| CFRG | S. Goldberg |
| Internet-Draft | L. Reyzin |
| Intended status: Standards Track | Boston University |
| Expires: March 18, 2019 | D. Papadopoulos |
| Hong Kong University of Science and Techology | |
| J. Vcelak | |
| NS1 | |
| September 14, 2018 |
Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-03
A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private key can compute the hash, but anyone with public key can verify the correctness of the hash. VRFs are useful for preventing enumeration of hash-based data structures. This document specifies several VRF constructions that are secure in the cryptographic random oracle model. One VRF uses RSA and the other VRF uses Eliptic Curves (EC).
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A Verifiable Random Function (VRF) [MRV99] is the public-key version of a keyed cryptographic hash. Only the holder of the private VRF key can compute the hash, but anyone with corresponding public key can verify the correctness of the hash.
A key application of the VRF is to provide privacy against offline enumeration (e.g. dictionary attacks) on data stored in a hash-based data structure. In this application, a Prover holds the VRF private key and uses the VRF hashing to construct a hash-based data structure on the input data. Due to the nature of the VRF, only the Prover can answer queries about whether or not some data is stored in the data structure. Anyone who knows the public VRF key can verify that the Prover has answered the queries correctly. However no offline inferences (i.e. inferences without querying the Prover) can be made about the data stored in the data strucuture.
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].
The following terminology is used through this document:
A VRF comes with a key generation algorithm that generates a public VRF key PK and private VRF key SK.
The prover hashes an input alpha using the private VRF key SK to obtain a VRF hash output beta
The VRF_hash algorithm is deterministic, in the sense that it always produces the same output beta given a pair of inputs (SK, alpha). The prover also uses the private key SK to construct a proof pi that beta is the correct hash output
The VRFs defined in this document allow anyone to deterministically obtain the VRF hash output beta directly from the proof value pi as
Notice that this means that
and thus this document will specify VRF_prove and VRF_proof_to_hash rather than VRF_hash.
The proof pi allows a Verifier holding the public key PK to verify that beta is the correct VRF hash of input alpha under key PK. Thus, the VRF also comes with an algorithm
that outputs (VALID, beta = VRF_proof_to_hash(pi)) if pi is valid, and INVALID otherwise.
VRFs are designed to ensure the following security properties.
Uniqueness means that, for any fixed public VRF key and for any input alpha, there is a unique VRF output beta that can be proved to be valid. Uniqueness must hold even for an adversarial Prover that knows the VRF private key SK.
More precisely, "full uniqueness" states that a computationally-bounded adversary cannot choose a VRF public key PK, a VRF input alpha, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha, pi1) outputs (VALID, beta1), VRF_verify(PK, alpha, pi2) outputs (VALID, beta2), and beta1 is not equal to beta2.
A slightly weaker security property called "trusted uniqueness" sufficies for many applications. Trusted uniqueness is the same as full uniqueness, but it must hold only if the VRF keys PK and SK were generated in a trustworthy manner. In other words, uniqueness might not hold if keys were generated in an invalid manner or with bad randomness.
Like any cryprographic hash function, VRFs need to be collision resistant. Collison resistance must hold even for an adversarial Prover that knows the VRF private key SK.
More precisely, "full collision resistance" states that it should be computationally infeasible for an adversary to find two distinct VRF inputs alpha1 and alpha2 that have the same VRF hash beta, even if that adversary knows the private VRF key SK.
For most applications, a slightly weaker security property called "trusted collision resistance" suffices. Trusted collision resistance is the same as collision resistance, but it holds only if PK and SK were generated in a trustworthy manner.
Pseudorandomness ensures that when an adversarial Verifier sees a VRF hash output beta without its corresponding VRF proof pi, then beta is indistinguishable from a random value.
More precisely, suppose the public and private VRF keys (PK, SK) were generated in a trustworthy manner. Pseudorandomness ensures that the VRF hash output beta (without its corresponding VRF proof pi) on any adversarially-chosen "target" VRF input alpha looks indistinguishable from random for any computationally bounded adversary who does not know the private VRF key SK. This holds even if the adversary also gets to choose other VRF inputs alpha' and observe their corresponding VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the "target" VRF input alpha at any time, even after it observes VRF outputs beta' and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property which suffices in many applications. Here, the adversary must choose the target VRF input alpha independently of the public VRF key PK, and before it observes VRF outputs beta' and proofs pi' on inputs alpha' of its choice.
