B), containing everything in B except the actual payset. As in the original protocol, we can ask the potential leaders to propagate the headers of their proposed blocks together with the hashes, to allow users to start a round r +1 before seeing the actual block B. In this case, v and I are of the same length as possible hash-header pairs. We will ignore the headers in the protocol description below.

9.4 The Actual Protocol Algorand2 In the protocol below, blocks and potential leaders' credentials from different periods are selectively propagated as in Section 5.
Round r, Period 1 The following are period 1 instructions for a generic user i. If user i is not in the committee of a specific step, he still computes his vote in that step, but does not propagate it.
The moment user i starts his own round-r, he starts Period 1 and resets clock, to 0.
STEP 1: [Block Proposal] User i does the following when clock, = 0.
= If i is a potential leader, then he prepares his proposed block Br", which contains his signature S/G,(Qr-1,r) to define his proposed seed r 1 He propagates H(B,' ), and right after also propagates the block itself.
STEP 2: [The Filtering Step] User i does the following when clock, = 2A.
= He identifies his leader i,r.p and soft-votes his leader's proposed block hash.43 STEP 3: [The Certifying Step] Let T A A. User i does the following when clock, E (2, T).
= If i sees a valid block B, together with tH soft-votes for H(B), then i cert-votes H(B).
STEP 4: [The First Finishing Step] User i does the following when clock, = T.
= If i has cert-voted some value v in Step 3,44 then he next-votes v;
= Else he next-votes I.
STEP 5: [The Second Finishing Step] User i does the following when clock, E
[T, T L), where L equals, say, 50A.45 = If i sees a valid block B, together with tH soft-votes for H(B), then i next-votes H(B).
STEPS s > 6: [Consecutive Finishing Steps]
= If s is even then i follows the same voting instruction as in Step 4 when clock, =
T (s ¨ 4)L/2.
= Else (i.e., s is odd), i follows the same voting instruction as in Step 5 when clock, E
[T (s ¨ 5)L/2, T (s ¨ 3)L/2).
42That is, Q = r)).
43It's not required that i has seen the actual block before soft-voting.
44If i is not in SV''1'3, he may still have cert-voted some v "for himself", without propagating his vote.
45If the Adversary cannot corrupt users dynamically, then it is enough to have Step 5 lasting till time oo and no further steps, as in the permissioned version of our protocol. Future steps are used to ensure that the protocol recovers after a partition is resolved, even if the Adversary has pocketed all Step 4 and Step 5 messages and corrupted all Step 4 and Step 5 committee members so that they cannot resend their messages after the partition.
A smaller L helps speeding up partition resolution in the worst case, while a larger L reduces the amount of messages that need to be re-propagated once the partition is resolved.

