CA2594670A1 - Elliptic curve random number generation - Google Patents

Abstract

An elliptic curve random number generator avoids escrow keys by choosing a point Q on the elliptic curve as verifiably random. An arbitrary string is chosen and a hash of that string computed. The hash is then converted to a field element of the desired field, the field element regarded as the x-coordinate of a point Q on the elliptic curve and the x-coordinate is tested for validity on the desired elliptic curve. If valid, the x-coordinate is decompressed to the point Q, wherein the choice of which is the two points is also derived from the hash value. Intentional use of escrow keys can provide for back up functionality. The relationship between P and Q is used as an escrow key and stored by for a security domain. The administrator logs the output of the generator to reconstruct the random number with the escrow key.

Description

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4 FIELD OF TH.E INVENTION:
6 [0001] The present invention relates to systems and methods for cryptographic random 7 number generation.

[0002] Random numbers are utilised in many cryptographic operations to provide underlying 11 security. In public key infrastructures, for example, the private key of a key pair is generated by a 12 random number generator and the corresponding public key mathematically derived therefrom.
13 A new key pair may be generated for each session and the randomness of the generator therefore 14 is critical to the security of the cryptographic system.

[0003] To provide a secure source of random numbers, cryptographically secure 16 pseudorandom bit generators have been developed in which the security of each generator relies 17 on a presumed intractability of the underlying number-theoretical problem.
The American 18 National Standards Institute (ANSI) has set up an Accredited Standards Committee (ASC) X9 19 for the fmancial services industry, which is preparing a American National Standard (ANS) X9. 82 for cryptographic random number generation (RNG). One of the RNG
methods in the 21 draft of X9.82, called Dual EC DRBG, uses elliptic curve cryptography (ECC) for its security.
22 Dual EC DRBG will hereinafter be referred to as elliptic curve random number generation 23 (ECRNG).

24 [0004] Elliptic curve cryptography relies on the intractability of the discrete log problem in cyclic subgroups of elliptic curve groups. An elliptic curve E is the set of points (x, y) that satisfy 26 the defining equation of the elliptic curve. The defining equation is a cubic equation, and is non-27 singular. The coordinates x and y are elements of a field, which is a set of elements that can be 28 added, subtracted and divided, with the exception of zero. Examples of fields include rational 1 numbers and real numbers. There are also finite fields, which are the fields most often used in 2 cryptography. An example of a fmite field is the set of integers modulo a prime q.

3 [0005] Without the loss of generality, the defining equation of the elliptic curve can be in the 4 Weierstrass form, which depends on the field of the coordinates. When the field F is integers modulo a prime q > 3, then the Weierstrass equation takes the form y2 = x3 +
ax + b, where a and 6 b are elements of the field F.

7 [0006] The elliptic curve E includes the points (x, y) and one further point, namely the point 8 0 at infinity. The elliptic curve E also has a group structure, which means that the two points P
9 and Q on the curve can be added to form a third point P + Q. The point 0 is the identity of the group, meaning P + 0= 0+ P = P, for all points P. Addition is associative, so that P + (Q + R) 11 =(P + Q) + R, and commutative, so that P+ Q= Q+ R, for all points P, Q and R. Each point P
12 has a negative point P, such that P + (-P) = O. When the curve equation is the Weierstrass 13 equation of the form y2 = x3 + ax + b, the negative of P=(x, y) is determined easily as 14 P=(x, -y). The formula for adding points P and Q in terms of their coordinates is only moderately complicated involving just a handful of field operations.

16 [0007] The ECRNG uses as input two elliptic curve points P and Q that are fixed. These 17 points are not assumed to be secret. Typically, P is the standard generator of the elliptic curve 18 domain parameters, and Q is some other point. In addition a secret seed is inserted into the 19 ECRNG.

