WO2013152725A1 - New cryptographic systems using pairing with errors - Google Patents

Abstract

Using the same mathematical principle of paring with errors, which can be viewed as an extension of the idea of the LWE problem, this invention gives constructions of a new key exchanges system, a new key distribution system and a new identity-based encryption system. These new systems are efficient and have very strong security property including provable security and resistance to quantum computer attacks.

Description

NEW CRYPTOGRAPHIC SYSTEMS USING PAIRING WITH ERRORS

Background

[1] The present disclosure claims priority to the U.S. provisional patent application with Ser. No. 61623272, entitled " New methods for secure communications and secure information systems" , filed April 12, 2012, which is incorporated herein by reference in its entirety and for all purposes.

[2] This invention is related to the construction of cryptographic systems, in particular, key exchange (KE) systems, key distribution (KD) systems and identity-based-encryption (IBE) systems, which are based on essentially the same mathematical principle, pairing with errors.

[3] In our modern communication systems like Internet, cell phone, and etc, to protect the secrecy of the information concerned, we need to encrypt the message. There are two different ways to do this. In the first case, we use symmetric cryptosystems to perform this task, where the sender uses the same key to encrypt the message as the key that the receiver uses to decrypt the message. Symmetric systems demand that the sender and the receiver have a way to exchange such a shared key securely. In an open communication channel without any central authority, like wireless communication, this demands a way to perform such a key exchange (KE) in the open between two parties. In a system with a central server, like a cell phone system within one cell company, this demands an efficient and scalable key distribution (KD) system such that any two users can derive a shared key via the key distribution (KD) system established by the central server. Therefore it is important and desirable that we have secure and efficient KE systems and KD systems. The first KE system was proposed by Dime and Hellman [DiHe] , whose security is based on the hardness of discrete logarithm problems. This system can be broken by future quantum computers as showed in the work of Shor [SHO] . There are many key-distribution systems including the system using pairing over quadratic forms [BSHKVY] , and the one based on bilinear paring over elliptic curves by Boneh and Boyen (in USA Patent 7,590,236) . But the existing systems have either the problem of computation efficiency or scalability. For instance, the bilinear paring over elliptic curves is very computationally intensive.

[4] In the second case, we use asymmetric systems, namely public key cryptographic systems, for encryption, where the receiver has a set of a public key and a private key, and the sender has only the public key. The sender uses the public key to encrypt messages, the receiver uses the private key to decrypt the messages and only the entity who has the private key can decrypt the messages. In an usual public key system, we need to make sure the authenticity of the public keys and therefore each public key needs to have a certificate, which is a digital signature provided by a trusted central authority. The certificate is used to verify that the public key belongs to the legitimate user, the receiver of a message. To make public key encryption system fully work, we need to use such a system, which is called a public key infrastructure (PKI) system.

[5] In 1984, Shamir proposed another kind of public key encryption system [SHA] . In this new system, a person or an entity's public key is generated with a public algorithm from the information that can identify the person or the entity uniquely. For example, in the case of a person, the information may include the person's name, residential address, birthday, finger print information, e-mail address, social security number and etc. Since the public key is determined by the public information that can identify the person, this type of public key cryptosystem is called an identity-based encryption (IBE) system.

[6] There are a few Identity-based-encryption (IBE) public key cryptosystems, and currently, the (best) one being practically used is the IBE system based on bilinear paring over elliptic curves invented by Boneh and Franklin ( in USA Patent: 7,113,594). In IBE systems, a sender encrypts a message for a given receiver using the receiver's public key based on the identity of the receiver. The receiver decrypts the message using the the receiver's private key. The receiver obtains the private key from a central server, which has a system to generate and distribute the IBE private key for the legitimate user securely. An IBE system does not demand the sender to search for the receiver's public key, but rather, a sender in an IBE system derives any receiver's corresponding public key using an algorithm on the information that identifies the receiver, for example, an email address, an ID number or other information. Current IBE systems are very complicated and not efficient in terms of computations, since the bilinear paring over elliptic curves is very computationally intensive. These systems based on pairing over elliptic curves can also be broken efficiently if we have a quantum computer as showed in the work of Shor [SHO] . There are also constructions based on lattices, but those are also rather complicated systems for applications [ABB] [ABVVW] [BKPW] . Therefore it is important and desirable that we have secure and efficient IBE systems.

[7] Clearly, there are still needs for more efficient and secure KE, KD and IBE systems for practical applications.

BRIEF SUMMARY OF THE INVENTION

[8] This invention first contains a novel method for two parties A and B to perform an secure KE over an open communication channel. This method is based on the computation of pairing of the same bilinear form in two different ways but each with different small errors. In the KE process, each users will choose a private matrix SA, SB respectively with small entries following certain error distributions secretly and a public matrix M randomly. Then each user will compute the multiplication of the user's secret matrix with the publicly chosen matrix but with small errors, exchange the new matrices, and then perform the computation of pairing of SA and ¾ over the same bilinear form based on M in two different ways but each with different small errors. This kind of mathematical computation is called pairing with errors. The shared key is derived from the pairings with a rounding technique. This method can be viewed as an extension of the idea of the learning with errors (LWE) problem discovered by Regev in 2005 [Reg]. The security of this system depends the hardness of certain lattice problem, which can be mathematically proven hard [DiLi]. This system involves only matrix multiplication and therefore is very efficient. Such a system can also resist the future quantum computer attacks.

[9] This invention second contains a novel method to build a KD system with a central server or authority. In this system, the central server or authority assigns each user i a public ID as a matrix with small entries or establish the ID of each user as a matrix A^ with small entries following certain error distributions with the information that can identify the user uniquely, and, in a secure way, gives each user a private key based on certain multiplication of this ID matrix with the central server or authority's secret master key M, another matrix, but with small errors. Then any two users in the system will compute the pairing of the two ID matrices of the users with the same bilinear form based on the master key matrix M in two different ways but each with different small errors to derive a shared key between these two users with certain rounding technique. This method can be viewed as an extension of the idea of the learning with error problem discovered by Regev in 2005 [Reg]. The security of this system depends on the hardness of the problem related to pairing with errors. This system involves only matrix multiplication and therefore is very efficient.