It is important to remember that the VRF output beta does not look random to the Prover, or to any other party that knows the private VRF key SK! Such a party can easily distinguish beta from a random value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that knows valid VRF proof pi corresponding to the VRF input alpha, even if this party does not know the private VRF key SK. Such a party can easily distinguish beta from a random value by checking whether VRF_verify(PK, alpha, pi) returns (VALID, beta).
Also, the VRF output beta may not look random if VRF key generation was not done in a trustworthy fashion. (For example, if VRF keys were generated with bad randomness.)
Pseudorandomness, as defined in Section 3.3, does not hold if the VRF keys were generated adversarially. For instance, if an adversary outputs VRF keys that are deterministically generated (or hard-coded and publicly known), then the outputs are easily derived by anyone.
There is, however, a different type of unpredictability that is desirable in certain VRF applications (such as [GHMVZ17] and [KRDO17]). This property is similar to the unpredictability achieved by an (ordinary, unkeyed) cryptographic hash function: if the input has enough entropy (i.e., cannot be predicted), then the correct output is indistinguishable from uniform.
Although neither formal definitions nor proofs of this property have appeared in cryptographic literature, the VRF schemes presented in this specification are believed to satisfy this property if the public key was generated in a trustworthy manner. Additionally, the ECVRF also satisifies this property even if the public key was not generated in a trustworthy manner, as long as the public key satisfies the key validation procedure in Section 5.6.
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. Its security follows from the standard RSA assumption in the random oracle model. Formal security proofs are in [PWHVNRG17].
The VRF computes the proof pi as a deterministic RSA signature on input alpha using the RSA Full Domain Hash Algorithm [RFC8017] parametrized with the selected hash algorithm. RSA signature verification is used to verify the correctness of the proof. The VRF hash output beta is simply obtained by hashing the proof pi with the selected hash algorithm.
The key pair for RSA-FDH-VRF MUST be generated in a way that it satisfies the conditions specified in Section 3 of [RFC8017].
In this document, the notation from [RFC8017] is used.
Parameters used:
Fixed options:
Primitives used:
RSAFDHVRF_prove(K, alpha_string)
Input:
Output:
Steps:
RSAFDHVRF_proof_to_hash(pi_string)
Input:
Output:
Important note:
Steps:
RSAFDHVRF_verify((n, e), alpha_string, pi_string)
Input:
Output:
Steps:
The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that satisfies the trusted uniqueness, trusted collision resistance, and full pseudorandomness properties defined in Section 3. The security of this VRF follows from the decisional Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are in [PWHVNRG17].
To additionally satisfy "full uniqueness" and "full collision resistance", the Verifier MUST additionally perform the validation procedure specified in Section 5.6 upon receipt of the public VRF key.
Fixed options (specified in Section 5.5):
Notation and primitives used:
Type conversions:
Parameters used (the generation of these parameters is specified in Section 5.5):
Note: this function must have the VRF private key SK as input. Below we make it more efficient by supplying it also with the secret scalar x and the public key Y as additional inputs; however, each of these can be computed from SK if desired.
ECVRF_prove(Y, x, alpha_string)
Input:
Output:
Steps:
ECVRF_proof_to_hash(pi_string)
Input:
Output:
Important note:
Steps:
ECVRF_verify(Y, pi_string, alpha_string)
Input:
Output:
Steps:
The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and converts it to H, an EC point in G. This algorithm is the only place the VRF input alpha is used in for proving and verfying. See Section 7.6 for further discussion.
The algorithms in this section are not compatible with each other; the choice of algorithm is made in Section 5.5.
The following ECVRF_hash_to_curve_try_and_increment(suite_string, Y, alpha_string) algorithm implements ECVRF_hash_to_curve in a simple and generic way that works for any elliptic curve.
The running time of this algorithm depends on alpha_string. For the ciphersuites specified in Section 5.5, this algorithm is expected to find a valid curve point after approximately two attempts (i.e., when ctr=1) on average.
However, because the running time of algorithm depends on alpha_string, this algorithm SHOULD be avoided in applications where it is important that the VRF input alpha remain secret.
ECVRF_hash_to_try_and_increment(suite_string, Y, alpha_string)
Input:
Output:
Steps:
The following ECVRF_hash_to_curve_elligator2_25519(suite_string, Y, alpha_string) algorithm implements ECVRF_hash_to_curve using the elligator2 algorithm from Section 5 of [BHKT13] (see also [I-D.irtf-cfrg-hash-to-curve]) exclusively for the Ed25519 elliptic curve (which the Edwards equivalent of Curve25519). It can be implemented with running time that is independent of the input alpha (so-called "constant-time").