[0008] The ECRNG has a state, which may be considered to be an integer s. The state s is 21 updated every time the ECRNG produces an output. The updated state is computed as u = z(sP), 22 where z() is a function that converts an elliptic curve point to an integer. Generally, z consists of 23 taking the x-coordinate of the point, and then converting the resulting field element to an integer.
24 Thus u will typically be an integer derived from the x-coordinate of the point s.

[0009] The output of the ECRNG is computed as follows: r = t(z(sQ)), where t is a truncation 26 function. Generally the truncation function removes the leftmost bits of its input. In the 1 ECRNG, the number of bits tn.mcated depends on the choice of elliptic curve, and typically may 2 be in the range of 6 to 19 bits.

3 [0010] Although P and Q are known, it is believed that the output r is random and cannot be 4 predicted. Therefore successive values will have no relationship that can be exploited to obtain private keys and break the cryptographic functions. The applicant has recognised that anybody 6 who knows an integer d such that Q = dP, can deduce an integer e such that ed = 1 mod n, where 7 n is the order of G, and thereby have an integer e such that P= eQ. Suppose U = sP and R = sQ, 8 which are the precursors to the updated state and the ECRNG output. With the integer e, one can 9 compute U from R as U= eR. Therefore, the output N= t(z(R)), and possible values of R can be determined from r. The truncation fixnction means that the truncated bits of R
would have to be 11 guessed. The z function means.that only the x-coordinate is available, so that decompression 12 would have to be applied to obtain the full point R. In the case of the ECRNG, there would be 13 somewhere between about 26 = 64 and 219 (i.e. about half a million) possible points R which 14 correspond to r, with the exact number depending on the curve and the specific value of r.

[0011] The full set of R values is easy to determine from r, and as noted above, 16 determination of the correct value for R determines U = eR, if one knows e.
The updated state is 17 u = z( U), so it can be determined from the correct value of R. Therefore knowledge of r and e 18 allows one to determine the next state to within a number of possibilities somewhere between 26 19 and 219. This uncertainty will invariably be eliminated once another output is observed, whether directly or indirectly through a one-way function.

21 [0012] Once the next state is determined, all future states of ECRNG can be determined 22 because the ECRNG is a deterministic function. (at least unless additional random entropy is fed 23 into the ECRNG state) All outputs of the ECRNG are determined from the determined states of 24 the ECRNG. Therefore knowledge of r and e, allows one to determine all future outputs of the ECRNG.

26 [0013] It has therefore been identified by the applicant that this method potentially possesses 27 a trapdoor, whereby standardizers or implementers of the algorithm may possess a piece of 28 information with which they can use a single output and an instantiation of the RNG to 1 determine all future states and output of the RNG, thereby completely compromising its security.
2 It is therefore an object of the present invention to obviate or mitigate the above mentioned 3 disadvantages.

4 SUMMARY OF THE .1NV.ENTION

[0014] In one aspect, the present invention provides a method for computing a verifiably 6 random point Q for use with another point P in an elliptic curve random number generator 7 comprising computing a hash including the point P as an input, and deriving the point Q from the 8 hash.

9 [0015] In another aspect, the present invention provides a method for producing an elliptic curve random number comprising generating an output using an elliptic curve random number 11 generator, and truncating the output to generate the random number.

12 [0016] In yet another aspect, the present invention provides a method for producing an 13 elliptic curve random number comprising generating an output using an elliptic curve random 14 number generator, and applying the output to a one-way function to generate the random_ number.

16 [0017] In yet another aspect, the present invention provides a method of backup fiulctionality 17 for an elliptic curve random number generator, the method comprising the steps of computing an 18 escrow key e upon determination of a point Q of the elliptic curve, whereby P = eQ, P being 19 another point of the elliptic curve; instituting an administrator, and having the administrator store the escrow key e=, having members with an elliptic curve random number generator send to the 21 administrator, an output r generated before an output value of the generator; the administrator 22 logging the output r for future determination of the state of the generator.

2 [0018] An embodiment of the invention will now be described byway of example only with 3 reference to the appended drawings wherein:

4 [0019] Figure 1 is a schematic representation of a cryptographic random number generation scheme.