[10] This invention third contains a novel method to build a IBE system with a central server or authority. In this system, the central server or authority assigns each user i a public ID Ai as a matrix with small entries following certain certain error distributions or establish the ID of each user as a matrix with small entries following certain certain error distributions with the information that can identify the user uniquely. Each user is given by the central server or authority a private key ¾ based on certain multiplication of this ID matrix with the central server or authority's master private key S, another matrix, but with errors related to one part of the master public key M, another matrix. The central server or authority will establish another half of the mater key as the multiplication of M and S with small errors, which we call M_\. Then any user who wishes to send the user i a message in the system will compute public key of i which consists of M and a paring of M and Ai of the bilinear form based on the master secret key matrix S, then encrypt the message using the encryption system based on the MLWE problem, and the user i will use the secret key ¾ to decrypt the message. This method can be viewed as an extension of the idea of the learning with error problem discovered by REGEV in 2005. The security of this system depends the harness of certain lattice problem, which can be mathematically proven hard. This system involves only matrix multiplication and therefore is very efficient.

[11] In our constructions, we can replace matrices by elements in ideal lattice, and we can also use other type of rounding techniques. We can also build the system in a distributed way where several servers can work together to build KD and IBE systems.

[12] In short, we use the same mathematical principle of paring with errors, which can be viewed as an extension of the idea of the LWE problem, to build secure and more efficient KE, KD and IBE systems.

[13] Though this invention has been described with specific embodiments thereof, it is clear that many variations, alternatives, modifications will become apparent to those who are skilled in the art of cryptography. Therefore, the preferred embodiments of the invention as set forth herein, are intended to be illustrative, not limiting. Various changes may be made without departing from the scope and spirit of the invention as set forth herein and defined in the claims. The claims in this invention are based on the U.S. provisional patent application with Ser. No. 61623272, entitled "New methods for secure communications and secure information systems" , filed April 12, 2012, only more technical details are added.

DETAILED DESCRIPTION OF THE INVENTION 1.1 The basic idea of pairing with errors [14] The learning with errors (LWE) problem, introduced by Regev in 2005 [Reg] , and its extension, the ring learning with errors (RLWE) problem [LPR] have broad application in cryptographic constructions with some good provable secure properties. The main claim is that they are as hard as certain worst-case lattice problems and hence the related cryptographic constructions.

[15] A LWE problem can be described as follows. First, we have a parameter n, a (prime) modulus q , and an error probability distribution κ on the finite ring (field) F_q with q elements. To simplify the exposition, we will take q to be a odd prime and but we can also work on any whole number except that we may need to make slight modifications.

[16] In F_q, each element is represented by the set {— (q— l)/2, .., 0, (q— l)/2}. In this exposition, by " an error" distribution, we mean a distribution we mean a distribution such that there is a high probability we will select an element, which is small. There are many such selections and the selection directly affect the security of the system. One should select good error distribution to make sure the system works well and securely.

[17] Let ΤΙ ^on F_q be the probability distribution obtained by selecting an element A in F™ randomly and uniformly, choosing e G F_q according to κ, and outputting (A, < A, S > +e), where + is the addition that is performed in F_q. An algorithm that solves the LWE problem with modulus q and error distribution κ, if, for any S in F_q , with an arbitrary number of independent samples from Tls,_K, it outputs S (with high probability).

[18] To achieve the provable security of the related cryptographic constructions based on the LWE problem, one chooses q to be specific polynomial functions of n, that is q is replaced by a polynomial functions of n, which we will denote as q(n), κ to be certain discrete version of normal distribution centered around 0 with the standard deviation σ = aq >

and elements of F_q are represented by integers in the range [—(q— l)/2, (q— l)/2)], which we denote clS

[19] In the original encryption system based on the LWE problem, one can only encrypt one bit a time, therefore the system is rather inefficient and it has a large key size. To further improve the efficiency of the cryptosystems based on the LWE problem, a new problem, which is a LWE problem based on a quotient ring of the polynomial ring F_q[x] [LPR], was proposed. This is called the ring LWE (RLWE) problem. In the cryptosystems based on the RLWE problem, their security is reduced to hard problems on a subclass of lattices, the class of ideal lattices, instead of general lattices.

[20] Later, a new variant of LWE was proposed in [ACPS] . This variant of the LWE problem is based on the LWE problem. We will replace a vector A with a matrix A of size m x n, and S also with a matrix of size n x 1, such that they are compatible to perform matrix multiplication A x S. We also replace e with a compatible matrix of size m x 1. We will work on the same finite field with q elements.

[21] To simplify the exposition, we will only present, in detail, for the case where A is a square matrices of the size n x n and, S and e of the size n x 1.

[22] Let Tls ^over F_q be the probability distribution obtained by selecting an n x n matrix A, whose each entry are chosen in F_q uniformly and independently, choosing e as a n x l vector over F_q with entries chosen according to certain error distribution κ_η, for example, each entries follows an error distribution κ independently, and outputting (A, Ax S+e), where + is the addition that is performed in F_q . An algorithm that solves a LWE with modulus q and error distribution κ_η, if, for any vector S in F_q , with any number of independent sample (s) from ¾_)(ίη , it outputs S (with high probability). [23] For the case that we choose a small S, namely entries of S are chosen independently according to also the error distribution κ_η, we call this problem a small LWE problem (SLWE) . If we further impose the condition A to be symmetric, we call it a small symmetric LWE problem (SSLWE) . If we choose the secret S randomly and independently from the set —ζ, , . , 0, l.., z with z a fixed small positive integer, we call such a problem uniformly small LWE problem (USLWE) .

[24] For practical applications, we can choose S and e with different kind of error distributions.

[25] Due to the results in [ACPS] , we know If the secret 5"s coordinates and the error e's entries are sampled independently from the LWE error distribution κ_σ, the corresponding LWE problem is as hard as LWE with a uniformly random secret S. This shows that the SLWE problem is as hard as the corresponding LWE problem. The same is true for the case of the RLWE problem that if one can solve the Ring LWE problem with a small secret namely the element S being small, then one can solve it with an uniform secret.

[26] We further extend the problem to a full matrix form.

[27] Let ΤΙ ,κ ₂ ^over F_q be the probability distribution obtained by selecting an n x n matrix A, whose each entry are chosen in F_q uniformly and independently, choosing e as a n x n matrix over F_q with entries following certain error distribution κ_η2 , for exmaple, an distribution chosen according to the error distribution κ independently, and outputting (A, A x S + e) , where + is the addition that is performed in F™ . An algorithm that solves a LWE with modulus q and error distribution κ_η2 , if, for any n x n matrix S in F™, with any number of independent sample(s) from Tls,_{K 2 >} it outputs S (with a high probability) .

