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diff --git a/doc/rfc/rfc9381.txt b/doc/rfc/rfc9381.txt new file mode 100644 index 0000000..e25111b --- /dev/null +++ b/doc/rfc/rfc9381.txt @@ -0,0 +1,2495 @@ + + + + +Internet Research Task Force (IRTF) S. Goldberg +Request for Comments: 9381 Boston University +Category: Informational L. Reyzin +ISSN: 2070-1721 Boston University and Algorand + D. Papadopoulos + Hong Kong University of Science and Technology + J. Včelák + NS1 + August 2023 + + + Verifiable Random Functions (VRFs) + +Abstract + + A Verifiable Random Function (VRF) is the public key version of a + keyed cryptographic hash. Only the holder of the secret key can + compute the hash, but anyone with the public key can verify the + correctness of the hash. VRFs are useful for preventing enumeration + of hash-based data structures. This document specifies VRF + constructions based on RSA and elliptic curves that are secure in the + cryptographic random oracle model. + + This document is a product of the Crypto Forum Research Group (CFRG) + in the IRTF. + +Status of This Memo + + This document is not an Internet Standards Track specification; it is + published for informational purposes. + + This document is a product of the Internet Research Task Force + (IRTF). The IRTF publishes the results of Internet-related research + and development activities. These results might not be suitable for + deployment. This RFC represents the consensus of the Crypto Forum + Research Group of the Internet Research Task Force (IRTF). Documents + approved for publication by the IRSG are not candidates for any level + of Internet Standard; see Section 2 of RFC 7841. + + Information about the current status of this document, any errata, + and how to provide feedback on it may be obtained at + https://www.rfc-editor.org/info/rfc9381. + +Copyright Notice + + Copyright (c) 2023 IETF Trust and the persons identified as the + document authors. All rights reserved. + + This document is subject to BCP 78 and the IETF Trust's Legal + Provisions Relating to IETF Documents + (https://trustee.ietf.org/license-info) in effect on the date of + publication of this document. Please review these documents + carefully, as they describe your rights and restrictions with respect + to this document. + +Table of Contents + + 1. Introduction + 1.1. Requirements + 1.2. Terminology + 2. VRF Algorithms + 3. VRF Security Properties + 3.1. Full Uniqueness + 3.2. Full Collision Resistance + 3.3. Trusted Uniqueness and Trusted Collision Resistance + 3.4. Full Pseudorandomness or Selective Pseudorandomness + 3.5. Unpredictability under Malicious Key Generation + 4. RSA Full Domain Hash VRF (RSA-FDH-VRF) + 4.1. RSA-FDH-VRF Proving + 4.2. RSA-FDH-VRF Proof to Hash + 4.3. RSA-FDH-VRF Verifying + 4.4. RSA-FDH-VRF Ciphersuites + 5. Elliptic Curve VRF (ECVRF) + 5.1. ECVRF Proving + 5.2. ECVRF Proof to Hash + 5.3. ECVRF Verifying + 5.4. ECVRF Auxiliary Functions + 5.4.1. ECVRF Encode to Curve + 5.4.2. ECVRF Nonce Generation + 5.4.3. ECVRF Challenge Generation + 5.4.4. ECVRF Decode Proof + 5.4.5. ECVRF Validate Key + 5.5. ECVRF Ciphersuites + 6. IANA Considerations + 7. Security Considerations + 7.1. Key Generation + 7.1.1. Uniqueness and Collision Resistance under Malicious Key + Generation + 7.1.2. Pseudorandomness under Malicious Key Generation + 7.1.3. Unpredictability under Malicious Key Generation + 7.2. Security Levels + 7.3. Selective vs. Full Pseudorandomness + 7.4. Proper Pseudorandom Nonce for the ECVRF + 7.5. Side-Channel Attacks + 7.6. Proofs Provide No Secrecy for the VRF Input + 7.7. Prehashing + 7.8. Hash Function Domain Separation + 7.9. Hash Function Salting + 7.10. Futureproofing + 8. References + 8.1. Normative References + 8.2. Informative References + Appendix A. Test Vectors for the RSA-FDH-VRF Ciphersuites + A.1. RSA-FDH-VRF-SHA256 + A.2. RSA-FDH-VRF-SHA384 + A.3. RSA-FDH-VRF-SHA512 + Appendix B. Test Vectors for the ECVRF Ciphersuites + B.1. ECVRF-P256-SHA256-TAI + B.2. ECVRF-P256-SHA256-SSWU + B.3. ECVRF-EDWARDS25519-SHA512-TAI + B.4. ECVRF-EDWARDS25519-SHA512-ELL2 + Contributors + Authors' Addresses + +1. Introduction + + A Verifiable Random Function (VRF) [MRV99] is the public key version + of a keyed cryptographic hash. Only the holder of the VRF secret key + can compute the hash, but anyone with the corresponding public key + can verify the correctness of the hash. + + A key application of the VRF is to provide privacy against offline + dictionary attacks (also known as enumeration attacks) on data stored + in a hash-based data structure. In this application, a Prover holds + the VRF secret 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 VRF public 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 structure. + + This document defines VRFs based on RSA and elliptic curves. The + choices of VRFs for inclusion in this document were based, in part, + on synergy with existing RFCs and commonly available implementations + of individual components that are used within the VRFs. + + The particular choice of the VRF for a given application depends on + the desired security properties, the availability of + cryptographically strong implementations, efficiency constraints, and + the trust one places in RSA and elliptic curve Diffie-Hellman + assumptions (and the trust in a particular choice of curve in the + case of elliptic curves). Differences in the security properties + provided by the different options are discussed in Sections 3 and 7. + + This document represents the consensus of the Crypto Forum Research + Group (CFRG). + +1.1. Requirements + + The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", + "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and + "OPTIONAL" in this document are to be interpreted as described in + BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all + capitals, as shown here. + +1.2. Terminology + + The following terminology is used throughout this document: + + SK: The secret key for the VRF. (Note: The secret key is also + sometimes called a "private key".) + + PK: The public key for the VRF. + + alpha or alpha_string: The input to be hashed by the VRF. + + beta or beta_string: The VRF hash output. + + pi or pi_string: The VRF proof. + + Prover: Holds the VRF secret key SK and public key PK. + + Verifier: Holds the VRF public key PK. + + Adversary: Potential attacker; often used to define a security + property. + + Malicious (or adversarial): Performed by an adversary. + +2. VRF Algorithms + + A VRF comes with a key generation algorithm that generates a VRF + public key PK and secret key SK. + + The Prover hashes an input alpha using the VRF secret key SK to + obtain a VRF hash output beta: + + beta = VRF_hash(SK, alpha) + + The VRF_hash algorithm is deterministic, in the sense that it always + produces the same output beta, given the same pair of inputs (SK, + alpha). + + The Prover also uses the secret key SK to construct a proof pi that + beta is the correct hash output: + + pi = VRF_prove(SK, alpha) + + The VRFs defined in this document