6 [0020] Figure 2 is a flow chart illustrating a selection prbcess for choosing elliptic curve 7 points.

8 [0021] Figure 3 is a block diagram, similar to figure 1 showing a further embodiment 9 [0022] Figure 4 is flow chart illustrating the process implemented by the apparatus of Figure 3.

11 [0023] Figure 5 is a block diagram showing a further embodiment.

12 [0024] Figure 6 is a flow chart illustrating yet another embodiment of the process of Figure 13 2.

14 [0025] Figure 7 is schematic representation of an administrated cryptographic random number generation scheme.

16 [0026] Figure 8 is a flow chart illustrating an escrow key selection process.

17 [0027] Figure 9 is a flow chart illustrating a method for securely utilizing an escrow key.

[0028] Referring therefore to Figure 1, a cryptographic random number generator (ECRNG) 21 10 includes an arithmetic unit 12 for performing elliptic curve computations. The ECRNG also 22 includes a secure register 14 to retain a state value s and has a pair of inputs 16, 18 to receive a

-5-1 pair of initialisation points P, Q. The points P, Q are elliptic curve points that are assumed to be 2 known. An output 20 is provided for communication of the random integer to a cryptographic 3 module 22. The initial contents of the register 14 are provided by a seed input S.

4 [0029] This input 16 representing the point P is in a first embodiment, selected from a known value published as suitable for such use.

6 [0030] The input 18 is obtained from the output of a one way function in the form of a hash

7 function 24 typically a cryptographically secure hash function such as SHAl or SHA2 that

8 receives as inputs the point P. The function 24 operates upon an arbitrary bit string A to produce

9 a hashed output 26. The output 26 is applied to arithmetic unit 12 for further processing to provide the input Q.

11 [0031] In operation, the ECRNG receives a bit string as a seed, which is stored in the register 12 14. The seed is maintained secret and is selected to meet pre-established cryptographic criteria, 13 such as randomness and Hamming weight, the criteria being chosen to suit the particular 14 application.

[0032] In order to ensure that d is not likely to be known (e.g. such that P=
dQ, and ed =1 16 mod n); one or both of the inputs 16, 18 is chosen so as to be verifiably random. In the 17 embodiment of Figure 1, Q is chosen in a way that is verifiably random by deriving it from the 18 output of a hash-function 24 (preferably one-way) whose input includes the point P. As shown 19 in Figure 2 an arbitrary string A is selected at step 202, a hash H ofA is computed at step 204 with P and optionally S as inputs to a hash-based function FH(), and the hash H is then converted 21 by the arithmetic unit 12 to a field element Xof a desired field F at step 206. P may be pre-22 computed or fixed, or may also be chosen to be a verifiably random.chosen value. The field 23 element Xis regarded as the x-coordinate of Q (thus a "compressed"
representation of Q). The x-24 coordinate is then tested for validity on the desired elliptic curve E at step 208, and whether or not Xis valid, is determined at step 210. If valid, the x-co ordinate provided by element X is 26 decompressed to provide point Q at step 212. The choice of which of two possible values of the 27 y co-ordinate is generally derived from the hash value.

1 [0033] The points P and Q are applied at respective inputs 16, 18 and the arithmetic unit 12 2 computes the point sQ where s is the current value stored in the register 14. The arithmetic unit 3 12 converts the x-coordinate of the point (in this example point sQ) to an integer and truncates 4 the value to obtain r = t(z(sQ)). The truncated value r is provided to the output 20.

[0034] The arithmetic unit 12 similarly computes a value to update the register 14 by 6 computing sP, where s is the value of the register 14, and converting the x-coordinate of the 7 point sP to an integer u. The integer u is stored in the register to replace s for the next iteration.
8 {ditto above}

9 [0035] As noted above, the point P may also be verifiably random, but may also be an established or fixed value. Therefore, the embodiment of Figure 1 may be applied or retrofitted 11 to systems where certain base points (e.g. P) are already implemented in hardware. Typically, 12 the base point P will be some already existing base point, such as those recommended in Federal 13 Information Processing Standard (FIPS) 186-2. In such cases, P is not chosen to be verifiably 14 random.