Round r, Period p> 2 The following are period-p instructions for a generic user i. Again, if user i is not in the committee of a specific step, he still computes his vote in the step but does not propagate it.
User i starts period p the moment he receives tH next-votes for some value v (which might be equal to I) for the same step s of period p ¨ 1, and only if he has not yet started a period > p. User i sets his starting value for period p, sty, to v. The moment i starts period p, he finishes all previous periods and resets clock i to 0.
STEP 1: [Block Proposal] If user i is a potential leader, then he does the following when clock =
0.
= If i has seen tH next-votes for I for the same step s of period p ¨ 1,46 then i proposes a new block Bir'P, which defines his proposed seed qz: as H(Qr-1, 0.47 He propagates H(B"), and right after also propagates the block itself;
= Else (i.e., i only received tH next-votes for some v I for the same period s of period p ¨ 1, and stli) = v), i proposes stli) by propagating it;48 STEP 2: [The Filtering Step] User i does the following when clock = 2A.
= If i has seen tH next-votes for I for the same step s of period p ¨ 1,49 then i identifies his leader fi,p for period p and soft-votes the value v proposed by tim;
= Else (i.e., i only received tH next-votes for v I for the same step s of period p ¨ 1, and stP = v), i soft-votes stP.
STEP 3: [The Certifying Step] User i does the following when clock i E (2A, T), T = A A.
= If i sees a valid block B, together with tH soft-votes for H(B), then i cert-votes H(B).
STEP 4: [The First Finishing Step] User i does the following when clock i = T.
= If i has cert-voted some value v in Step 3, then he next-votes v;
= Else if i has seen tH next-votes for I for the same steps of period p¨ 1, he next-votes I.
= Else he next-votes stP
i =
STEP 5.1: [The Second Finishing Step] User i does the following when clock i E
[T,T L).
= If i sees a valid block B, together with tH soft-votes for H(B), then i next-votes H(B).
STEP 5.2: [The Second Finishing Step] User i does the following when clock i E
[T,T L).5 = If i has not cert-voted in Step 3 and he sees tH next-votes for I for the same step s of period p ¨ 1, then i next-votes I.
461n extreme cases, a user i may simultaneously see tH next-votes for I and tH
next-votes for some v I for period p ¨1. He is free to set his starting value to either I or v in this case, but the Block Proposal step always gives priority to I.
47That is, a newly proposed block for a period p> 2 is as if there were no leader for it, and the corresponding seed Q is defined in the same way as in the original Algorand protocol when B' is the empty block.
481n this case, enough honest users have seen a valid block B such that v =
H(B), and it is not required that i herself has seen B.
49Note that these votes may reach i during his time (0, 2)], thus i's starting value is not necessarily I.
50Steps 5.1 and 5.2 are made into different steps so that they have different committees.

STEPS s > 6: [Consecutive Finishing Steps]
= If s is even then i follows the same voting instruction as in Step 4 when clock =
T (s ¨ 4)L/2.
= Else (i.e., s is odd), i follows the same voting instructions as in Steps 5.1 and 5.2 in parallel when clock E [T (s ¨ 5)L/2, T (s ¨ 3)L/2).
Changing Rounds Instructions for every user i E PK:
= At any point while i is working on a round r, if i sees a string v I and a period T.' .p with > r such that ¨ i has received at least tH cert-votes for v for period r'.p,51 and ¨ i has received the block header corresponding to v,52 then i cert-votes v for period r'.p,53 sets his own CERTr' to be the set of cert-votes for v and starts round r' 1.
Referring to FIG. 6, a diagram 20' shows a first plurality of computing workstations 22a-22c connected to a first portion 24' of a data network, such as the Internet, and a second plurality of computing workstations 22a'-22c' connected to a second portion 24" of the data network. The workstations 22a-22c communicate with each other via the first portion 24' of the network and the workstations 22a'-22c' communicate with each other via the second portion 24" of the network. However, unlike the diagram 20 of FIG. 1, the workstations 22a-22c do not communicate with the workstations 22a'-22c' because the network has been partitioned into the first part 24' and the second part 24" by a partition mechanism 26. The partition mechanism 26 could be any mechanism that suppresses communication between the parts 24', 24" of the network. Note that the partition mechanism 26 could be purposefully inserted into the network by an adversary or could occur due to unanticipated network interruption caused by natural or man-made phenomena, such as a power outage or a network switching error. In the system described herein, each of the workstations 22a-22c, 22a'-22c' performs the steps set forth above, but the first plurality of workstations 22a-22c does not communicate with the second plurality of workstations 22a'-22c'. That is, each of the entities 22a-22c manages a transaction system locally within the part 24' while each of the entities 22a'-22c' manages a transaction system locally within the part 24". Only one of the parts 24', 24", at most, has enough users (entities) to generate certified blocks. After the partition mechanism 26 is removed and the parts 24', 24"
converge to a single network (like the network 24, discussed above in connection with FIG. 1), all of the workstations 22a-22c, 22a'-22c' communicate with each other and are able to resume operation at the same point (block) of the chain. After the partition mechanism 26 is removed, each of the entities 22a-22c, 22a'-22c' manages a single transaction system in the network where all of the entities 22a-22c, 22a'-22c' communicate with each other.
51A cert-vote from a player j is counted even if player i has also received a message from j cert-voting for a different value for the same period. As shown in the analysis, this is to ensure that all honest users know CERT
within time A from each other.
52The header defines the seed Q'' and, as mentioned earlier, strictly speaking the header is part of v itself.
53For the theoretical analysis it is not necessary for i to propagate his vote, because he ,already helped propagating the tH cert-votes he received. However, having i propagate his vote when i E SV" 'P'3 helps other users see a certificate faster in reality.