allow anyone to deterministically + obtain the VRF hash output beta directly from the proof value pi by + using the function VRF_proof_to_hash: + + beta = VRF_proof_to_hash(pi) + + Thus, for the VRFs defined in this document, VRF_hash is defined as + + VRF_hash(SK, alpha) = VRF_proof_to_hash(VRF_prove(SK, alpha)), + + and therefore 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 VRFs defined in this document also come with an algorithm + + VRF_verify(PK, alpha, pi) + + that outputs ("VALID", beta = VRF_proof_to_hash(pi)) if pi is valid, + and "INVALID" otherwise. + +3. VRF Security Properties + + VRFs are designed to ensure the following security properties: + uniqueness (full or trusted), collision resistance (full or trusted), + and pseudorandomness (full or selective). Some are designed to also + ensure unpredictability under malicious key generation. We now + describe these properties. + +3.1. Full Uniqueness + + Uniqueness means that, for any fixed VRF public key and for any input + alpha, it is infeasible to find proofs for more than one VRF output + beta. + + More precisely, "full uniqueness" means that an adversary cannot find + + * 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. + +3.2. Full Collision Resistance + + Like cryptographic hash functions, VRFs are collision resistant. + Collision resistance means that it is infeasible to find two + different inputs alpha1 and alpha2 with the same output beta. + + More precisely, "full collision resistance" means that an adversary + cannot find + + * a VRF public key PK, + + * two VRF inputs alpha1 and alpha2 that are not equal to each other, + and + + * two proofs pi1 and pi2 + + such that + + * VRF_verify(PK, alpha1, pi1) outputs ("VALID", beta1), + + * VRF_verify(PK, alpha2, pi2) outputs ("VALID", beta2), and + + * beta1 is equal to beta2. + +3.3. Trusted Uniqueness and Trusted Collision Resistance + + Full uniqueness and full collision resistance hold even if the VRF + keys are generated maliciously. For some applications, it is + sufficient for a VRF to possess weaker security properties than full + uniqueness and full collision resistance. These properties are + called "trusted uniqueness" and "trusted collision resistance"; they + are the same as full uniqueness and full collision resistance, + respectively, but are not guaranteed to hold if the adversary gets to + choose the VRF public key PK. Instead, they are guaranteed to hold + only if the VRF keys PK and SK are generated as specified by the VRF + key generation algorithm and then given to the adversary. In other + words, they are guaranteed to hold even if the adversary has + knowledge of SK and PK but are not guaranteed to hold if the + adversary has the ability to choose SK and PK. + + As further discussed in Section 7.1.1, some of the VRFs specified in + this document satisfy only trusted uniqueness and trusted collision + resistance. VRFs in this document that satisfy only trusted + uniqueness and trusted collision resistance MUST NOT be used in + applications that need protection against adversarial VRF key + generation. + +3.4. Full Pseudorandomness or Selective Pseudorandomness + + Pseudorandomness ensures that when someone who does not know SK sees + a VRF hash output beta without its corresponding VRF proof pi, beta + is indistinguishable from a random value. + + More precisely, suppose that the public and secret VRF keys (PK, SK) + were generated correctly. 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 adversary who does not know the VRF secret key + SK. This holds even if the adversary sees VRF hash outputs beta' and + proofs pi' for multiple other inputs alpha' (and even if those other + inputs alpha' are chosen by the adversary). + + The "full pseudorandomness" security property holds even against an + adversary who 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 that + suffices in many applications. This security property holds against + an adversary who chooses the target VRF input alpha first, before it + learns the VRF public key PK and obtains VRF outputs beta' and proofs + pi' on other inputs alpha' of its choice. + + As further discussed in Section 7.3, the VRFs specified in this + document satisfy both full pseudorandomness and selective + pseudorandomness, but their quantitative security against the + selective pseudorandomness attack is stronger. + + It is important to remember that the VRF output beta is always + distinguishable from random by the Prover or by any other party that + knows the VRF secret key SK. Such a party can easily distinguish + beta from a random value by comparing beta to the result of + VRF_hash(SK, alpha). In particular, if the key is generated + maliciously, even parties other than the Prover may know SK, and thus + pseudorandomness cannot be guaranteed. + + Similarly, the VRF output beta is always distinguishable from random + by any party that knows a valid VRF proof pi corresponding to the VRF + input alpha, even if this party does not know the VRF secret key SK. + Such a party can easily distinguish beta from a random value by + checking to see whether VRF_verify(PK, alpha, pi) returns ("VALID", + beta). + + Additionally, the VRF output beta may be distinguishable from random + if VRF key generation was not done correctly (for example, if VRF + keys were generated with bad randomness). + +3.5. Unpredictability under Malicious Key Generation + + As explained in Section 3.4, pseudorandomness cannot hold against + malicious key generation. 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 and are + therefore not pseudorandom. + + There is, however, a different type of unpredictability that is + desirable in certain VRF applications (such as leader selection in + the consensus protocols of [GHMVZ17] and [DGKR18]), called + "unpredictability under malicious key generation". 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 uniformly random, no matter how the VRF keys are generated. + + A formal definition of this property appears in Section 3.2 of + [DGKR18]. As further discussed in Section 7.1.3, only some of the + VRFs specified in this document satisfy this property. + +4. RSA Full Domain Hash VRF (RSA-FDH-VRF) + + The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that, for + suitable key lengths, satisfies the "trusted uniqueness", "trusted + collision resistance", and "full pseudorandomness" properties defined + in Section 3, as further discussed in Section 7. Its security + follows from the standard RSA assumption in the random oracle model. + Formal security proofs are provided in [PWHVNRG17]. + + The VRF computes the proof pi as a deterministic RSA signature on + input alpha using the RSA Full Domain Hashing algorithm [RFC8017] + parameterized 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 the RSA-FDH-VRF MUST satisfy the conditions + specified in Section 3 of [RFC8017]. + + In this section, the notation from [RFC8017] is used. + + Parameters used: + + (n, e): RSA public key + + K: RSA private key (its representation is implementation + dependent) + + k: length, in octets, of the RSA modulus n (k must be less than + 2^32) + + Fixed options (specified in Section 4.4): + + Hash: cryptographic hash function + + hLen: output length, in octets, of hash function Hash + + suite_string: an octet string specifying the RSA-FDH-VRF + ciphersuite, which determines the above options + + Primitives used: + + I2OSP: Conversion of a non-negative integer to an octet string as + defined in Section 4.1 of [RFC8017] (given an integer and a + length (in octets), produces a big-endian representation of the + integer, zero-padded to the desired length) + + OS2IP: Conversion of an octet string to a non-negative integer as + defined in Section 4.2 of [RFC8017] (given a big-endian + encoding of an integer, produces the integer) + + RSASP1: RSA signature primitive as defined in Section 5.2.1 of + [RFC8017] (given a private key and an input, raises the input + to the private RSA exponent modulo n) + + RSAVP1: RSA verification primitive as defined in Section 5.2.2 of + [RFC8017] (given a public key and an input, raises the input to + the public RSA exponent modulo n) + + MGF1: Mask generation function based on the hash function Hash as + defined in Appendix B.2.1 of [RFC8017] (given an input, + produces a random-oracle-like output of desired length) + + ||: octet string concatenation + +4.1. RSA-FDH-VRF Proving + + RSAFDHVRF_prove(K, alpha_string[, MGF_salt]) + + Input: + + K: RSA private key + + alpha_string: VRF hash input, an octet string + + Optional input: + + MGF_salt: a public octet string used as a hash function salt; + this input is not used when MGF_salt is specified as part of + the ciphersuite + + Output: + + pi_string: proof, an octet string of length k + + Steps: + + 1. mgf_domain_separator = 0x01 + + 2. EM = MGF1(suite_string || mgf_domain_separator || MGF_salt || + alpha_string, k - 1) + + 3. m = OS2IP(EM) + + 4. s = RSASP1(K, m) + + 5. pi_string = I2OSP(s, k) + + 6. Output pi_string + +4.2. RSA-FDH-VRF Proof to Hash + + RSAFDHVRF_proof_to_hash(pi_string) + + Input: + + pi_string: proof, an octet string of length k + + Output: + + beta_string: VRF hash output, an octet string of length hLen + + Important note: + + RSAFDHVRF_proof_to_hash should be run only on a pi_string value + that is known to have been produced by RSAFDHVRF_prove, or from + within RSAFDHVRF_verify as specified in Section 4.3. + + Steps: + + 1. proof_to_hash_domain_separator = 0x02 + + 2. beta_string = Hash(suite_string || + proof_to_hash_domain_separator || pi_string) + + 3. Output beta_string + +4.3. RSA-FDH-VRF Verifying + + RSAFDHVRF_verify((n, e), alpha_string, pi_string[, MGF_salt]) + + Input: + + (n, e): RSA public key + + alpha_string: VRF hash input, an octet string + + pi_string: proof to be verified, an octet string of length k + + Optional input: + + MGF_salt: a public octet string used as a hash function salt; + this input is not used when MGF_salt is specified as part of + the ciphersuite + + Output: + + ("VALID", beta_string), where beta_string is the VRF hash output, + an octet string of length hLen, or + + "INVALID" + + Steps: + + 1. s = OS2IP(pi_string) + + 2. m = RSAVP1((n, e), s); if RSAVP1 returns "signature + representative out of range", output "INVALID" and stop + + 3. mgf_domain_separator = 0x01 + + 4. EM' = MGF1(suite_string || mgf_domain_separator || MGF_salt || + alpha_string, k - 1) + + 5. m' = OS2IP(EM') + + 6. If m and m' are equal, output ("VALID", + RSAFDHVRF_proof_to_hash(pi_string)); else output "INVALID" + +4.4. RSA-FDH-VRF Ciphersuites + + This document defines RSA-FDH-VRF-SHA256 as follows: + + * suite_string = 0x01. + + * The hash function Hash is SHA-256 as specified in [RFC6234], with + hLen = 32. + + * MGF_salt = I2OSP(k, 4) || I2OSP(n, k). + + This document defines RSA-FDH-VRF-SHA384 as follows: + + * suite_string = 0x02. + + * The hash function Hash is SHA-384 as specified in [RFC6234], with + hLen = 48. + + * MGF_salt = I2OSP(k, 4) || I2OSP(n, k). + + This document defines RSA-FDH-VRF-SHA512 as follows: + + * suite_string = 0x03. + + * The hash function Hash is SHA-512 as specified in [RFC6234], with + hLen = 64. + + * MGF_salt = I2OSP(k, 4) || I2OSP(n, k). + +5. Elliptic Curve VRF (ECVRF) + + The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that, + for suitable parameter choices, satisfies the "full uniqueness", + "trusted collision resistance", and "full pseudorandomness" + properties defined in Section 3. If the validate_key parameter given + to ECVRF_verify is TRUE, then the ECVRF additionally satisfies "full + collision resistance" and "unpredictability under malicious key + generation". See Section 7 for further discussion. Formal security + proofs are provided in [PWHVNRG17]. + + Notation used: + + Elliptic curve operations are written in additive notation, with + P+Q denoting point addition and x*P denoting scalar multiplication + of a point P by a scalar x + + x^y: x raised to the power y + + x*y: x multiplied by y + + s || t: concatenation of octet strings s and t + + 0xMN (where M and N are hexadecimal digits): a single octet with + value M*16+N; equivalently, int_to_string(M*16+N, 1), where + int_to_string is as defined below + + Fixed options (specified in Section 5.5): + + F: finite field + + fLen: length, in octets, of an element in F encoded as an octet + string + + E: elliptic curve (EC) defined over F + + ptLen: length, in octets, of a point on E encoded as an octet + string + + G: subgroup of E of large prime order + + q: prime order of group G + + qLen: length of q, in octets, i.e., the smallest integer such + that 2^(8qLen) > q + + cLen: length, in octets, of a challenge value used by the VRF + (note that in the typical case, cLen is qLen/2 or close to it) + + cofactor: number of points on E divided by q + + B: generator of group G + + Hash: cryptographic hash function + + hLen: output length, in octets, of Hash (hLen must be at least + cLen; in the typical case, it is at least qLen) + + ECVRF_encode_to_curve: a function that hashes strings to points + on E + + ECVRF_nonce_generation: a function that derives a pseudorandom + nonce from SK and the input as part of ECVRF proving + + suite_string: an octet string specifying the ECVRF ciphersuite, + which determines the above options as well as type conversions + and parameter generation + + Type conversions (specified in Section 5.5): + + int_to_string(a, len): conversion of non-negative integer a to + octet string of length len + + string_to_int(a_string): conversion of an octet string a_string + to a non-negative integer + + point_to_string: conversion of a point on E to a ptLen-octet + string + + string_to_point: conversion of a ptLen-octet string to a point on + E. string_to_point returns "INVALID" if the octet string does + not convert to a valid EC point on the curve E + + Note that with certain software libraries (for big integer and + elliptic curve arithmetic), the int_to_string and point_to_string + conversions are not needed when the libraries encode integers and + EC points in the same way as required by the ciphersuites. For + example, in some implementations, EC point operations will take + octet strings as inputs and produce octet strings as outputs, + without introducing a separate elliptic curve point type. + + Parameters used (the generation of these parameters is specified + in Section 5.5): + + SK: VRF secret key + + x: VRF secret scalar, an integer. Note: Depending on the + ciphersuite used, the VRF secret scalar may be equal to SK; + else it is derived from SK + + Y = x*B: VRF public key, a point on E + + PK_string = point_to_string(Y): VRF public key represented as an + octet string + + encode_to_curve_salt: a public value used as a hash function salt + +5.1. ECVRF Proving + + ECVRF_prove(SK, alpha_string[, encode_to_curve_salt]) + + Input: + + SK: VRF secret key + + alpha_string: input alpha, an octet string + + Optional input: + + encode_to_curve_salt: a public salt value, an octet string; this + input is not used when encode_to_curve_salt is specified as + part of the ciphersuite + + Output: + + pi_string: VRF proof, an octet string of length ptLen+cLen+qLen + + Steps: + + 1. Use SK to derive the VRF secret scalar x and the VRF public key Y + = x*B + + (this derivation depends on the ciphersuite, as per Section 5.5; + these values can be cached, for example, after key generation, + and need not be rederived each time) + + 2. H = ECVRF_encode_to_curve(encode_to_curve_salt, alpha_string) + (see Section 5.4.1) + + 3. h_string = point_to_string(H) + + 4. Gamma = x*H + + 5. k = ECVRF_nonce_generation(SK, h_string) (see Section 5.4.2) + + 6. c = ECVRF_challenge_generation(Y, H, Gamma, k*B, k*H) (see + Section 5.4.3) + + 7. s = (k + c*x) mod q + + 8. pi_string = point_to_string(Gamma) || int_to_string(c, cLen) || + int_to_string(s, qLen) + + 9. Output pi_string + +5.2. ECVRF Proof to Hash + + ECVRF_proof_to_hash(pi_string) + + Input: + + pi_string: VRF proof, an octet string of length ptLen+cLen+qLen + + Output: + + "INVALID", or + + beta_string: VRF hash output, an octet string of length hLen + + Important note: + + ECVRF_proof_to_hash should be run only on a pi_string value that + is known to have been produced by ECVRF_prove, or from within + ECVRF_verify as specified in Section 5.3. + + Steps: + + 1. D = ECVRF_decode_proof(pi_string) (see Section 5.4.4) + + 2. If D is "INVALID", output "INVALID" and stop + + 3. (Gamma, c, s) = D + + 4. proof_to_hash_domain_separator_front = 0x03 + + 5. proof_to_hash_domain_separator_back = 0x00 + + 6. beta_string = Hash(suite_string || + proof_to_hash_domain_separator_front || point_to_string(cofactor + * Gamma) || proof_to_hash_domain_separator_back) + + 7. Output beta_string + +5.3. ECVRF Verifying + + ECVRF_verify(PK_string, alpha_string, pi_string[, + encode_to_curve_salt, validate_key]) + + Input: + + PK_string: public key, an octet string + + alpha_string: VRF input, an octet string + + pi_string: VRF proof, an octet string of length ptLen+cLen+qLen + + Optional input: + + encode_to_curve_salt: a public salt value, an octet string; this + input is not used when encode_to_curve_salt is specified as + part of the ciphersuite + + validate_key: a boolean. An implementation MAY support only the + option of validate_key = TRUE, or only the option of + validate_key = FALSE, in which case this input is not needed. + If an implementation supports only one option, it MUST specify + which option it supports + + Output: + + ("VALID", beta_string), where beta_string is the VRF hash output, + an octet string of length hLen, or + + "INVALID" + + Steps: + + 1. Y = string_to_point(PK_string) + + 2. If Y is "INVALID", output "INVALID" and stop + + 3. If validate_key, run ECVRF_validate_key(Y) (Section 5.4.5); if + it outputs "INVALID", output "INVALID" and stop + + 4. D = ECVRF_decode_proof(pi_string) (see Section 5.4.4) + + 5. If D is "INVALID", output "INVALID" and stop + + 6. (Gamma, c, s) = D + + 7. H = ECVRF_encode_to_curve(encode_to_curve_salt, alpha_string) + (see Section 5.4.1) + + 8. U = s*B - c*Y + + 9. V = s*H - c*Gamma + + 10. c' = ECVRF_challenge_generation(Y, H, Gamma, U, V) (see + Section 5.4.3) + + 11. If c and c' are equal, output ("VALID", + ECVRF_proof_to_hash(pi_string)); else output "INVALID" + + Note that the first three steps need to be performed only once for a + given public key. + +5.4. ECVRF Auxiliary Functions + +5.4.1. ECVRF Encode to Curve + + The ECVRF_encode_to_curve algorithm takes a public salt (see + Section 7.9) and 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 for proving and verifying. See Section 7.7 for further + discussion. + + This section specifies a number of such algorithms; these algorithms + are not compatible with each other and are intended for use with the + various ciphersuites specified in Section 5.5. + + Input: + + encode_to_curve_salt: public salt value, an octet string + + alpha_string: value to be hashed, an octet string + + Output: + + H: hashed value, a point in G + +5.4.1.1. ECVRF_encode_to_curve_try_and_increment + + The ECVRF_encode_to_curve_try_and_increment(encode_to_curve_salt, + alpha_string) algorithm implements ECVRF_encode_to_curve in a simple + and generic way that works for any elliptic curve. To use this + algorithm, hLen MUST be at least fLen. + + 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 algorithm's running time depends on + alpha_string, this algorithm SHOULD be avoided in applications where + it is important that the VRF input alpha remain secret. + + ECVRF_encode_to_curve_try_and_increment(encode_to_curve_salt, + alpha_string) + + Fixed option (specified in Section 5.5): + + interpret_hash_value_as_a_point: a function that attempts to + convert a cryptographic hash value to a point on E; may output + "INVALID" + + Steps: + + 1. ctr = 0 + + 2. encode_to_curve_domain_separator_front = 0x01 + + 3. encode_to_curve_domain_separator_back = 0x00 + + 4. H = "INVALID" + + 5. While H is "INVALID" or H is the identity element of the elliptic + curve group: + + a. ctr_string = int_to_string(ctr, 1) + + b. hash_string = Hash(suite_string || + encode_to_curve_domain_separator_front || + encode_to_curve_salt || alpha_string || ctr_string || + encode_to_curve_domain_separator_back) + + c. H = interpret_hash_value_as_a_point(hash_string) + + d. If H is not "INVALID" and cofactor > 1, set H = cofactor * H + + e. ctr = ctr + 1 + + 6. Output H + + Note that even though the loop is infinite as written and + int_to_string(ctr, 1) may fail when ctr reaches 256, each of the + options for the interpret_hash_value_as_a_point function specified in + Section 5.5 will succeed on roughly half hash_string values. Thus, + the loop is expected to stop after two iterations, and ctr is + overwhelmingly unlikely (probability about 2^-256) to reach 256. + +5.4.1.2. ECVRF_encode_to_curve_h2c_suite + + The ECVRF_encode_to_curve_h2c_suite(encode_to_curve_salt, + alpha_string) algorithm implements ECVRF_encode_to_curve using one of + the several hash-to-curve options defined in [RFC9380]. The specific + choice of the hash-to-curve option (called the Suite ID in [RFC9380]) + is given by the h2c_suite_ID_string