[0036] In general, inclusion of the point P in the input to the hash function ensures that P
16 was determined before Q is determined, by virtue of the one-way property of the hash function 17 and since Q is derived from an already determined P. Because P was determined before Q, it is 18 clearly understood that P could not have been chosen as a multiple of Q
(e.g. where P = eQ), and 19 therefore finding d is generally as hard as solving a random case of the discrete logarithm problem.

21 [0037] Thus, having a seed value S provided and a hash-based function F() provided, a 22 verifier can determine that Q = F(S,P), where P may or may not be verifiably random.
23 Similarly, one could compute P = F(S,Q) with the same effect, though it is presumed that this is 24 not necessary given that the value of P in the early drafts of X9. 82 were identical to the base points specified in FIPS 18 6-2.

26 [0038] The generation of Q from a bit string as outlined above may be performed externally 27 of the ECRNG 10, or, preferably, internally using the arithmetic unit 12.
Where both P and Q

1 are required to be verifiably random, a second hash function 24 shown in ghosted outline in 2 Figure 1 is incorporated to generate the coordinate of point P from the bit string A. By providing 3 a hash function for at least one of the inputs, a verifrably random input is obtained.

4 [0039] It will also be noted that the output generated is derived from the x coordinate of the point sP. Accordingly, the inputs 16, 18 may be the x coordinates of P and Q
and the 6 corresponding values of sP and sQ obtained by using Montgomery multiplication techniques 7 thereby obviating the need for recovery of the y coordinates.

8 [0040] An alternative method for choosing Q is to choose Q in some canonical form, such 9 that its bit representation contains some string that would be difficult to produce by generating Q= dP for some known d and P for example a representation of a name. It will be appreciated 11 that intermediate forms between this method and the preferred method may also exist, where Q is 12 partly canonical and partly derived verifiably at random. Such selection of Q, whether veriflably ' 13 random, canonical, or some intermediate, can be called verifiable.

14 [0041] Another alternative method for preventing a key escrow attack on the output of an ECRNG, shown in Figures 3 and 4 is to add a truncation function 28 to ECRNG 10 to truncate 16 the ECRNG output to approximately half the length of a compressed elliptic curve point.
17 Preferably, this operation is done in addition to the preferred method of Figure 1 and 2, however, 18 it will be appreciated that it may be performed as a primary measure for preventing a key escrow 19 attack. The benefit of truncation is that the list of R values associated with a single ECRNG
output r is typically infeasible to search. For example, for a 160-bit elliptic curve group, the 21 number of potential points R in the list is about 280, and searching the list would be about as hard 22 as solving the discrete logarithm problem. The cost of this method is that the ECRNG is made 23 half as efficient, because the output length is effectively halved.

24 [0042] Yet another alternative method shown in Figure 5 and 6 comprises filtering the output of the ECRNG through another one-way function FH2,, identified as 34, such as a hash function 26 to generate a new output. Again, preferably, this operation is performed in addition to the 27 preferred method shown in Figure 2, however may be performed as a primary measure to prevent 28 key escrow attacks. The extra hash is relatively cheap compared to the elliptic curve operations 1 performed in the arithmetic unit 12, and does not significantly diminish the security of the 2 ECRNG.

3 [0043] As discussed above, to effectively prevent the existence of escrow keys, a verifiably 4 random Q should be accompanied with either a verifiably random P or a pre-established P. A
pre-established P may be a point P that has been widely publicized and accepted to have been 6 selected before the notion of the ECRNG 12, which consequently means that P
could not have 7 been chosen as .P = eQ because Q was not created at the time when P was established.