[28] We call this problem matrix LWE problem(MLWE) . For the case where we choose a small S, namely entries of S also follows the error distribution κ_η2 , we call this problem a small MLWE problem (SMLWE) . If we further impose the condition A to be symmetric, we call it a small symmetric MLWE problem (SSMLWE) . If we choose the secret S randomly and independently from the set —z, .., 0, l .. , z with z a fixed small positive integer, we call such a problem uniformly small MLWE problem (USMLWE) . It is clear the MLWE problem is nothing but put n LWE problem together and sharing the same matrices. Therefore it is as hard as the corresponding LWE problem.

[29] We can use different error distributions for S and e.

[30] The mathematical principle behind our construction comes from the fact of associativity of matrices multiplications of three matrices A, B and C:

A x B x C = (A x B) x C = A x (B x C) .

Such a product can be mathematically viewed as computing the bilinear paring of the row vectors of A with column vectors of C.

[31] For two matrices A and B with small entries following certain error distributions, for example, with entries following some error distributions, instead of computing this product directly, we can first compute

AB + E_a,

then compute

(AB + E_A)C or (AB + E_A)C + E_AC,

or we will compute

BC + E_C, then compute

A(BC + E_C) or (AB + E_A) C + E_BC,

where EA , EB , EA_C, EB_C ^are matrices with small entries following the same ( or different) error distributions. Then we have two way to compute the product ABC with small errors or differences between these two matrices. We call such a computation pairing with errors. All our constructions depends on such a paring with errors and on the fact that the two different paring are close to each other if A and C are also small.

[32] We can mathematically prove the theorem that an MLWE problem is as hard as the corresponding LWE problem with the same parameters. This provides the foundation of the provable security of our constructions

1.2 The construction of the new KE systems based on paring with errors

[33] Two parties Alice and Bob decide to do a key exchange (KE) over an open channel. This means that the communication of Alice and Bob are open to anyone including malicious attackers. To simplify the exposition, we will assume in this part all matrices involves are n x n matrices. But they do not have to be like this, and they can be matrices of any sizes except that we need to choose the compatible sizes such that the matrix multiplications performed are well defined.

[34] Their key change protocol will go step by step as follows.

(1) Alice and Bob will first publicly select F_q, n and a n x n matrix M over F_q uniformly and randomly, where q is of size of a polynomial of n, for example q i¾ n³, and an error distribution κ_ηι to be a distribution over n x n matrices over F_q, for example, a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_σ as in the case of LWE, namely a discrete normal distribution over F_q center around 0 with standard deviation approximately

All the information above is public. They jointly and publicly choose a small (prime) integer t (t « n) .

(2) Then each party chooses its own secret Si ( i = A, B) as a n x n matrix chosen according to the error distribution κ_ηι , also as a n x n matrix following the error distribution. For Alice, she computes

M_A = MS_A + te_A,

where t is a small integer (t « n).

For Bob, he computes

M_B = *¾ + ie_B.

(3) Both parties exchange Mi in the open communication channel. This means both Mi ( i = A, B) are public, but keep Si and ej (i = A, B ), secret.

(4) Alice computes:

K_A = S_A x M_B = S^M'SB + tS_Ae_B.

Bob computes:

K_B =

x S_B = SAM'SB + te_AS_B. (5) Both of them will perform a rounding technique to derive the shared key as follows:

(a) Bob will make a Ust Ti of all positions of the entries of KB such that these entries are in the range of [— q— l)/4, q— l)/4] and a list T₂ of all positions which are not in the range of [—(q— l)/4, (q— l)/4] . Then Bob will send to Alice the list

Ά.

(b) Then each party will compute the residues of these entries modular t in T_\, and for the entries not in T_{1 ;} which is in T₂, they will add q— l)/2 to each entry and compute the residue modular q first (into the range of [— q— l)/4, q— 1)/4]^⁾ then the residue modular t. That gives a shared key between these two users.

[35] The reason that Alice and Bob can derive from K_A and KB a shared secret to be the exchanged key via certain rounding techniques as in the case above is exactly that and Si are small, therefore KA and KB are close. We call this system a SMLWE key exchange protocol. We can derive the provable security of this more efficient system [Dili] .

[36] In term of both communication and computation efficiency, the new system is very good. The two parties need to exchange n² entries in F_q, and each perform 2η²·⁸ computations (with Strassen fast matrix multiplication [STR]) to derive n² bits if t = 2.

[37] Si and ¾ can follow different kind of error distributions.

[38] We can prove the theorem that if we choose the same system parameters, namely n and q, the matrix SLWE key exchange protocol is provably secure if the error distribution is properly chosen [DiLi] . The proof relies on the the mathematical hardness of the following pairing with error problem.

[39] Assume that we are given

(1) an n x n matrix M, a prime integer q, a small positive integer t, and an error distribution κ_η and ;

(2)

M_A' = MS_A' + te_A

and

MB' = *¾ + ie_B,

where 1 vector follows the error distribution κ_η and the entries of n x 1 vectors S also follows the same error distribution;

(3) and the fact that

K_B ^L = M_A ^t x SB^L = (S^' M'SB' + t e_A , S_B' >

is in the range of [—(q— l)/4, (q— l)/4] or not;

the problem is to find an algorithm to derive

KA' = (S_A' y x M_B = (S' M'SB' + t < S_A' , e_B >

modular t if K_B' is in the range of [—(q— l)/4, (q— l)/4], otherwise K_A' + q— l)/2 first modular q then modular t, with a high probability. We call such a problem a pairing with error problem (PEP). [40] The proof follows from the fact that the SMLWE problem is as hard as the SLWE problem, since the matrix version can be viewed as just assembling multiple SLWE samples into one matrix SLWE sample.

[41] We note here that we can choose also rectangular matrix for the construction as long as we make sure the sizes are matching in terms of matrix multiplications, but parameters need to be chosen properly to ensure the security.

[42] Similarly we can build a key exchange system based on the ring learning with errors problem (RLWE) [LPR] , we will a variant of the RLWE problem described in [LNV] .