parameter. + + ECVRF_encode_to_curve_h2c_suite(encode_to_curve_salt, alpha_string) + + Fixed option (specified in Section 5.5): + + h2c_suite_ID_string: a hash-to-curve Suite ID, encoded in ASCII + (see discussion below) + + Steps: + + 1. string_to_be_hashed = encode_to_curve_salt || alpha_string + + 2. H = encode(string_to_be_hashed) + + (the encode function is discussed below) + + 3. Output H + + The encode function is provided by the hash-to-curve suite (as + specified in Section 8 of [RFC9380]) whose ID is h2c_suite_ID_string. + The domain separation tag DST, a parameter in the hash-to-curve + suite, SHALL be set to + + "ECVRF_" || h2c_suite_ID_string || suite_string + + where "ECVRF_" is represented as a 6-byte ASCII encoding (in + hexadecimal, octets 45 43 56 52 46 5F). + +5.4.2. ECVRF Nonce Generation + + The following algorithms generate the nonce value k in a + deterministic pseudorandom fashion. This section specifies a number + of such algorithms; these algorithms are not compatible with each + other. The choice of a particular algorithm from the options + specified in this section depends on the ciphersuite, as specified in + Section 5.5. + +5.4.2.1. ECVRF Nonce Generation from RFC 6979 + + ECVRF_nonce_generation_RFC6979(SK, h_string) + + Input: + + SK: an ECVRF secret key + + h_string: an octet string + + Output: + + k: an integer nonce between 1 and q-1 + + The ECVRF_nonce_generation function is implemented according to the + process specified in Section 3.2 of [RFC6979], where + + * Input m is set equal to h_string. + + * The "suitable for DSA or ECDSA" check in Step h.3 is omitted. + + * The hash function H is Hash, and its output length hlen (in bits) + is set as hLen*8 (note that hlen is not to be confused with hLen, + which is used in this document to represent the length of the + output of Hash in octets). + + * The secret key x is set equal to the VRF secret scalar x. + + * The prime q is the same as in this specification. + + * qlen is the binary length of q, i.e., the smallest integer such + that 2^qlen > q (this qlen is not to be confused with qLen, which + is used in this document to represent the length of q in octets). + + * All the other values and primitives are as defined in [RFC6979]. + +5.4.2.2. ECVRF Nonce Generation from RFC 8032 + + The following is derived from Steps 2 and 3 in Section 5.1.6 of + [RFC8032]. To use this algorithm, hLen MUST be at least 64. + + ECVRF_nonce_generation_RFC8032(SK, h_string) + + Input: + + SK: an ECVRF secret key + + h_string: an octet string + + Output: + + k: an integer nonce between 0 and q-1 + + Steps: + + 1. hashed_sk_string = Hash(SK) + + 2. truncated_hashed_sk_string = + hashed_sk_string[32]...hashed_sk_string[63] + + 3. k_string = Hash(truncated_hashed_sk_string || h_string) + + 4. k = string_to_int(k_string) mod q + +5.4.3. ECVRF Challenge Generation + + ECVRF_challenge_generation(P1, P2, P3, P4, P5) + + Input: + + P1, P2, P3, P4, P5: EC points + + Output: + + c: challenge value, an integer between 0 and 2^(8*cLen)-1 + + Steps: + + 1. challenge_generation_domain_separator_front = 0x02 + + 2. Initialize str = suite_string || + challenge_generation_domain_separator_front + + 3. For PJ in [P1, P2, P3, P4, P5]: + + str = str || point_to_string(PJ) + + 4. challenge_generation_domain_separator_back = 0x00 + + 5. str = str || challenge_generation_domain_separator_back + + 6. c_string = Hash(str) + + 7. truncated_c_string = c_string[0]...c_string[cLen-1] + + 8. c = string_to_int(truncated_c_string) + + 9. Output c + +5.4.4. ECVRF Decode Proof + + ECVRF_decode_proof(pi_string) + + Input: + + pi_string: VRF proof, an octet string (ptLen+cLen+qLen octets) + + Output: + + "INVALID", or + + Gamma: a point on E + + c: an integer between 0 and 2^(8*cLen)-1 + + s: an integer between 0 and q-1 + + Steps: + + 1. gamma_string = pi_string[0]...pi_string[ptLen-1] + + 2. c_string = pi_string[ptLen]...pi_string[ptLen+cLen-1] + + 3. s_string = pi_string[ptLen+cLen]...pi_string[ptLen+cLen+qLen-1] + + 4. Gamma = string_to_point(gamma_string) + + 5. If Gamma = "INVALID", output "INVALID" and stop + + 6. c = string_to_int(c_string) + + 7. s = string_to_int(s_string) + + 8. If s >= q, output "INVALID" and stop + + 9. Output Gamma, c, and s + +5.4.5. ECVRF Validate Key + + ECVRF_validate_key(Y) + + Input: + + Y: public key, a point on E + + Output: + + "VALID" or "INVALID" + + Important note: + + The public key Y provided as input to this procedure MUST be a + valid point on E. + + Steps: + + 1. Let Y' = cofactor*Y + + 2. If Y' is the identity element of the elliptic curve group, output + "INVALID" and stop + + 3. Output "VALID" + + Note that if the cofactor = 1, then Step 1 simply sets Y'=Y. In + particular, for the P-256 curve, ECVRF_validate_key simply ensures + that Y is not the point at infinity. + + Any algorithm with identical input-output behavior MAY be used in + place of the above steps. For example, if the total number of Y + values that could cause Step 2 to output "INVALID" is small, it may + be more efficient to simply check Y against a fixed list of such + values. For example, the following algorithm MAY be used for the + edwards25519 curve: + + 1. PK_string = point_to_string(Y) + + 2. oneTwentySeven_string = 0x7F + + 3. y_string[31] = y_string[31] & oneTwentySeven_string + + (this step clears the high-order bit of octet 31) + + 4. bad_pk[0] = int_to_string(0, 32) + + 5. bad_pk[1] = int_to_string(1, 32) + + 6. bad_y2 = 2707385501144840649318225287225658788936804267575313519 + 463743609750303402022 + + 7. bad_pk[2] = int_to_string(bad_y2, 32) + + 8. bad_pk[3] = int_to_string(p-bad_y2, 32) + + 9. bad_pk[4] = int_to_string(p-1, 32) + + 10. bad_pk[5] = int_to_string(p, 32) + + 11. bad_pk[6] = int_to_string(p+1, 32) + + 12. If y_string is in the list [bad_pk[0],...,bad_pk[6]], output + "INVALID" and stop + + 13. Output "VALID" + + (This algorithm works for the following reason. Note that there are + eight bad points -- namely, the points whose order is 1, 2, 4, or 8 + -- on the edwards25519 curve. Their y-coordinates happen to be 0 + (two points of order 4), 1 (one point of order 1), bad_y2 (two points + of order 8), p-bad_y2 (two points of order 8), and p-1 (one point of + order 2). They can be obtained by converting the points specified in + [X25519] to Edwards coordinates. Thus, bad_pk[0] (of order 4), + bad_pk[2] (of order 8), and bad_pk[3] (of order 8) each match two bad + points, depending on the sign of the x-coordinate. This sign is + cleared in Step 3 in order to make sure that it does not affect the + comparison. bad_pk[1] (of order 1) and bad_pk[4] (of order 2) each + match one bad point, because the x-coordinate is 0 for these two + points. Note that the first five list elements cover the eight bad + points. However, to cover the case when the y-coordinate of the + public key Y has not been modular reduced by p, the list also + includes bad_pk[5] and bad_pk[6], which are simply bad_pk[0] and + bad_pk[1] shifted by p. There is no need to shift the other bad_pk + values by p (or any bad_pk values