ECVRF_hash_to_curve_elligator2_25519(suite_string, Y, alpha_string)
Input:
Output:
Fixed options:
Constraints on options:
Steps:
In order to make this algorithm run in time that is (almost) independent of the input alpha_string (so-called "constant-time"), implementers should pay particular attention to Steps 10 and 11 above. These steps can be implemented using the following approach:
The first step will produce a value e that is either 1 or p-1 (it is guaranteed not to be any other value, because w is guaranteed to be nonzero). Implementers should also ensure that the second step runs in the same amount of time regardless of e by ensuring that arithmetic in constant time.
Alternatively, let CMOV(result_if_1, result_if_0, selector) be the function that returns result_if_1 when selector is 1 and result_if_0 when selector is 0. If CMOV is implemented in constant time, then steps 12 and 13 above can be implemented as follows:
(Note that after the first step, e is either 0 or 2, and only the least significant byte of e is needed in the second step). CMOV can be implemented in constant time a variety of ways; for example, by expanding b from a single bit to an all-0 or all-1 string (accomplished by negating b in standard two's-complement arithmetic) and then applying bitwise XOR and AND operations as follows: other_x XOR ((x XOR other_x) AND b)
If having this algorithm run in constant time is not important, then there are much faster algorithms to compute the Legendre symbol (which is the same as the Jacobi symbol because p is a prime). See, for example, Section 12.3 of [ntb].
The following ECVRF_hash_to_curve_Simplified_SWU(suite_string, Y, alpha_string) algorithm implements ECVRF_hash_to_curve using the simplified Shallue-Woestijne [SW06] and Ulas [Ulas07] algorithm from Section 7 of [BCIMRT10] (see also [I-D.irtf-cfrg-hash-to-curve]). It can be implemented with running time that is independent of the input alpha (so-called "constant-time"). Generally, this method can be used for any curve with prime p that is congruent to 3 modulo 4; however, the (very unlikely) case of d=0 in step 6 below may need to be handled differently depending on the curve equation, to ensure that the result is a point on the curve.
ECVRF_hash_to_curve_Simplified_SWU(suite_string, Y, alpha_string)
Input:
Output:
Fixed options:
Steps:
In order to make this algorithm run in time that is (almost) independent of the input (so-called "constant-time"), implementers should pay particular attention to Steps 10 and 11 above. These steps can be implemented using the following approach. Let CMOV(result_if_1, result_if_0, selector) be the function that returns result_if_1 when selector is 1 and result_if_0 when selector is 0. If arithmetic and CMOV are implemented in constant time, then steps 9 and 10 above can be implemented as follows:
CMOV can be implemented in constant time a variety of ways; for example, by expanding b from a single bit to an all-0 or all-1 string (accomplished by negating b in standard two's-complement arithmetic) and then applying bitwise XOR and AND operations as follows: other_x XOR ((x XOR other_x) AND b)
If having this algorithm run in constant time is not important, then there are much faster algorithms to compute the Legendre symbol (which is the same as the Jacobi symbol because p is a prime). See, for example, Section 12.3 of [ntb].
The following subroutines generate the nonce value k in a deterministic pseudorandom fashion.
ECVRF_nonce_generation_RFC6979(SK, h_string)
Input:
Output:
The ECVRF_nonce_generation function is as specified in [RFC6979] Section 3.2 where
The following is from Steps 2-3 of Section 5.1.6 in [RFC8032].
ECVRF_nonce_generation_RFC8032(SK, h_string)
Input:
Output:
Steps:
ECVRF_hash_points(P1, P2, ..., PM)
Input:
Output:
Steps:
ECVRF_decode_proof(pi_string)
Input:
Output:
Steps:
This document defines ECVRF-P256-SHA256-TAI as follows:
This document defines ECVRF-P256-SHA256-SWU as follows:
This document defines ECVRF-ED25519-SHA512-TAI as follows:
This document defines ECVRF-ED25519-SHA512-Elligator2 as follows:
The ECVRF as specified above is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. In order to obtain "full uniqueness" and "full collision resistance" (which provide protection against a malicious VRF public key), the Verifier MUST perform the following additional validation procedure upon receipt of the public VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID".
Note that this procedure is not sufficient if the elliptic curve E or the point B, the generator of group G, is untrusted. If the prover is untrusted, the Verifier MUST obtain E and B from a trusted source, such as a ciphersuite specification, rather than from the prover.
This procedure supposes that the public key provided to the Verifier is an octet string. The procedure returns "INVALID" if the public key in invalid. Otherwise, it returns Y, the public key as an EC point.