8 [0044] Whilst the above techniques ensure the security of the system using the ECRNG by 9 "closing" the trap door, it is also possible to take advantage of the possible interdependence of P
and Q, namely where P= eQ, through careful use of the existence of e.

11 [0045] In such a scenario, the value e may be regarded as an escrow key. If P and Q are 12 established in a security domain controlled by an administrator, and the entity who generates Q
13 for the domain does so with knowledge of e (or indirectly via knowledge of d). The administrator 14 will have an escrow key for every ECRNG that follows that standard.~

[0046] Escrow keys are known to have advantages in some contexts. They can provide a 16 backup functionality. If a cryptographic key is lost, then data encrypted under that key is also 17 lost. However, encryption keys are generally the output of random number generators.
18 Therefore, if the ECRNG is used to generate the encryption key K, then it may be possible that 19 the escrow key e can be used to recover the encryption key K. Escrow keys can provide other functionality, such as for use in a wiretap. In this case, trusted law enforcement agents may need 21 to decrypt encrypted traffic of criminals, and to do this they may want to be able to use an 22 escrow key to recover an encryption key.

23 [0047] Figure 7 shows a domain 40 having a number of ECRNG's 10 each associated with a 24 respective member of the domain 40. The domain 40 communicates with other domains 40a, 40b, 40c through a network 42, such as the internet. Each ECRNG of a domain has a pair of 26 identical inputs P,Q. The domain 40 includes an administrator 44 who maintains in a secure 27 manner an escrow key e.

1 [0048] The administrator 44 chooses the values of P and Q such that he knows an escrow 2 key e such that Q = eP. Other members of the domain 40 use the values of P
and Q, thereby 3 giving the administrator 44 an escrow key e that works for all the members of the organization.
4 [0049] This is most useful in its backup functionality for protecting against the loss of encryption keys. Escrow keys e could also be made member-specific so that each member has 6 its own escrow e' from points selected by the administrator 44.

7 [0050] As generally denoted as numeral 400 in Figure 8, the administrator initially selects a 8 point P which will generally be chosen as the standard generator P for the desired elliptic curve 9 402. The administrator then selects a value d and the point Q will be determined as Q= dP 404, for some random integer d of appropriate size. The escrow key e is computed as e= d"1 mod n 11 406, where n is the order of the generator P and stored by the administrator.

12 [0051] The secure use of such an escrow key 34e is generally denoted by numeral 500 and 13 illustrated in Figure 9. The administrator 44 is first instituted 502 and an escrow keys e would be 14 chosen and stored 504 by the administrator44 [0052] In order for the escrow key to fu.nction with full effectiveness, the escrow 16 administrator 44 needs direct access to an ECRNG output value r that was generated before the 17 ECRNG output value k (i.e. 16) which is to be recovered. It is not sufficient to have indirect 18 access to r via a one-way function or an encryption algorithm. A formalized way to achieve 19 this is to have each member with an ECRNG 12 communicate with the administrator 44 as indicated at 46 in figure 7. and step 506 in figure 9. This may be most useful for encrypted file 21 storage systems or encrypted email accounts. A more seamless method may be applied for 22 cryptographic applications. For example, in the SSL and TLS protocols, which are used for 23 securing web (HTTP) traffic, a client and server perform a handshake in which their first actions 24 are to exchange random values sent in the clear.

[0053] Many other protocols exchange such random values, often called nonces.
If the 26 escrow administrator observes these nonces, and keeps a log of them 508, then later it may be 27 able to determine the necessary r value. This allows the administrator to determine the

-10-1 subsequent state of the ECRNG 12 of the client or server 510 (whoever is a member of the 2 domain), and thereby recover the subsequent ECRNG 12 values. In particular, for the client who 3 generally generates a random pre-master secret from which is derived the encryption key for the 4 SSL or TLS session, the escrow key may allow recovery of the session key.
Recovery of the session key allows recovery of the whole SSL or TLS session.