by a larger multiple of p), because + their y-coordinates would exceed 2^255, and the algorithm ensures + that y_string corresponds to an integer less than 2^255 in Step 3.) + +5.5. ECVRF Ciphersuites + + This document defines ECVRF-P256-SHA256-TAI as follows: + + * suite_string = 0x01. + + * The EC group G is the NIST P-256 elliptic curve, with the finite + field and curve parameters as specified in Section 3.2.1.3 of + [SP-800-186] and Section 2.6 of [RFC5114]. For this group, fLen = + qLen = 32 and cofactor = 1. + + * cLen = 16. + + * The key pair generation primitive is specified in Section 3.2.1 of + [SECG1] (q, B, SK, and Y in this document correspond to n, G, d, + and Q in Section 3.2.1 of [SECG1]). In this ciphersuite, the + secret scalar x is equal to the secret key SK. + + * encode_to_curve_salt = PK_string. + + * The ECVRF_nonce_generation function is as specified in + Section 5.4.2.1. + + * The int_to_string function is the I2OSP function specified in + Section 4.1 of [RFC8017]. (This is big-endian representation.) + + * The string_to_int function is the OS2IP function specified in + Section 4.2 of [RFC8017]. (This is big-endian representation.) + + * The point_to_string function converts a point on E to an octet + string according to the encoding specified in Section 2.3.3 of + [SECG1] with point compression on. This implies that ptLen = fLen + + 1 = 33. (Note that certain software implementations do not + introduce a separate elliptic curve point type and instead + directly treat the EC point as an octet string per the above + encoding. When using such an implementation, the point_to_string + function can be treated as the identity function.) + + * The string_to_point function converts an octet string to a point + on E according to the encoding specified in Section 2.3.4 of + [SECG1]. This function MUST output "INVALID" if the octet string + does not decode to a point on the curve E. + + * The hash function Hash is SHA-256 as specified in [RFC6234], with + hLen = 32. + + * The ECVRF_encode_to_curve function is as specified in + Section 5.4.1.1, with interpret_hash_value_as_a_point(s) = + string_to_point(0x02 || s). + + This document defines ECVRF-P256-SHA256-SSWU as identical to ECVRF- + P256-SHA256-TAI, except that + + * suite_string = 0x02. + + * The ECVRF_encode_to_curve function is as specified in + Section 5.4.1.2, with h2c_suite_ID_string = P256_XMD:SHA- + 256_SSWU_NU_ (the suite is defined in Section 8.2 of [RFC9380]). + + This document defines ECVRF-EDWARDS25519-SHA512-TAI as follows: + + * suite_string = 0x03. + + * The EC group G is the edwards25519 elliptic curve, with the finite + field and curve parameters as defined in Table 1 in Section 5.1 of + [RFC8032]. For this group, fLen = qLen = 32 and cofactor = 8. + + * cLen = 16. + + * The secret key and generation of the secret scalar and the public + key are specified in Section 5.1.5 of [RFC8032]. + + * encode_to_curve_salt = PK_string. + + * The ECVRF_nonce_generation function is as specified in + Section 5.4.2.2. + + * The int_to_string function is implemented as specified in the + first paragraph of Section 5.1.2 of [RFC8032]. (This is little- + endian representation.) + + * The string_to_int function interprets the string as an integer in + little-endian representation. + + * The point_to_string function converts a point on E to an octet + string according to the encoding specified in Section 5.1.2 of + [RFC8032]. This implies that ptLen = fLen = 32. (Note that + certain software implementations do not introduce a separate + elliptic curve point type and instead directly treat the EC point + as an octet string per the above encoding. When using such an + implementation, the point_to_string function can be treated as the + identity function.) + + * The string_to_point function converts an octet string to a point + on E according to the encoding specified in Section 5.1.3 of + [RFC8032]. This function MUST output "INVALID" if the octet + string does not decode to a point on the curve E. + + * The hash function Hash is SHA-512 as specified in [RFC6234], with + hLen = 64. + + * The ECVRF_encode_to_curve function is as specified in + Section 5.4.1.1, with interpret_hash_value_as_a_point(s) = + string_to_point(s[0]...s[31]). + + This document defines ECVRF-EDWARDS25519-SHA512-ELL2 as identical to + ECVRF-EDWARDS25519-SHA512-TAI, except that + + * suite_string = 0x04. + + * The ECVRF_encode_to_curve function is as specified in + Section 5.4.1.2, with h2c_suite_ID_string = edwards25519_XMD:SHA- + 512_ELL2_NU_ (the suite is defined in Section 8.5 of [RFC9380]). + +6. IANA Considerations + + This document has no IANA actions. + +7. Security Considerations + +7.1. Key Generation + + Implementations of the VRFs defined in this document MUST ensure that + they generate VRF keys correctly and use good randomness. However, + in some applications, keys may be generated by an adversary who does + not necessarily implement this document. We now discuss the + implications of this possibility. + +7.1.1. Uniqueness and Collision Resistance under Malicious Key + Generation + + See Section 3 for definitions of uniqueness and collision resistance + properties. + + The RSA-FDH-VRF satisfies only the "trusted" variants of uniqueness + and collision resistance. Thus, for the RSA-FDH-VRF, uniqueness and + collision resistance may not hold if the keys are generated + adversarially (specifically, if the RSA function specified in the + public key is not bijective because the modulus n or the exponent e + are chosen without complying with [RFC8017]); thus, the RSA-FDH-VRF + as defined in this document does not have "full uniqueness" and "full + collision resistance". Therefore, if malicious key generation is a + concern, the RSA-FDH-VRF has to be enhanced by additional + cryptographic checks (such as zero-knowledge proofs) to ensure that + its public key has the right form. These enhancements are left for + future specifications. + + For the ECVRF, the Verifier MUST obtain E and B from a trusted + source, such as a ciphersuite specification, rather than from the + Prover. If the Verifier does so, then the ECVRF satisfies "full + uniqueness", ensuring uniqueness even under malicious key generation. + The ECVRF also satisfies "trusted collision resistance". It + additionally satisfies "full collision resistance" if the + validate_key parameter given to ECVRF_verify is TRUE. This setting + of ECVRF_verify ensures collision resistance under malicious key + generation. + +7.1.2. Pseudorandomness under Malicious Key Generation + + Without good randomness, the "pseudorandomness" properties of the VRF + (defined in Section 3.4) 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. + +7.1.3. Unpredictability under Malicious Key Generation + + Unpredictability under malicious key generation (defined in + Section 3.5) does not hold for the RSA-FDH-VRF. (Specifically, the + VRF output may be predictable if the RSA function specified in the + public key is far from bijective because the modulus n or the + exponent e are chosen without