ECVRF_validate_key(PK_string)
Input:
Output:
Steps:
Note that if the cofactor = 1, then Step 3 need not multiply Y by the cofactor; instead, it suffices to output "INVALID" if Y is the point at infinity. Moreover, when cofactor>1, it is not necessary to verify that Y is in the subgroup G; Step 3 suffices. Therefore, if the cofactor is small, the total number of points that could cause Step 3 to output "INVALID" may be small, and it may be more efficient to simply check Y against a fixed list of such points. For example, the following algorithm can be used for the Ed25519 curve:
(bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys, depending on the sign of the x-coordinate, which was cleared in step 5, in order to make sure that it does not affect the comparison. bad_pk[1] and bad_pk[4] each match one bad public key, because x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6] are simply bad_pk[0] and bad_pk[1] shifted by p, in case the y-coordinate had not been modular reduced by p. There is no need to shift the other bad_pk values by p, because they will exceed 2^255. These bad keys, which represent all points of order 1, 2, 4, and 8, have been obtained by converting the points specified in [X25519] to Edwards coordinates.)
A reference implementation of ECVRF-P256-SHA256-TAI, ECVRF-P256-SHA256-SWU, ECVRF-ED25519-SHA512-TAI, ECVRF-ED25519-SHA512-Elligator2 is available at <https://github.com/reyzin/ecvrf>. This implementation is neither secure nor especially effecient, but can be used to generate test vectors.
An implementation of the RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256-TAI was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5] and is available at <http://github.com/fcelda/nsec5-crypto>. These implementations may be out of date as this spec has evolved.
The Key Transparency project at Google uses a VRF implemention that is similar to the ECVRF-P256-SHA256-TAI, with a few minor changes including the use of SHA-512 instead of SHA-256. Its implementation is available <https://github.com/google/keytransparency/blob/master/core/vrf/vrf.go>
An implementation by Yahoo! similar to the ECVRF is available at <https://github.com/r2ishiguro/vrf>.
An implementation similar to ECVRF is available as part of the CONIKS implementation in Golang at <https://github.com/coniks-sys/coniks-go/tree/master/crypto/vrf>.
Open Whisper Systems also uses a VRF very similar to ECVRF-ED25519-SHA512-Elligator, called VXEdDSA, and specified here: <https://whispersystems.org/docs/specifications/xeddsa/>
Applications that use the VRFs defined in this document MUST ensure that that the VRF key is generated correctly, using good randomness.
The ECVRF as specified in Section 5.1-Section 5.5 statisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. If the Verifier trusts the VRF keys are generated correctly, it MAY use the public key Y as is.
However, if the ECVRF uses keys that could be generated adversarially, then the the Verfier MUST first perform the validation procedure ECVRF_validate_key(PK) (specified in Section 5.6) upon receipt of the public key PK as an octet string. If the validation procedure outputs "INVALID", then the public key MUST not be used. Otherwise, the procedure will output a valid public key Y, and the ECVRF with public key Y satisfies the "full uniqueness" and "full collision resistance" properties.
The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. These properties may not hold if the keys are generated adversarially (e.g., if RSA is not permutation). Meanwhile, the "full uniqueness" and "full collision resistance" are properties that hold even if VRF keys are generated by an adversary. The RSA-FDH-VRF defined in this document does not have these properties. However, if adversarial key generation is a concern, the RSA-FDH-VRF may be modifed to have these properties by adding additional cryptographic checks that its public key has the right form. These modifications are left for future specification.
Without good randomness, the "pseudorandomness" properties of the VRF may not hold. Note that it is not possible to guarantee pseudorandomness in the face of adversarially generated VRF keys. This is because an adversary can always use bad randomness to generate the VRF keys, and thus, the VRF output may not be pseudorandom.
[PWHVNRG17] presents cryptographic reductions to an underlying hard problem (e.g. Decisional Diffie Hellman for the ECVRF, or the standard RSA assumption for RSA-FDH-VRF) that prove the VRFs specificied in this document possess full pseudorandomness as well as selective pseudorandomness. However, the cryptographic reductions are tighter for selective pseudorandomness than for full pseudorandomness. This means the the VRFs have quantitavely stronger security guarentees for selective pseudorandomness.
Applications that are concerned about tightness of cryptographic reductions therefore have two options.
The security of the ECVRF defined in this document relies on the fact that nonce k used in the ECVRF_prove algorithm is chosen uniformly and pseudorandomly modulo q, and is unknown to the advesrary. Otherwise, an adversary may be able to recover the private VRF key x (and thus break pseudorandomness of the VRF) after observing several valid VRF proofs pi. The nonce generation methods specified in the ECVRF ciphersuites of Section 5.5 are designed with this requirement in mind.