6 [0054] If the session was logged, then it may be recovered. This does not compromise long-7 term private keys, just session keys obtained from the output of the ECRNG, which should 8 alleviate any concern regarding general suspicions related to escrows.

9 [0055] Whilst escrow keys are also known to have disadvantages in other contexts, their control within specific security domains may alleviate some of those concerns.
For example,

11 with digital signatures for non-repudiation, it is crucial that nobody but the signer has the signing

12 key, otherwise the signer may legitimately argue the repudiation of signatures. The existence of

13 escrow keys means the some other entity has access to the signing key, which enables signers to

14 argue that the escrow key was used to obtain their signing key and subsequently generate their signatures. However, where the domain is limited to a particular organisation or part of an 16 organisation it may be sufficient that the organisation cannot repudiate the signature. Lost 17 signing keys do not imply lost data, unlike encryption keys, so there is little need to backup 18 signing keys.

19 [0056] Although the invention has been described with reference to certain specific embodiments, various modifications thereof will be apparent to those skilled in the art without 21 departing from the spirit and scope of the invention as outlined in the claims appended hereto.

Claims (19)

1. A method of computing a random number for use in a cryptographic operation comprising the steps of providing a pair of inputs to an elliptic curve random number generator with each input representative of at least one coordinate of an elliptic curve point and with at least one of said inputs being verifiably random.

2. A method according to claim 1 wherein said at least one input is obtained from an output of a hash function.

3. A method according to claim 2 wherein the other of said inputs is utilized as an input to said hash function.

4. A method according to claim 1 wherein said random number generator has a secret value and said secret value is used to compute scalar multiples of said points represented by said inputs.

5. A method according to claim 4 wherein one of said scalar multiples is used to derive said random number and the other of said scalar multiples is used to change said secret value for subsequent use.

6. A method according to claim 2 wherein said output of said hash function is validated as a coordinate of a point on an elliptic curve prior to utilization as said input.

7 A method according to claim 6 wherein another coordinate of said point is obtained from said one coordinate for inclusion as said input.

8. A method according to claim 7 wherein said other input is a representation of an elliptic curve point.

9. A method according to claim 5 wherein said random number is derived from said scalar multiple by selecting one coordinate of said point represented by said scalar multiple and truncating said coordinate to a bit string for use as said random number.

10. A method according to claim 9 wherein said one coordinate is truncated in the order of one half the length of a representation of an elliptic curve point representation.

11 A method according to claim 5 wherein said random number is derived from said scalar multiple by selecting one coordinate of said point represented by said scalar multiple and hashing said one coordinate to provide a bit string for use as said random number.

12. A method according to claim 1 wherein said verifiably random input is chosen to be of a canonical form whereby a predetermined relationship between said inputs is difficult to maintain.

13 A method of computing a random number for use in a cryptographic operation, said method comprising the steps of providing a pair of inputs, each representative of at least one coordinate of a pair of elliptic curve points to an elliptic curve random number generator, obtaining an output representative of at least one coordinate of a scalar multiple of an elliptic curve point and passing said output through a one way function to obtain a bit string for use as a random number.

14. A method according to claim 13 wherein said one way function is a hash function.

15. An elliptic curve random number generator having a pair of inputs each representative of at least one coordinate of a pair of elliptic curve points and an output for use as a random number in a cryptographic operation, at least one of said inputs being verifiably random.

16. An elliptic curve random number generator according to claim 15 wherein said one input is derived from an output of a one way function.

17. An elliptic curve random number generator according to claim 16 wherein said one way function is a hash function.

18. An elliptic curve random number generator according to claim 17 wherein the other of said inputs is provided as an input to said hash function.

19. A method of establishing an escrow key for a security domain within a network, said method comprising the steps of establishing a pair of points PQ as respective inputs to an elliptic curve random number generator with a relationship between said point such that P= eQ, storing said relationship e as an escrow key with an administrator and generating from said elliptic curve random number generator a random number for use in cryptographic operations within said domain.