[43] For the RLWE problem, we consider the rings 1Z = Z[x] / f (x) , and lZ_q = TZ/qTZ, where f x) is a degree n polynomial in Z[x] , Z is the ring of integers, and q is a prime integer. Here q is an odd (prime) and elements in Z_q = F_q = Z/q are represented by elements: — q— l)/2, —1 , 0, 1 , .., (g— l)/2, which can be viewed as elements in Z when we talk about norm of an element. Any element in lZ_q is represented by a degree n— 1 polynomial, which can also be viewed as a vector with its corresponding coefficients as its entries. For an element

a{x) = io + a_\x + ... + α_η-_\χ^{η~ λ} ,

we define

|| || = max|<¾ [ ,

the l_∞ norm of the vector (ao, <¾ , a_n-i) and we treat this vector as an element in Zⁿ and <2j an element in Z. We can also choose q to be even positive number and things need slight modification.

[44] The RLWE/_j9jX problem is parameterized by an polynomial f x) of degree n, a prime number q and an error distribution χ over lZ_q. It is defined as follows.

[45] Let the secret s be an element in TZ_q, a uniformly chosen random ring element. The problem is to find s, given any polynomial number of samples of the pair

where <¾ is uniformly random in lZ_q and ¾ is selected following certain error distribution χ.

[46] The hardness of such a problem is based on the fact that the ¾ are computationally indistinguishable from uniform in lZ_q. One can show [LPR] that solving the RLWE/_j9jX problem above is known to give us a quantum algorithm that solves short vector problems on ideal lattices with related parameters. We believe that the latter problem is exponentially hard.

[47] We will here again use the facts in [ACPS] , [LPR] that the RLWE/_j9jX problem is equivalent to a variant where the secret s is sampled from the error distribution χ rather than being uniform in lZ_q and the error element ¾ are multiples of some small integer t.

[48] To derive the provable security, we need consider the RLWE problem with specific choices of the parameters.

• We choose f x) to be the cyclotomic polynomial xⁿ + 1 for n = 2^U, a power of two;

• The error distribution χ is the discrete Gaussian distribution Z½_™j(T for some n » σ > w( ^og ) > 1 ;

• q = 1 (mod 2n) and q a polynomial of n and q i¾ n³;

• t a small prime and t « n « q.

We can also use other parameters for practical applications.

[49] There are two key facts in the RLWE/_j9jX setting defined above, which are needed for our key exchange system. (1) The length of a vector drawn from a discrete Gaussian of with standard deviation σ is bounded by an, namely,

for X chosen according to χ.

(2) The multiplication in the ring lZ_q increases from the norms of the constituent elements in a reasonable scale, that is,

l| x y(mod f(x)) \\ < n|| ||||y||,

for 1, 7 G lZ_q and the norm is the l_∞ norm defined above.

[50] With the RLWEJ_j9jX setting above, we are now ready to have two parties Alice and Bob to do a key exchange over an open channel. It goes step by step as follows.

(1) Alice and Bob will first publicly select all the parameters for the RLWEf_tqtX including n³ or similar polynomial functions of n), n, f (x) and χ. In addition, they will select a random element M over lZ_q uniformly. All the information above is public.

(2) Then each party chooses its own secret Sj as an element in lZ_q according to the error distribution χ , and independently also as an element following the error distribution χ, but jointly choose a small prime integer t (t « n)

For Alice, she computes

M_A = Ms_A + te_A,

where t is a small integer (t « n).

For Bob, he computes

M_B = Ms_B + te_B.

(3) Both parties exchange Mi. This means both Mi are public, but certainly keep Sj and Ci secret.

(4) Alice computes:

KA = SA X M_b = s_AMs_B + te_Bs_A.

Bob computes:

K_B = M_A x s_B = s_AMs_B + te_As_B.

(5) Both of them will perform a rounding technique to derive the shared key as follows:

(a) Bob will then make a list of size n, and this list consists of pairs in the form of where i = 0, n— 1, and j = 1 if the x^% coefficient of K_B is in the range of [—(q— l)/4, (q— l)/4] otherwise j = 0.

(b) Then Bob will send this list to Alice. Then each will compute the residue of the corresponding entries modular t in the following way:

for an element of the list

1) if j = 1, each will compute the i-th entry of K_A and K_B modular t respectively;

= 0_; each will add q— l)/2 to the i-th entry of K_A and K_B modular q back to range of [—(q— l)/4, (q— l)/4] then compute the residues modular t.

[51] We can use different distributions for Sj and

[52] That will give a shared key between these two users. We call this system a RLWE key exchange system. We can deduce that there is a very low probability of failure of this key exchange system. We note here that the commutativity and the associativity of the ring lZ_q play a key role in this construction.

[53] In terms of security analysis, we can show the provable security of the system following the hardness of the RLWE/_j9jX problem by using a similar PEP over the ring R_q[OiU].

[54] Assume that we are given

• a random element M in R_q, prime integers t, q and the error distribution χ with parameters selected as in the RLWEJ_j9jX above;

• MA = MSA + t&A and MB = MSB + te e , where follows the error distribution χ and Si also follows the error distribution χ;

• and the fact that the coefficients x^% of KB = MA x SB = SAMSB + t&A^B is in the range of [—(q— l)/4, (q— l)/4] or not;

the problem is to find an algorithm to derive KB (or KA) modular t or KB + (q— l) /2 (or KA + (q— l) /2) modular q (into the range of [-(q-l)/4, (q-l)/4]) and then modular t with a high probability. We call such a problem a pairing with error problem over a ring(RPE).

[55] It is nearly a parallel extension of the proof of the provable security of the case of SLWE key exchange system to the RLWE key exchange system. We conclude that the RLWE key exchange system is provable secure based on the hardness of the RLWE/_j9jX problem.

[56] With the same parameters q and n, this system can be very efficient due to the possibility doing fast multiplication over the ring lZ_q using FFT type of algorithms.

1.3 The construction of the new KD systems based on paring with errors

[57] Over a large network, key distribution among the legitimate users is a critical problem. Often, in the key distribution systems, a difficult problem is how to construct a system, which is truly efficient and scalable. For example, in the case of the constructions of [BSHKVY] , the system can be essentially understood as that the master key of a central server is a symmetric matrix M of size n x n and each user's identity can be seen as a row vector Hi of size n. The central server gives each user the secret Hi x M. Then two users can derive the shared key as Hi x M x H¹-. The symmetric property of M ensures that

However, large number of users can collaborate to derive the master key. If one can collect enough (essentially n) Hi x M, which then can be used to find the master key M and therefore break the system.