Side channel attacks on cryptographic primatives are an important issue. Here we discuss only one such side channel: timing attacks that can be used to leak information about the VRF input alpha. Implementers should take care to avoid side-channel attacks that leak information about the VRF private key SK (and the nonce k used in the ECVRF).
The ECVRF_hash_to_curve_try_and_increment algorithm defined in Section 5.4.1.1 SHOULD NOT be used in applications where the VRF input alpha is secret and is hashed by the VRF on-the-fly. This is because the algorithm's running time depends on the VRF input alpha, and thus creates a timing channel that can be used to learn information about alpha. That said, for most inputs the amount of information obtained from such a timing attack is likely to be small (1 bit, on average), since the algorithm is expected to find a valid curve point after only two attempts. However, there might be inputs which cause the algorithm to make many attempts before it finds a valid curve point; for such inputs, the information leaked in a timing attack will be more than 1 bit.
Meanwhile, ECVRF-P256-SHA256-SWU and ECVRF-ED25519-SHA512-Elligator2 can be made to run in time constant in alpha.
The VRF proof pi is not designed to provide secrecy and, in general, may reveal the VRF input alpha. Anyone who knows PK and pi is able to perform an offline dictionary attack to search for alpha, by verifying guesses for alpha using VRF_verify. This is in contrast to the VRF hash output beta which, without the proof, is pseudorandom and thus is designed to reveal no information about alpha.
The VRFs specified in this document allow for read-once access to the input alpha for both signing and verifying. Thus, additional prehashing of alpha (as specified, for example, in [RFC8032] for EdDSA signatures) is not needed, even for applications that need to handle long alpha or to support the Initialized-Update-Finalize (IUF) interface (in such an interface, alpha is not supplied all at once, but rather in pieces by a sequence of calls to Update). The ECVRF, in particular, uses alpha only in ECVRF_hash_to_curve. The curve point H becomes the representative of alpha thereafter. Note that the suite_string octet and the public key are hashed together with alpha in ECVRF_hash_to_curve, which ensures that the curve (including the generator B) and the public key are included indirectly into subsequent hashes.
Hashing is used for different purposes in the two VRFs (namely, in the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in hash_to_curve, nonce_generation, hash_points, and proof_to_hash). The theoretical analysis assumes each of these functions is a separate random oracle. This analysis still holds even if the same hash function is used, as long as the four queries made to the hash function for a given SK and alpha are overwhelmingly unlikely to equal each other or to any queries made to the hash function for the same SK and different alpha. This is indeed the case for the RSA-FDH-VRF defined in this document, because the first octets of the input to the hash function used in MGF1 and in proof_to_hash are different. This is also the case for the ECVRF ciphersuites defined in this document, because:
For the RSA VRF, if future designs need to specify variants of the design in this document, such variants should use different first octets in inputs to MGF1 and to the hash funciton used in proof_to_hash, in order to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant
For the elliptic curve VRF, if future designs need to specify variants (e.g., additional ciphersuites) of the design in this document, then, to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant, they should specify a different suite_string constant. This way, the inputs to the hash_to_curve hash function used in producing H are guaranteed to be different; since all the other hashing done by the prover depends on H, inputs all the hash functions used by the prover will also be different as long as hash_to_curve is collision resistant.
Note to RFC Editor: if this document does not obsolete an existing RFC, please remove this appendix before publication as an RFC.
This document also would not be possible without the work of Moni Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf Ziv (Facebook). Shumon Huque, David C. Lawerence, Trevor Perrin, Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova, Tony Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood, Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, and Gary Belvin provided valuable input to this draft.
The test vectors in this section were genereated using the reference implementation at <https://github.com/reyzin/ecvrf>.
These two example secret keys and messages are taken from Appendix A.2.5 of [RFC6979].