Analysis. Even when the network is partitioned, the new embodiment remains secure that is, at most one block is certified for each round r. At a high level, this is because honest users cert-vote at most once in each period of round r, and the choices of the committee size n and the threshold tH guarantee that in each period r.p, at most one hash value H(B) of a valid block B can get a certificate. To see it from a different direction, assume a period r.p has generated tH cert-votes for one valid block B and tH cert-votes for another valid block B' relative to B , ... , Br-1-, then the conditions for how n and tH are chosen imply that at least one honest user has cert-voted for the hash values of both B and B', which contradicts the fact that an honest user cert-votes at most once in period r.p.
If in some period r.p, a valid block B has gotten a certificate _______ that is, at least tH cert-votes for its hash value H(B), then in all future periods p' > p (if they are ever reached), B will be the only block that may get a certificate in period r.p'. Indeed, an honest user does not next-vote for I in period r.p if he has cert-voted for H(B) in r.p. Thus by the same choices of n and tH, in no step s > 4 of period r.p will there be tH next-votes for I
or for any other value v H(B). So an honest user moves to period r.(p 1) only if he has received at least tH
next-votes for H(B). Accordingly, H(B) will be the only value that is (re-)proposed in step 1 of period r.(p 1), the only value that honest users will soft-vote for in step 2 of period r.(p 1), and consequently the only value they will cert-vote in step 3 and next-vote in step s > 4 of period r.(p 1). By an inductive argument, the same is true for all consecutive periods.
Notice that, when the network is partitioned, B having gotten a certificate in a period r.p does not mean that the honest users will receive the certificate. Indeed, during a network partition an adversary controls how messages are delivered in the system. He may, for example, allow all messages to be delivered properly except the cert-votes, where he does not allow cert-votes from one group of users to be delivered to other groups. Nevertheless, B
having gotten a certificate implies that enough honest users have cert-voted for it and will not next-vote for anything else, which prevents any other block from being certified in period r.p and any future period.
The efficiency of the new embodiment comes from two parts. First, when the network is not partitioned, consensus about the rth block is reached quickly. Indeed, if the leader of step 1 of period r.1 is honest, then all honest users immediately cert-vote for his proposed block B, B
has gotten a certificate after step 3 of period r.p, and all honest users finish round r afterward.
Similarly, if round r has reached a period p> 2 and the leader f for period p is honest, then the block newly proposed or re-proposed by f is certified in step 3 and all honest users finish round r afterward. This is so because, if f has seen tH next-votes for I from period p ¨ 1 and proposed a new block, then those next-votes will reach all honest users within time A and they will all soft-vote for L's proposal. Otherwise, f has only seen tH next-votes for the hash H(B) of a valid block B from period p ¨ 1 and has re-proposed H(B). All honest users' starting values of period p are either I or H(B), and no matter which is the case, they soft-vote for H(B) _______________________________________________________________________ because they are following the leader's proposal in the former case, and because they are voting for their own starting values in the latter case.
Moreover, if a certificate for some valid block B is generated in a period r.p, then all honest users finish round r soon after that. This is so because, if at least tH cert-votes for B come from honest users, then all honest users will receive them within time A and will finish round r with B being the rth block. If any set of tH cert-votes for B contains at least one malicious user, the malicious users may choose to not send their cert-votes and the honest users do not receive a certificate for B right away. However, the choices of n and tH ensure that a certificate for B