[58] We will build a truly scalable key distribution system using the pairing with error with a trusted central server, which can be viwed as a combination of the idea above and the idea of the LWE.

[59] We work again over the finite field F_q, whose elements are represented by— q— l) /2, ..., 0, ..., (g - l) /2. We cho OSe Q ~ n³ or other similar polynomial function of n, we choose again κ_ηι to be an error distribution over the space of n x n matrices, for example, an distribution each component are independent, and each component follows error distribution κ_σ, the discrete distribution as in the case of LWE, namely a discrete normal distribution over F_q centered around 0 with standard deviation approximately

The choice of these parameters can be modified. [60] The key distribution system is set up step by step as follows.

(1) We have a central server, which will select a symmetric randomly chosen n x n matrix S, as a master key, whose entries are in F_q:

S = S*.

(2) For each user index as i, the central server gives it a ( in general not symmetric) matrix Ai ( as an ID) with small entries following error distribution κ_ηι . The ID matrix of each user is public and it can also be generated with information that can identify the user like email address, name and etc.

(3) For each user, the central server distribute securely a secret:

where is a matrix (not symmetric) selected following certain error distribution, such as κ_η2. This is kept private for each user.

[61] To obtain a secret key shared between the user i and the user j , the user i computes

K_i = E_i x A_j ^t = A_iSA_j ^t + te_iA);

and the user j computes

K₃ = Ai x {E_jY = AiS¹ A + tA_te = A_tSA + tA_te .

This is possible because the IDs are public. They then can use the following simple rounding method to derive a shared key between the two users.

• When the user j wants to establish a shared key with the user i, the user j will collect all the entries (including their positions in the matrix) in K_j that are in the range of (—(q— l)/4, (q— l)/4), namely those entries which are closer to 0 than (q— l)/2. Then user j will send to the user i a list of the positions of the entries in the matrix ( only the position not the values of the entries themselves) that are randomly selected from the collection, which is tagged by 0, and a list of entries not in the list tagged by 0. Then the user i will select the same entries in its own matrix Ei x A_j . Now they have a shared list of common entry positions, therefore the corresponding entries of the matrix. Then each user will compute the residue of these entries modular t tagged by 1 and compute the residue of the sum of each of these entries tagged by 0 with (q— l)/2 to build a new identical ordered list of values, which will be their shared secret key.

[62] Because S symmetric, we have that

Λ . a A t — Λ . t A t therefore the user j derives

AiSA_j -\- tAiC_j. The difference between the results computed by the two users is:

= SA + te_tA) - ( SA + tA_te))

= tciA_j—

This difference is small since t is small and &iA- and ^_j are small, which is due to the fact that ej, e_j, Ai and A_j are all small. This allows us to get a common key for i and j by certain rounding techniques and therefore build a key distribution system.

[63] Since the error terms for both matrices, are small, the corresponding selected entries with tag 1 in AiSA_j (without

are essentially within the range of [(— (q— l)/4, (q— l)/4] or very close. Therefore the error terms will not push those selected terms in AiSA_j over either (—(q— l)/2 or (q— l)/2), that is when added the error terms, those selected entries will not need any further modular q operation but just add them as integers, since each element is represented as an integer in the range of [(— (q— l)/2(q— l)/2)]. The same argument goes with entries tagged by 0. These ensures that the process give a shared key between these two users.

[64] From the way matrices Ki, K_j are constructed, we know that each entry of Ki and K_j follows uniform distribution. Therefore we expect that each time the size of the first list selected by the user j from the matrix K_j should be around n². Therefore this system can provide the shared secret with enough bits if we choose proper n.

[65] Also we can build a version of this system with none symmetric matrices, in this case, the central serve needs to compute more matrices like AiS + e and A S + e'. Then it is possible, we can do the same kind of key distribution. This system again is less efficient.

[66] On the other hand, since the RLWE problem can be viewed as a specialized commutative version of matrix-based LWE since an element in the ring can be view as a homomorphism on the ring. We can use the RLWE to build a key distribution in the same way.

[67] Now let us look at why this key distribution is scalable. Clearly each user will have a pair Ai and I¾ = A^S + tei, and many users together can get many pairs, then to find the secret master key S is to solve the corresponding MLWE problem, except that, in this case, we impose the symmetric condition on the secret S. It is not difficult to argue again that this problem is as hard as a LWE problem, since given a LWE problem, we can convert it also into such a MLWE problem with symmetric secret matrix. Therefore, it is easy to see that this system is indeed scalable.

[68] In terms of the provable security of the system, the situation is similar to the work done in the paper [DiLi]. We can give a provable security argument along the same line.

[69] As we said before, since RLWE can be viewed as a special case MLWE, we will use the RLWE to build a very simple key distribution system.

[70] We will choose the ring lZ_q to be F_q[x]/xⁿ + 1. To ensure the provable security, we need to choose parameter properly n, q, properly, for exmaple n = 2^k, q = lmo<i(2n) [LPR]. For provable secure systems, we assume that we will follow the conventional assumptions on these parameters, and the assumption on the error distribution like χ in [LPR].

[71] This construction is essentially based on the systems of above. We assume that we have a ring lZ_q with a properly defined learning with error problem on the ring lZ_q with erro distribution χ. The problem is defined as follows: We are given a pair (A, E), where

E = A x S + te',

A, S where e' are elements in R, t is small integer, e' is an error element following the distribution of χ, S is a fixed element and A is select randomly following uniform distribution, and the problem is to find the secret S.

[72] With a central server, we can build a simple key distribution system as follows.

(1) The central server will also select a random element M in lZ_q following uniform distribution.

(2) For each user, the central server will assign an public ID as Ai, where Ai should be in the form of a chosen small element in TZ_q, namely following an error distribution like χ.

(3) Each member is given a secret key by the central server:

where ¾ follows an error distribution χ.

(4) // two user i and j wants to build a shared key, one user, say i can use the ID matrix of j, namely A_i; the its secret key to build a shared key with j by computing

K_t = A_j x Si = A_jMAi + tA_jei,

and j can use its secret key to build a shared key with i by computing

K_j = Ai x S_j = A_jMAi + tA_te_j,

then derive the shared key with the rounding teachnique as follows:

(a) i will then make a list of size n, and this list consists of pairs in the form of ( , b), where a = 0, n— 1, and b = 1 if the x^a coefficient of Ki is in the range °f [- (<? - !)/⁴, (q - l)/4] otherwise 6 = 0.