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
try_and_increment succeded on ctr = 0
H = 02e2e1ab1b9f5a8a68fa4aad597e7493095648d3473b213bba120fe42d1a595f3e
k = c1aba586552242e6b324ab4b7b26f86239226f3cfa85b1c3b675cc061cf147dc
U = k*B = 02007fe22a3ed063db835a63a92cb1e487c4fea264c3f3700ae105f8f3d3fd391f
V = k*H = 03d0a63fa7a7fefcc590cb997b21bbd21dc01304102df183fb7115adf6bcbc2a74
pi = 029bdca4cc39e57d97e2f42f88bcf0ecb1120fb67eb408a856050dbfbcbf57c524193b7a850195ef3d5329018a8683114cb446c33fe16ebcc0bc775b043b5860dcb2e553d91268281688438df9394103ab
beta = 59ca3801ad3e981a88e36880a3aee1df38a0472d5be52d6e39663ea0314e594c
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 74657374 (ASCII "test")
try_and_increment succeded on ctr = 0
H = 02ca565721155f9fd596f1c529c7af15dad671ab30c76713889e3d45b767ff6433
k = 7fc43fbc2aa51886139614792c613e672624b3fb8d0cf3fa6f52543d6a2fc26c
U = k*B = 037cd79cd8ab7b0324600b4d76c673a0726f3f1ff26b4b850d0c14b3aa272bf841
V = k*H = 02372cc8b56281e8d7f21ab7503c10af22164a4227e7433de0953a0df5e7a609bd
pi = 03873a1cce2ca197e466cc116bca7b1156fff599be67ea40b17256c4f34ba2549c9c8b100049e76661dbcf6393e4d625597ed21d4de684e08dc6817b60938f3ff4148823ea46a47fa8a4d43f5fa6f77dc8
beta = dc85c20f95100626eddc90173ab58d5e4f837bb047fb2f72e9a408feae5bc6c1
This example secret key and message are taken from Appendix L.4.2 of [ANSI.X9-62-2005].
SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65206f66204543445341207769746820616e736970323536723120616e64205348412d323536 (ASCII "Example of ECDSA with ansip256r1 and SHA-256")
try_and_increment succeded on ctr = 1
H = 02141e41d4d55802b0e3adaba114c81137d95fd3869b6b385d4487b1130126648d
k = 111e1505c8531c885dab6607a0962cd40a0af77637cdf183c7c9fb799dded43e
U = k*B = 02b3aceb619b90e811c8e50de73b27e65dd84669821055cb60dc1fa47199396c74
V = k*H = 02f117fe7daa8942d5492cc968784ced16025161b2dad374808eb7fbaf5eda5331
pi = 02abe3ce3b3aa2ab3c6855a7e729517ebfab6901c2fd228f6fa066f15ebc9b9d41fd212750d9ff775527943049053a77252e9fa59e332a2e5d5db6d0be734076e98befcdefdcbaf817a5c13d4e45fbf9bc
beta = e880bde34ac5263b2ce5c04626870be2cbff1edcdadabd7d4cb7cbc696467168
These two example secret keys and messages are taken from Appendix A.2.5 of [RFC6979].
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
In SWU: t = f1523667d029b9119a319a5bb316ff846691600e3552514ec4f93f9c84d65a4f
In SWU: w = d8125c3ae82fc2b7f1c326b6f3dbfdf3583272336a60cb08efb84e002e98a3b3
In SWU: e = -1
H = 027827143876a58c2189402306c6ff6f7f9a7271067f3ed28eb63790d58a84fdd6
k = e15d8e7677f9473ae922d36977dbcc305a4ffd8149499dccfcb44fa097a2200c
U = k*B = 035dc0dee6903dba1a3e7cae4e2a960609e873e6e696cc8d5e56dfa8efccdfc97a
V = k*H = 03bec301c2930d69ed359eab2a54349d1431625c7a3ee0cfc2643ae5f8a21c3add
pi = 021d684d682e61dd76c794eef43988a2c61fbdb2af64fbb4f435cc2a842b0024c35641fe838a72d0d9bc1bcf032f895f3b3f4c79d0f8f9d5705d83181fe82e19f49619eb8290930809b2b9651786e4f945
beta = 143f36bf7175053315693cfcfdff5aebb13e5eb9c47f897f53f81561993cfcd2
SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 74657374 (ASCII "test")
In SWU: t = e20da1d7386cb673deffec63d47ec65862dce55f113be168fa45cba2a6c1ddbc
In SWU: w = 0eed10be2937c902c9612d80b8ea5b0783f81c419faedd57efc84e6dfcfe2c72
In SWU: e = 1
H = 020e6c14efc8bc7150a3467aafa78be9856a2c6e405bdcc50f767fe638569d0172
k = 2addf2924eea6557e87acd635f08b54156cba70d718d8a3b6268af795cfdb7f2
U = k*B = 028347e823490cb50becb229cf059942afff39d6b276b987a384e45f29d1bf0dc3
V = k*H = 034759927f83caf0ed5d7ad1505844548051ef90a2f29de30efaf8eb811afb3342
pi = 0376b758f457d2cabdfaeb18700e46e64f073eb98c119dee4db6c5bb1eaf677806895ab451335f6adb792d40c68351929fce44068ffdcbbeac12f058b0365856ed5d86aadba1f54c9db13f9c8759589609
beta = 6b5bb622a6bc1387a7dcc4f46cfdcc3bce67669b32f3bc39e047c3b6cd3e65d9
This example secret key and message are taken from Appendix L.4.2 of [ANSI.X9-62-2005].
SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65206f66204543445341207769746820616e736970323536723120616e64205348412d323536 (ASCII "Example of ECDSA with ansip256r1 and SHA-256")
In SWU: t = e93da6ba2bca714061dc94c8c513343ad11bfc9678339e4a8bd86a08232aa6d7
In SWU: w = 76f564cca31934c80dd2a285ba43543df63a078b132c8f34d2ab1b7089cb3401
In SWU: e = -1
H = 02429690b91e1783cd0d7e393db07cc44b48c226cb837adb2282251cabf431a484
k = 2ba4cd5e9f7be946659555b3d052af618d2d3a14f5a756c8d56fab058d1831ff
U = k*B = 025943c217f48354e297156ac7dce8d2b50f63867acc23ba1aea87a66578392ca8
V = k*H = 0399ccafce764acf0db264183d97c1ae968daf661b9931a145bb8cbfc5f3f8f9a5
pi = 035e844533a7c5109ab3dffd04f2ef0d38d679101124f15243199ce92f0f29477cd29f8754f3bbdea3dd129560e9ba0c73ae7894a8d0c0e1ac01e5c2685da67009d96e6ccdb634c7e0c5f38fa3e4908c02
beta = be1dcb17e9815ac6acf819e7ad4b75e575eafad25915c2608959d780364fc912
These three example secret keys and messages are taken from Section 7.1 of [RFC8032].
SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
try_and_increment succeded on ctr = 0
H = 5b2c80db3ce2d79cc85b1bfb269f02f915c5f0e222036dc82123f640205d0d24
k = 647ac2b3ca3f6a77e4c4f4f79c6c4c8ce1f421a9baaa294b0adf0244915130f7067640acb6fd9e7e84f8bc30d4e03a95e410b82f96a5ada97080e0f187758d38
U = k*B = a21c342b8704853ad10928e3db3e58ede289c798e3cdfd485fbbb8c1b620604f
V = k*H = 426fe41752f0b27439eb3d0c342cb645174a720cae2d4e9bb37de034eefe27ad
pi = 9275df67a68c8745c0ff97b48201ee6db447f7c93b23ae24cdc2400f52fdb08a1a6ac7ec71bf9c9c76e96ee4675ebff60625af28718501047bfd87b810c2d2139b73c23bd69de66360953a642c2a330a
beta = a64c292ec45f6b252828aff9a02a0fe88d2fcc7f5fc61bb328f03f4c6c0657a9d26efb23b87647ff54f71cd51a6fa4c4e31661d8f72b41ff00ac4d2eec2ea7b3
SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
try_and_increment succeded on ctr = 4
H = 08e18a34f3923db32e80834fb8ced4e878037cd0459c63ddd66e5004258cf76c
k = 627237308294a8b344a09ad893997c630153ee514cd292eddd577a9068e2a6f24cbee0038beb0b1ee5df8be08215e9fc74608e6f9358b0e8d6383b1742a70628
U = k*B = 18b5e500cb34690ced061a0d6995e2722623c105221eb91b08d90bf0491cf979
V = k*H = 87e1f47346c86dbbd2c03eafc7271caa1f5307000a36d1f71e26400955f1f627
pi = 84a63e74eca8fdd64e9972dcda1c6f33d03ce3cd4d333fd6cc789db12b5a7b9d03f1cb6b2bf7cd81a2a20bacf6e1c04e59f2fa16d9119c73a45a97194b504fb9a5c8cf37f6da85e03368d6882e511008
beta = cddaa399bb9c56d3be15792e43a6742fb72b1d248a7f24fd5cc585b232c26c934711393b4d97284b2bcca588775b72dc0b0f4b5a195bc41f8d2b80b6981c784e
SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
alpha = af82 (2 bytes)
x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
try_and_increment succeded on ctr = 0
H = e4581824b70badf0e57af789dd8cf85513d4b9814566de0e3f738439becfba33
k = a950f736af2e3ae2dbcb76795f9cbd57c671eee64ab17069f945509cd6c4a74852fe1bbc331e1bd573038ec703ca28601d861ad1e9684ec89d57bc22986acb0e
U = k*B = 5114dc4e741b7c4a28844bc585350240a51348a05f337b5fd75046d2c2423f7a
V = k*H = a6d5780c472dea1ace78795208aaa05473e501ed4f53da57e1fb13b7e80d7f59
pi = aca8ade9b7f03e2b149637629f95654c94fc9053c225ec21e5838f193af2b727b84ad849b0039ad38b41513fe5a66cdd2367737a84b488d62486bd2fb110b4801a46bfca770af98e059158ac563b690f
beta = d938b2012f2551b0e13a49568612effcbdca2aed5d1d3a13f47e180e01218916e049837bd246f66d5058e56d3413dbbbad964f5e9f160a81c9a1355dcd99b453
These three example secret keys and messages are taken from Section 7.1 of [RFC8032].
SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
In Elligator: r = 9ddd071cd5837e591a3a40c57a46701bb7f49b1b53c670d490c2766a08fa6e3d
In Elligator: w = c7b5d6239e52a473a2b57a92825e0e5de4656e349bb198de5afd6a76e5a07066
In Elligator: e = -1
H = 1c5672d919cc0a800970cd7e05cb36ed27ed354c33519948e5a9eaf89aee12b7
k = 868b56b8b3faf5fc7e276ff0a65aaa896aa927294d768d0966277d94599b7afe4a6330770da5fdc2875121e0cbecbffbd4ea5e491eb35be53fa7511d9f5a61f2
U = k*B = c4743a22340131a2323174bfc397a6585cbe0cc521bfad09f34b11dd4bcf5936
V = k*H = e309cf5272f0af2f54d9dc4a6bad6998a9d097264e17ae6fce2b25dcbdd10e8b
pi = b6b4699f87d56126c9117a7da55bd0085246f4c56dbc95d20172612e9d38e8d7ca65e573a126ed88d4e30a46f80a666854d675cf3ba81de0de043c3774f061560f55edc256a787afe701677c0f602900
beta = 5b49b554d05c0cd5a5325376b3387de59d924fd1e13ded44648ab33c21349a603f25b84ec5ed887995b33da5e3bfcb87cd2f64521c4c62cf825cffabbe5d31cc
SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
In Elligator: r = 92181bd612695e464049590eb1f9746750d6057441789c9759af8308ac77fd4a
In Elligator: w = 7ff6d8b773bfbae57b2ab9d49f9d3cb7d9af40a03d3ed3c6beaaf2d486b1fe6e
In Elligator: e = 1
H = 86725262c971bf064168bca2a87f593d425a49835bd52beb9f52ea59352d80fa
k = fd919e9d43c61203c4cd948cdaea0ad4488060db105d25b8fb4a5da2bd40e4b8330ca44a0538cc275ac7d568686660ccfd6323c805b917e91e28a4ab352b9575
U = k*B = 04b1ba4d8129f0d4cec522b0fd0dff84283401df791dcc9b93a219c51cf27324
V = k*H = ca8a97ce1947d2a0aaa280f03153388fa7aa754eedfca2b4a7ad405707599ba5
pi = ae5b66bdf04b4c010bfe32b2fc126ead2107b697634f6f7337b9bff8785ee111200095ece87dde4dbe87343f6df3b107d91798c8a7eb1245d3bb9c5aafb093358c13e6ae1111a55717e895fd15f99f07
beta = 94f4487e1b2fec954309ef1289ecb2e15043a2461ecc7b2ae7d4470607ef82eb1cfa97d84991fe4a7bfdfd715606bc27e2967a6c557cfb5875879b671740b7d8
SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
alpha = af82 (2 bytes)
x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
In Elligator: r = dcd7cda88d6798599e07216de5a48a27dcd1cde197ab39ccaf6a906ae6b25c7f
In Elligator: w = 2ceaa2c2ff3028c34f9fbe076ff99520b925f18d652285b4daad5ccc467e523b
In Elligator: e = -1
H = 9d8663faeb6ab14a239bfc652648b34f783c2e99f758c0e1b6f4f863f9419b56
k = 8f675784cdc984effc459e1054f8d386050ec400dc09d08d2372c6fe0850eaaa50defd02d965b79930dcbca5ba9222a3d99510411894e63f66bbd5d13d25db4b
U = k*B = d6f8a95a4ce86812e3e50febd9d48196b3bc5d1d9fa7b6dfa33072641b45d029
V = k*H = f77cd4ce0b49b386e80c3ce404185f93bb07463600dc14c31b0a09beaff4d592
pi = dfa2cba34b611cc8c833a6ea83b8eb1bb5e2ef2dd1b0c481bc42ff36ae7847f6ab52b976cfd5def172fa412defde270c8b8bdfbaae1c7ece17d9833b1bcf31064fff78ef493f820055b561ece45e1009
beta = 2031837f582cd17a9af9e0c7ef5a6540e3453ed894b62c293686ca3c1e319dde9d0aa489a4b59a9594fc2328bc3deff3c8a0929a369a72b1180a596e016b5ded