contain at least one honest user i, and i has received tH soft-votes for B
before he cert-voted it. Since i propagated these soft-votes and the network is not partitioned, all honest users will receive them within time A and will next-vote for H(B) in step 5.1.
Accordingly, all honest users start period r.(p 1) with starting value H(B) and will soft-vote for it in step 2 of period r.(p 1), regardless whether the leader of period r.(p +1) is honest or not.
As a result, honest users will cert-vote for H(B) in step 3, B now has a certificate from honest users, and all honest users will receive them and finish round r within time A.
The choices of the seeds Qr and the cryptographic sortition used in the new embodiment ensure that each period p of round r has an honest leader with high probability, as in the original Algorand protocol, and the missing detailed analysis about the new embodiment's efficiency when there is no network partition follow from there.
Second, after the network partition is resolved, the protocol will recover and reach consensus quickly. Indeed, if some honest users have received a certificate for a block B in round r during the partition and moved to round r 1, then once the partition is resolved all honest users will receive such a certificate for B and move to round r 1. Moreover, let p be the furthest period in round r 1 where an honest user i has reached during the partition. Then all the next-votes that allowed i to move to period (r 1).p will reach other honest users after the partition is resolved, and they will also move to period (r +1).p. The protocol will then continue from there as usual, following the same analysis as when there is not network partition.
If no honest user moved from round r to round r 1 during the partition, then all honest users are in the same round but perhaps different periods. In this case, let p be the furthest period in round r where an honest user i has reached during the partition. Similarly, after the partition is resolved, all the next-votes that allowed i to move to period r.p will reach other honest users and they will also move to period r.p. Again the protocol will continue from there as usual.
To summarize, the new embodiment is secure and does not soft-fork even when the network is partitioned. It generates block efficiently when the network is not partitioned, and recovers quickly after a network partition is resolved.
10 Scope Note that the mechanism described herein is applicable to other blockchain systems where it is desirable to prevent more than one blocks are certified during a network partition and to restore liveness quickly after a partition is resolved. Thus, the system described herein may be adapted to other blockchain schemes, even schemes that do not relate directly to currency.
The system described herein may be adapted to be applied to and combined with mechanisms set forth in any or all of PCT/US2017/031037, filed on May 4, 2017, 15/551,678 filed August 17, 2017, 16/096,107 filed on October 24, 2018, PCT/U52018/053360 filed on September 28, 2018, PCT/U52018/054311 filed on October 4, 2018, 62/632,944 filed on February 20, 2018, 62/643,331 filed on March 15, 2018, 62/777,410 filed on December 10, 2018, and 62/778,482 filed on December 12, 2018, all of which are incorporated by reference herein.
Software implementations of the system described herein may include executable code that is stored in a computer readable medium and executed by one or more processors. The computer readable medium may be non-transitory and include a computer hard drive, ROM, RAM, flash memory, portable computer storage media such as a CD-ROM, a DVD-ROM, a flash drive, an SD card and/or other drive with, for example, a universal serial bus (USB) interface, and/or any other appropriate tangible or non-transitory computer readable medium or computer memory on which executable code may be stored and executed by a processor. The system described herein may be used in connection with any appropriate operating system.
Other embodiments of the invention will be apparent to those skilled in the art from a consideration of the specification or practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only, with the true scope and spirit of the invention being indicated by the following claims.