(b) i will send this list to j . Then each will compute the residue of the corresponding entries modular t in the following way:

for an element of the list (a, b),

1) if b = 1, each will compute the a-th entry of Ki and K_j modular t respectively;

2) if b = 0, each will add q— l)/2 to the a-th entry of Ki and K_j modular q back to range of [—(q— l)/4, (q— l)/4] then compute the residues modular t.

[73] Since Ai and are small elements in TZ_q, we have Ai x is also small. This ensures that we indeed have a shared secret key. This, therefore, gives an key-distribution system.

[74] Here we use very much the fact that in a RLWE problem that the multiplication is commutative. The key feature of our construction is that it is simple and straight forward. The provable security of the system is also straightforward.

1.4 The construction of the new IBE systems based on paring with errors [75] We will first build a new public key encryption based on MLWE. To build an encryption system, we choose similar parameter g ¾ n³ or n⁴ or similar polynomial functions of n, we choose again κ_ηι to be an error distribution, for example the error distribution with each component are independent, and each component follow the same discrete distribution κ_σ as in the case of LWE, namely a discrete normal distribution over F_q center around 0 with standard deviation approximately

Surely we can also select high dimensional Gaussian distribution, which should be very convenient for the purpose to provable security. We select this simple distribution to simplify the argument concerning the validity of the encryption system. We can surelt choose other parameters.

[76] With such a setting, we can build an encryption system as in the case of the MLWE problem as follows:

(1) We select an n x n matrix S, whose entries are small following an error distribution κ_ηι , for example, each entries independently and randomly follows the distribution

(2) In the setting of the MLWE, we will derive one output pair (A, E), where

E = A x S + e,

or

E = A x S + te,

and t is small, t « n, and they form the public key of our encryption system. Here e follow certain error distributions, for example the distribution we use above.

(3) S is the private key of the cryptosystem.

(4) A message m is represented as n x n matrix with binary entries of 0, 1 or n x n matrix with entries in the range modular t, namely 0, l.., t— 1.

(5) A sender chooses a n x n small matrix B similar to S namely following an error distribution κ_ηι , for example, each entries independently and randomly follows the distribution κ_σ . Then the sender compute the encrypted message as:

(D₁, D₂) = (B x A + e_{l 7} B x E + e₂ + m{q/2)), or

((£>ι, D₂) = (B x A + tei, B x E + te₂ + m,

where e_\ and e₂ are error matrices selected independently following some error distribution like e.

(6) To decrypt, the legitimate, in the first case, computes

D₂ - D₁ x S = (BE + e₂ + m(q/2) - (BA + eJS) = eE + e₂ - _eiS + m{q/2), where everything is done in F_q, and we can check on each entry of the matrix, if it is near 0, we output 0, and if it is near q— l)/2 we output 1, or we divide them by q— l)/2 performed as a real number division and round them to 0 or 1 and the output will be the plaintext m; or in the second case, the legitimate user computes

D₂ - D₁ x S = (BE + te₂ + m - (BA + =, teE + te₂ - te_xS + m, then modular t. This will be the plaintext m. [77] A, B, ei can follow different error distributions.

[78] With large n, the output can give us the right plaintext with as high probability as demanded. The reason we could decrypt with high probability comes from the following.

D₂ - D₁ x S

= BE + e₂ + m(q/2) - (BA + e)S

= B x (A x S + e) + e₂ + m(q/2) - (BA + ¾) x S

B x e + e₂— ei x S can be viewed as a error terms, which is determined by the distribution of the following random variable. With proper choice of parameters, like in the case of KE or KD systems, the decryption process will surely return the right answer when n is large enough. The same argument goes with the second case.

[79] One key point of this new method is that on average, we can do the encryption much faster in terms of per bit speed because we can use fast matrix multiplication [CW] to speed up the computation process.

[80] We note here that since matrix multiplication is not commutative, when we multiply two elements, the order is very important, unlike the case of the RLWE related systems.

[81] We can also use the same idea in the ring LWE (RLWE) [LPR] to do encryption, where all the elements are in the ring TZ_q, and we have

E = A x S + te,

t is small positive integer and the entries of S is also small following error distribution κ_ηι . We encrypt a message as

( , D₂) = {BA + te BE + te₂ + m).

Then we decrypt by computing

(BE + te₂ + m - B(AS + te₁)) (mod t).

This works because

D₂ - D₁ x S

= BE + te₂ + m - (BA + e^S

= B x (A x S + te) + te₂ + m - (BA + te_x) x S

= tB x e + te₂ — tei x S + m

Since the error terms are small, by modular t, we certainly should get back the original plaintext.

[82] For the MLWE problem, we surely need to choose the distribution accordingly when we need to obtain the provable security of the system.

[83] There are several versions of identity-based encryption systems based on lattice related problems including the LWE problem [ABB] ,[ABVVW] ,[BKPW] . But they all look rather complicated. We can use the MLWE to build an identity-based encryption system.

[84] With a central server, we can build a simple identity-based encryption system as follows. (1) The central server will first select a secret n x n matrix S as the secret master key, where S is selected as a small element following certain error distribution κ_ηι like error distributions like in KE and KD sytsems.

(2) The central server will also select a random element M following uniform distribution or similar distribution, but make sure that M has an inverse. If we could not find one first time, we will try again till we find one. We have a high probability of success to find such a M when q is large. Then the central serve will compute

where eis small following certain error distribution κ_ηι .

(3) Then the central server will publicize M and Mi as the master public key.

(4) For each user, the central server will assign an public ID as A_i; where is small following certain error distribution κ_ηι , and it can be generated from information that can identify the user.

(5) Each member is given a secret key:

S, = SA, + tM^~xe_t,

where 's entries are small following the error distribution κ. Surely this is the same as given

MS, = MSA, + te„

since M is public.

(6) Anyone can use the ID, namely Ai, and the master public key to build a new public key for the user with ID A_i; which is given as the pair (Ai, Β , where

A, = M

and

B, = M A, = MSA, + teAi,

and it is used as the public key to encrypt any message use the MLWE encryption system above.

This gives an identity based encryption system.

[85] S, Ai, ej, e can also follow different error distributions.

[86] Since A^ and e are small, we have A^ x e is also small. W also have that

Sinc is also small and tej— is also small. Therefore ¾ is a

h the pair (Ai, Bi) as the problem input. Therefore ¾ is indeed a secret key that could be used for decryption. Therefore the construction works. We need to choose parameters properly to ensure security.

[87] The key feature of our construction is that it is simple and straight forward. The provable security of the system is also straightforward.

[88] we can extend this construction using the RLWE problem. We will choose the ring R to be F_q[x]/xⁿ + 1. To ensure the provable security, we need to choose parameter properly n, q, properly, namely n = 2^k, q = lmo<i(2n) [LPR] . But we can select other parameters for secure applications.

[89] This construction is directly based on the encryption systems of the RLWE[LPR] , namely, we assume that we have a ring R with a properly defined learning with error problem on the ring R. The problem is defined as follows: we are given a pair (A, E), where

E = A x S + te',

A, S where e' are elements in TZ_q, t is small integer, e' is an error element following an error distribution χ, S is a fixed element and A is select randomly following uniform distribution, and the problem is to find the secret S. We also know that one can build a public key encryption systems using the RLWE problem[LPR] , where A, and E serve as the public key, and the secret S, which needs to be small, serves as the private key. We can use the fact that in a ring-LWE problem that the multiplication is commutative.

[90] With a central server, we can build a simple identity-based encryption system as follows.

(1) The central server will first select a secret S in R as the secret master key, where S is a selected small element follow certain error distributions χ.

(2) The central server will also select a random element M in R following uniform distribution and make sure that M has an inverse. If we could not find one first time, we will try again till we find one. We have a high probability of success to find such a M when q is large. Then the central serve will computer

where e is small and follows error distribution χ.

(3) Then the central server will publicize M and Mi as the master public key.

(4) For each user, the central server will assign an public ID as A_i; where Ai is a small element in TZ_q, and it follows error distribution χ.

(5) Each member is given a secret key:

S, = SA, + tM^~xe_t,

where ¾ small element in R, and it follow certain error distribution χ. Surely this is the same as given

MS, = MSA, + te,,

since M is public.

(6) Anyone can use the ID, namely Ai, and the master public key to build a new public key for the user with ID Ai, which is given as the pair (Ai, BA, where

A = M

and

Bi = Α_τΜ_λ = A_tMS + tA_te = MSA, + tA,e,

and it is used as the public key to encrypt any message.

This gives an identity based encryption system.

[91] The small elements like S, Ai, e, can follow different error distributions. [92] Since Ai and e are small elements in R, we have Ai x e is also small. We have that

= MSA, + tMM^e,) - MSA, + A_tt which is due to the fact that this is a commutative ring. Since e, Ai and are small, e— AiCi is also small and te— tAiCi is also small. Therefore ¾ is a solution to a ring LWE problem with the pair (Ai, BA as the problem input. Therefore ¾ is indeed a secret key that could be used for decryption.

[93] We can build easily a hierarchical IBE system using similar procedure, where each user can server as a central server.

[94] The key feature of our construction is that it is simple, straight forward and efficient. The provable security of the system is also straightforward.

[95] In the all the systems above using pairing with errors over the ring, one may use polynomials in the form of

f(^x) = Y[ ^X) + g(^x) ,

where each fi, g(x) is a extremely sparse matrix with very few terms, for example, 2 or 3 terms none-zero. Using this kind of polynomial can speed up the encryption and decryption computations.

LITERATURE CITED

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[ABVVW] S. Agrawal, X. Boyen, V. Vaikuntanathan, P. Voulgaris, H. Wee: Fuzzy Identity Based Encryption from Lattices. IACR Cryptology ePrint Archive 2011: 414 (2011)

[ACPS] B. Applebaum, D. Cash, C. Peikert, A. Sahai; Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems. Advances in Cryptology- CRYPTO 2009, Lecture Notes in Computer Science, Volume 5677 pp 595-618, 2009

[BKPW] M. Bellare, E. Kiltz, C. Peikert, B. Waters: Identity-Based (Lossy) Trapdoor Functions and Applications. In Proceedings of EUROCRYPT 2012, Lecture Notes in Computer Science, Volume 7237, pp 228-245 2012.

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[DiHe] W. Dime, M. Hellman, New directions in cryptography, IEEE Transactions on Information Theory 22 (6), pp 644-54, 1976. [DiLi] J. Ding, X. Lin, A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem, Cryptology ePrint Archive, Report 688, 2012

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Claims

CLAIM [1] Claim 1. Method for establishing a key exchange over an open channel between a first party A and a second party B, comprising: (1) openly selecting, by Party A and Party B together, parameters, n, q and small whole number t, (t « n), where q is an odd prime, and an error distribution κηι to be a distribution over n x n matrix over Fq, a n x n matrix M over Fq uniformly and randomly, where q is of size of a polynomial of n like n3 , and elements of Fq are represented by integers in the range [—(q— l)/2, (q— l)/2)]; (2) choosing, by each party privately, its own secret matrix Si ( i = A, B) a n x n matrix chosen according to the error distribution κηι , and error matrix ei , ( i = A, B ) as a n x n matrix following the error distribution κηι ; computing by Party A MA = MSA + teA, where t is a small integer (t « n); computing by Party B MB = *¾ + teB. (3) Both parties exchange Mi in the open communication channel; (4) computing by Patty A : KA = SA x MB = ¾ *¾ + tSAeB; computing by Party B: KB = MA x SB = ¾ *¾ + teASB; (5) performing by both Party A and Party B a rounding technique to derive the shared key, comprising: (a) making by Party B a list T\ of all positions of the entries of KB such that these entries are in the range of [—(q— l)/4, (q— l)/4] and a list T2 of all positions which are not in the range of [—(q— l)/4, (q— l)/4] then sending by Party B to Party A the list T\ . (b) computing by each party privately the residues of these entries modular t in T\, and for the entries not in T1; which is in T2, adding q— l)/2 to each entry and computing the residue modular q first (into the range of [—(q— l)/4, (q— 1)/4]^ then the residue modular t, which gives a shared key between these two parties. [2] Claim 2. Method, for a central server, building a key distribution (KD) system, comprising: (1) selecting, by the central server, parameters select parameters, n, q and small whole number t, (t « n), where q is an odd prime, q is of size of a polynomial of n like n3 and elements of Fq are represented by integers in the range [—(q— l)/2, (q— l)/2)] an error distribution κηι a distribution over n x n matrix over Fq, ; and selecting by the central server a symmetric randomly chosen n x n matrix S over Fq as a master key; (2) giving, by the central server, to each user index as i, a general matrix Ai as an ID with small entries following error distribution κηι , where the ID matrix of each user is public and the central server have also a choice to generate the ID with information that can identify the user; (3) distributing, by the central server, for each user securely a secret: where ¾ is a matrix selected following error distribution κηι and this is kept private for each user; obtaining a secret key shared between the User i and the User j comprising: computing by the User i: Ki = Ei Ajt = ASAjt + teiAjt and computing by the User j K3 = Ai x (EjY = AiS1 A + tAte) = AtSA) + tAte)- then two users deriving a shared key between the two users using the following simple rounding method, comprising: • assuming the User j wants to establish a shared key with the user i, collecting by the user j all the entries (including their positions in the matrix) in Kj that are in the range of (—(q— l)/4, (q— l)/4), namely those entries which are closer to 0 than (q— l)/2; sending by the User j to the user i a list of the positions of the entries in the matrix ( only the position not the values of the entries themselves) that are randomly selected from the collection, which is tagged by 0, and a list of entries not in the list tagged by 0; then selecting by the user i the same entries in its own matrix Ei x Aj, which gives them a shared list of common entry positions, therefore the corresponding entries of the matrix; then computing by each user the residue of these entries modular t tagged by 1 and compute the residue of the sum of each of these entries tagged by 0 with q— l)/2 which build a new identical ordered list of values, their shared secret key. [3] Claim 3. Method, for a central, building an identity-based encryption system, comprising:

(1) selecting by the central server parameters, n, q and small whole number t, (t « n), where q is an odd prime, q is of size of a polynomial of n like n³ and elements of F_q are represented by integers in the range [—(q— l)/2, (q— l)/2)] and an error distribution κ_ηι to be a distribution over n x n matrix over F_q; and selecting by the central server a secret n x n matrix S as the secret master key, where S is selected as a small element following certain error distribution κ_ηι ;

(2) selecting by the central server a random element M following uniform distribution, but making sure that M has an inverse: if the central server could not find one first time, it will try again till it finds one; then computing by the central server

where e is small following certain error distribution κ_ηι ;

(3) then publicizing by the central server M and M as the master public key;

(4) assigning by the central server for each user indexed by i an public ID as Ai, where Ai is small following certain error distribution κ_ηι , and the central server has a choice to generate it information that can identify the user i;

(5) giving by the central server for each user, namely, the User i, a secret key:

S, = SA, + tM^~xe_t,

where ¾ 's entries are small following the error distribution κ;

(6) then establishing by anyone using the ID, Ai, and the master public key, a new public key for the user with ID Ai, which is given as the pair (Ai, ΒΛ, where

A = M

and

B_t = Μ_λΑ_τ = MSA, + teAi;

and using by anyone as the public key to encrypt any message use the MLWE encryption system described in the Description section.

[4] Claim 4. The method according to Claim 1, wherein q is a polynomial function of degree 2 or higher, or a similar function, and is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_σ, namely a discrete normal distribution over F_q center around 0 with standard deviation approximately /n, or a similar distribution.

[5] Claim 5. The method according to Claim 2, wherein q is a polynomial function of degree 2 or higher, or a similar function, is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_σ, namely a discrete normal distribution over F_q center around 0 with standard deviation approximately

or a similar distribution.

[6] Claim 6. The method according to Claim 3, wherein q is a polynomial function of degree 2 or higher, or a similar function, is the a distribution that each component are independent and each component follow certain error distribution like the discrete error distribution κ_σ, namely a discrete normal distribution over F_q center around 0 with standard deviation approximately

or a similar distribution.

[7] Claim 7. The method according to Claim 1, wherein the matrices is rectangular as long as the matrix multiplication is compatible and the parameters are adjusted accordingly.

[8] Claim 8. The method according to Claim 2, wherein the matrices is rectangular as long as the matrix multiplication is compatible and the parameters are adjusted accordingly.

[9] Claim 9. The method according to Claim 1, wherein the matrices are replaced with elements of the ring TZ_q = F_q[x\/ f x) with f(x) = xⁿ + 1 and the parameters is adjusted accordingly.

[10] Claim 10. The method according to Claim 2, wherein the matrices are replaced with elements of the ring TZ_q = F_q[x\/ f x) with f (x) = xⁿ + 1 and the parameters is adjusted accordingly.

[11] Claim 11. The method according to Claim 3, wherein the matrices are replaced with elements of the ring lZ_q = F_q

with f (x) = xⁿ + 1 and the parameters is adjusted accordingly.

[12] Claim 12. The method according to Claim 2, wherein the procedure for two users i and j to derive a shared key is modified such that the roles of i and j are exchanged.

[13] Claim 13. The method according to Claim 2, wherein several central servers to work together to build a distributed KD system.

[14] Claim 14. The method according to Claim 3, wherein several central servers to work together to build a distributed IBE system.

[15] Claim 15. The method according to Claim 3, wherein the procedure is extended further to build a hierarchical IBE system, where each user servers as a lower level central server.

[16] Claim 16. The method according to Claim 1 , wherein the rounding technique is replaced with a similar technique.

[17] Claim 17. The method according to Claim 1, wherein the matrices are replaced

with elements of the ring lZ_q = F_q / f x) with f(x) = xⁿ + 1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f x) = Y[ fi(x) + g(x) , where each fi, g(x) is a sparse matrix with very few terms terms none-zero.

[18] Claim 18. The method g to Claim 2, wherein the matrices are replaced with elements of the ring lZ_q = F_q

with f(x) = xⁿ + 1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f x) = Y[ fi(x) + g(x) , where each fi, g(x) is a sparse matrix with very few terms terms none-zero.

[19] Claim 19. The method according to Claim 3, wherein the matrices are replaced with elements of the ring lZ_q = F_q [x]/ f (x) with f x) = xⁿ + 1, the parameters is adjusted accordingly, and the polynomial elements used are selected in the form of f x) = Y[ fi(x) + g(x) , where each fi, g(x) is a sparse matrix with very few terms terms none-zero.