Claims (20)

What is claimed is:

1. A method for an entity to manage a transaction system in which transactions are organized in a sequence of blocks that are certified by digital signatures of a sufficient number of verifiers, the method comprising:
the entity proposing a hash of a block B' that includes new valid transactions relative to a sequence of certified blocks B , ... , Br-1- if no rth block Br has been certified; and the entity proposing a hash of the block Br if the rth block Br has been verified by a sufficient number of other entities.

2. A method, according to claim 1, wherein a block is certified by the entity only in response to confirming transactions for the block and confirming that the block was constructed and propagated by an entity entitled to construct and propagate the block.

3. A method, according to claim 1, wherein the entity proposes a hash value by digitally signing the hash value to provide a digitally-signed version of the hash value and wherein the entity propagates the digitally-signed version of the hash value to a network that includes other entities.

4. A method, according to claim 3, wherein if no rth block Br has been certified, the entity also digitally signs and propagates the block B'.

5. A method, according to claim 1, wherein the entity determines a quantity Q
from the prior blocks and uses a secret key in order to compute a string S uniquely associated with Q
and computes from S a quantity T that is at least one of: S itself, a function of S, and hash value of S and wherein the entity determines whether to propose a hash value by determining whether T possesses a given property.

6. A method, according to claim 5, wherein S is a signature of Q under a secret key of the entity, T is a hash of S and T possesses the given property if T is less than a given threshold.

7. A method, according to claim 1, wherein the entity is part of a network of entities and wherein a particular one of the entities constructs and propagates the block B.
.

8. A method, according to claim 7, wherein the rth block Br is determined to be certified by the entity if the entity receives an indication that at least a predetermined number of the entities individually certify a hash value corresponding to the rth block B.
.

9. A method, according to claim 8, wherein, in response to the entity receiving the indication that a predetermined number of the entities individually certified the rth block Br , the entity increments r to begin adding additional blocks to the sequence of blocks.

10. A method, according to claim 7, wherein the particular one of the entities is individually chosen by a predetermined number of the entities to be a leader.

11. A method, according to claim 10, wherein the rth block Br is determined to be certifiable by the entity if the entity receives an indication that at least a predetermined number of the entities individually verify receiving an indication that the particular one of the entities has provided a hash value corresponding to the rth block Br to each of the predetermined number of the entities.

12. A method for an entity to manage a transaction system in which transactions are organized in a sequence of certified blocks, the method comprising:
the entity receiving a hash value of a block Br from an other entity that generated the block based on new valid transactions relative to a sequence of certified blocks B , ... , Br-1-;
the entity certifying the block Br in response to a sufficient number of other entities having indicated receipt of the hash value of the block Br from the other entity and the hash value being valid for the block Br ;
the entity generating a new block B' based on new valid transactions relative to a sequence of certified blocks B , ... , Br-1- in response to an insufficient number of the other entities indicating receipt of the hash value of the block Br from the other entity, wherein B' is different from Br ;
and the entity incrementing r to begin adding additional blocks to the sequence of blocks in response to the entity receiving the indication that a predetermined number of the entities individually certified the rth block Br or a predetermined number of the entities individually certified the new block B'.

13. A method, according to claim 12, wherein the blocks are certified by digital signatures.

14. A method, according to claim 12, wherein new blocks are proposed by different ones of the entities until receiving the indication that a predetermined number of the entities individually certified a previously proposed block.

15. A method, according to claim 12, wherein the entity provides an indication that a new block should be generated in response to the hash value not being valid for the block B.
.

16. A method, according to claim 15, wherein the entity generates a new block B' based on new valid transactions relative to a sequence of certified blocks B , . ..
, Br-1- in response to a sufficient number of the other entities providing an indication that a new block should be generated.

17. A method, according to claim 12, wherein the entity provides an indication that the hash value of the block Br should be propagated in response to a sufficient number of the other entities having indicated receipt of the hash value of the block Br from the other entity and the hash value being valid for the block B.
.

18. A method for an entity to verify a proposed hash value of a new block Br of transactions relative to a given a sequence of blocks, B , . .. , Br-1-, without access to the new block Br in a transaction system in which transactions are organized in blocks and blocks are certified by a set of digital signatures, the method comprising:
having the entity determine a quantity Q from the prior blocks;
having the entity compute a digital signature S of Q;
having the entity compute from S a quantity T that is at least one of: S
itself, a function of S, and hash value of S;
having the entity determine whether T possesses a given property; and if T possesses the given property, having the entity verify the proposed hash value of the new block Br independent of confirming whether the proposed hash value corresponds to the new block B.
.

19. A method, according to claim 18, wherein the entity propagates the proposed hash value of the new block Br prior to receiving the new block B.
.

20. Computer software, provided in a non-transitory computer-readable medium, comprising:
executable code that implements the method of one of the preceding claims 1-19.