complying with [RFC8017].) If + unpredictability under malicious key generation is desired, the RSA- + FDH-VRF has to be enhanced by additional cryptographic checks (such + as zero-knowledge proofs) to ensure that its public key has the right + form. These enhancements are left for future specifications. + + Unpredictability under malicious key generation holds for the ECVRF + if the validate_key parameter given to ECVRF_verify is TRUE. + +7.2. Security Levels + + As shown in [PWHVNRG17], the RSA-FDH-VRF satisfies the trusted + uniqueness property unconditionally. The security level of the RSA- + FDH-VRF, measured in bits, for the other two properties is as follows + (in the random oracle model for the functions MGF1 and Hash): + + For trusted collision resistance: approximately 8*min(k/2, hLen/2) + (as shown in [PWHVNRG17]). + + For selective pseudorandomness: approximately as strong as the + security, in bits, of the RSA problem for the key (n, e) (as shown + in [GNPRVZ15]). + + As shown in [PWHVNRG17], the security level of the ECVRF, measured in + bits, is as follows (in the random oracle model for the functions + Hash and ECVRF_encode_to_curve): + + For uniqueness (both trusted and full): approximately 8*min(qLen, + cLen). + + For collision resistance (trusted or full, depending on whether + validation is performed as explained in Section 7.1.1): + approximately 8*min(qLen/2, hLen/2). + + For selective pseudorandomness: approximately as strong as the + security, in bits, of the decisional Diffie-Hellman problem in the + group G (which is at most 8*qLen/2). + + See Section 3 for the definitions of these security properties and + Section 7.3 for the discussion of full pseudorandomness. + +7.3. Selective vs. Full Pseudorandomness + + [PWHVNRG17] presents cryptographic reductions to an underlying hard + problem (namely, the RSA problem for the RSA-FDH-VRF and the + decisional Diffie-Hellman problem for the ECVRF) to prove that the + VRFs specified in this document possess not only selective + pseudorandomness but also full pseudorandomness (see Section 3.4 for + an explanation of these notions). However, the cryptographic + reductions are tighter for selective pseudorandomness than for full + pseudorandomness. Specifically, the approximate provable security + level, measured in bits, for full pseudorandomness may be obtained + from the provable security level for selective pseudorandomness + (given in Section 7.2) by subtracting the binary logarithm of the + number of proofs produced for a given secret key. This holds for + both the RSA-FDH-VRF and the ECVRF. + + While no known attacks against full pseudorandomness are stronger + than similar attacks against selective pseudorandomness, some + applications may be concerned about tightness of cryptographic + reductions to ensure specific levels of provable security. Such + applications may consider the following three options: + + * They may limit the number of proofs produced for a given secret + key, to reduce the loss in the provable security level. + + * They may work to ensure that selective pseudorandomness is + sufficient for the application. That is, they may design the + application such that pseudorandomness of outputs matters only for + inputs that are chosen independently of the VRF key. + + * They may increase security parameters to make up for lossy + security reductions. For the RSA-FDH-VRF, this means increasing + the RSA key length. For the ECVRF, this means increasing the + cryptographic strength of the EC group G by specifying a new + ciphersuite. + +7.4. Proper Pseudorandom Nonce for the ECVRF + + The security of the ECVRF defined in this document relies on the fact + that the nonce k used in the ECVRF_prove algorithm is chosen + uniformly and pseudorandomly modulo q and is unknown to the + adversary. Otherwise, an adversary may be able to recover the VRF + secret scalar x (and thus break pseudorandomness of the VRF) after + observing several valid VRF proofs pi, using, for example, techniques + described in [BreHen19]. The nonce generation methods specified in + the ECVRF ciphersuites of Section 5.5 are designed with this + requirement in mind. + +7.5. Side-Channel Attacks + + Side-channel attacks on cryptographic primitives are an important + issue. Implementers should take care to avoid side-channel attacks + that leak information about the VRF secret key SK (and the nonce k + used in the ECVRF), which is used in VRF_prove. In most + applications, the VRF_proof_to_hash and VRF_verify algorithms take + only inputs that are public, and thus side-channel attacks are + typically not a concern for these algorithms. + + The VRF input alpha may also be a sensitive input to VRF_prove and + may need to be protected against side-channel attacks. Below, we + discuss one particular class of such attacks: timing attacks that can + be used to leak information about the VRF input alpha. + + The ECVRF_encode_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 + that 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. + + ECVRF-P256-SHA256-SSWU and ECVRF-EDWARDS25519-SHA512-ELL2 can be made + to run in time that is independent of alpha, following + recommendations in [RFC9380]. + + +7.6. Proofs Provide No Secrecy for the VRF Input + + 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. + +7.7. Prehashing + + 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 + Edwards-curve Digital Signature Algorithm (EdDSA) signatures) is not + needed, even for applications that need to handle long alpha or to + support the Initialize-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_encode_to_curve. The curve point H becomes the + representative of alpha thereafter. + +7.8. Hash Function Domain Separation + + Hashing is used for different purposes in the two VRFs. + Specifically, in the RSA-FDH-VRF, hashing is used in MGF1 and in + proof_to_hash; in the ECVRF, hashing is used in encode_to_curve, + nonce_generation, challenge_generation, and proof_to_hash. The + theoretical analysis treats each of these functions as a separate + hash function, modeled as a random oracle. This analysis still holds + even if the same hash function is used, as long as the inputs given + to the hash function for a given SK and alpha are overwhelmingly + unlikely to be equal to each other or to any inputs given 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 second + octets of the inputs 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 + + * Inputs to the hash function used in nonce_generation are unlikely + to equal inputs used in encode_to_curve, proof_to_hash, and + challenge_generation. This follows, since nonce_generation inputs + a secret to the hash function that is not used by honest parties + as input to any other hash function and is not available to the + adversary. + + * The second octets of the inputs to the hash function used in + proof_to_hash, challenge_generation, and + encode_to_curve_try_and_increment are all different. + + * The last octet of the inputs to the hash function used in + proof_to_hash, challenge_generation, and + encode_to_curve_try_and_increment is always zero and is therefore + different from the last octet of the input to the hash function + used in ECVRF_encode_to_curve_h2c_suite, which is set equal to the + nonzero length of the domain separation tag per [RFC9380]. + +7.9. Hash Function Salting + + If a hash collision is found, in order to make it more difficult for + the adversary to exploit such a collision, the MGF1 function for the + RSA-FDH-VRF and the ECVRF_encode_to_curve function for the ECVRF use + a public value in addition to alpha (as a so-called salt). This + value is determined by the ciphersuite. For the ciphersuites defined + in this document, it is set equal to the string representation of the + RSA modulus and EC public key, respectively. Implementations that do + not use one of the ciphersuites (see Section 7.10) MAY use a + different salt. For example, if a group of public keys shares the + same salt, then the hash of the VRF input alpha will be the same for + the entire group of public keys; this can be helpful for some + protocols that use the VRF. + +7.10. Futureproofing + + If future designs need to specify variants (e.g., additional + ciphersuites) of the RSA-FDH-VRF or the ECVRF as defined 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. The suite_string constants discussed in this document are + all single octets; if a future suite_string constant is longer than + one octet, then it should start with a different octet than the + suite_string constants discussed in this document. Then, for the + RSA-FDH-VRF, the inputs to the hash function used in MGF1 and + proof_to_hash will be different from other ciphersuites. For the + ECVRF, the inputs to the ECVRF_encode_to_curve hash function used in + producing H are then guaranteed to be different from other + ciphersuites; since all the other hashing done by the Prover depends + on H, inputs to all the hash functions used by the Prover will also + be different from other ciphersuites as long as ECVRF_encode_to_curve + is collision resistant. + +8. References + +8.1. Normative References + + [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate + Requirement Levels", BCP 14, RFC 2119, + DOI 10.17487/RFC2119, March 1997, + <https://www.rfc-editor.org/info/rfc2119>. + + [RFC5114] Lepinski, M. and S. Kent, "Additional Diffie-Hellman + Groups for Use with IETF Standards", RFC 5114, + DOI 10.17487/RFC5114, January 2008, + <https://www.rfc-editor.org/info/rfc5114>. + + [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms + (SHA and SHA-based HMAC and HKDF)", RFC 6234, + DOI 10.17487/RFC6234, May 2011, + <https://www.rfc-editor.org/info/rfc6234>. + + [RFC6979] Pornin, T., "Deterministic Usage of the Digital Signature + Algorithm (DSA) and Elliptic Curve Digital Signature + Algorithm (ECDSA)", RFC 6979, DOI 10.17487/RFC6979, August + 2013, <https://www.rfc-editor.org/info/rfc6979>. + + [RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, + "PKCS #1: RSA Cryptography Specifications Version 2.2", + RFC 8017, DOI 10.17487/RFC8017, November 2016, + <https://www.rfc-editor.org/info/rfc8017>. + + [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital + Signature Algorithm (EdDSA)", RFC 8032, + DOI 10.17487/RFC8032, January 2017, + <https://www.rfc-editor.org/info/rfc8032>. + + [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC + 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, + May 2017, <https://www.rfc-editor.org/info/rfc8174>. + + [RFC9380] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., + and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, + DOI 10.17487/RFC9380, August 2023, + <https://www.rfc-editor.org/info/rfc9380>. + + [SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1: + Elliptic Curve Cryptography", Version 2.0, May 2009, + <https://www.secg.org/sec1-v2.pdf>. + + [SP-800-186] + National Institute for Standards and Technology (NIST), + "Recommendations for Discrete Logarithm-based + Cryptography: Elliptic Curve Domain Parameters", NIST SP + 800-186, DOI 10.6028/NIST.SP.800-186, February 2023, + <https://nvlpubs.nist.gov/nistpubs/SpecialPublications/ + NIST.SP.800-186.pdf>. + +8.2. Informative References + + [ANSI.X9-62-2005] + American National Standards Institute (ANSI), "Public Key + Cryptography for the Financial Services Industry: the + Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI + X9.62, November 2005, + <https://standards.globalspec.com/std/1955141/ + ANSI%20X9.62>. + + [BreHen19] Breitner, J. and N. Heninger, "Biased Nonce Sense: Lattice + Attacks against Weak ECDSA Signatures in + Cryptocurrencies", Cryptology ePrint Archive, Paper + 2019/023, April 2019, <https://eprint.iacr.org/2019/023>. + + [DGKR18] David, B., Gaži, P., Kiayias, A., and A. Russell, + "Ouroboros Praos: An adaptively-secure, semi-synchronous + proof-of-stake blockchain", Cryptology ePrint Archive, + Paper 2017/573, April 2023, + <https://eprint.iacr.org/2017/573>. + + [GHMVZ17] Gilad, Y., Hemo, R., Micali, S., Vlachos, G., and N. + Zeldovich, "Algorand: Scaling Byzantine Agreements for + Cryptocurrencies", Cryptology ePrint Archive, Paper + 2017/454, September 2017, + <https://eprint.iacr.org/2017/454>. + + [GNPRVZ15] Goldberg, S., Naor, M., Papadopoulos, D., Reyzin, L., + Vasant, S., and A. Ziv, "NSEC5: Provably Preventing DNSSEC + Zone Enumeration", Cryptology ePrint Archive, Paper + 2014/582, December 2014, + <https://eprint.iacr.org/2014/582>. + + [MRV99] Micali, S., Rabin, M., and S. Vadhan, "Verifiable Random + Functions", FOCS '99: Proceedings of the 40th Annual + Symposium on Foundations of Computer Science, pp. 120-130, + DOI 10.1109/SFFCS.1999.814584, October 1999, + <https://dash.harvard.edu/handle/1/5028196>. + + [PWHVNRG17] + Papadopoulos, D., Wessels, D., Huque, S., Naor, M., + Včelák, J., Reyzin, L., and S. Goldberg, "Making NSEC5 + Practical for DNSSEC", Cryptology ePrint Archive, Paper + 2017/099, August 2022, <https://eprint.iacr.org/2017/099>. + + [X25519] Bernstein, D.J., "How do I validate Curve25519 public + keys?", 2006, <https://cr.yp.to/ecdh.html#validate>. + +Appendix A. Test Vectors for the RSA-FDH-VRF Ciphersuites + + The test vectors in this section were generated using code provided + at <https://github.com/reyzin/rsa-fdh-vrf>. + + There are three keys used in the nine examples below. First, we + provide the keys. They are shown in hexadecimal big-endian notation. + + 2048-bit key: + + p = efb52a568fa3038fffb853e2183791c6bc81ceee86d20e8f9b6401dc79a8f1 + f6248d3a25fdb3f99245fce41667da038f59745b87cc1aed8b4a9c1d74e7d5c16c + f7343f2b12f1b5055337369bf018fa07adc0d16f2164a516e80d2b4734f0c6563d + 6ee6d4a9e1a54e300cfe9ee679afc3d14a152dfb49b6cfb208bbf921f764af + q = ecbca5ee88bbc635d8263aaba84f6502fdb2b4998a40f7c149133d840b6b1b + d9a972fe2a981c770272b78fda213f76a062dd865dd116d4c8980975ee9347fe0f + 500567e51d78dbee4a34e626051cf018d7feb72f19189525d4f70b6467d0cef514 + 633ab08a9e7a9ec632064b7b5e3e82128fe563757a614092fc5cf624d10e1b + n = ddaba77202bafb796b85bcec98958aa58ae2d117cbc66a6e75c4c2af983985 + a3064eaef93e2b03393256d94d75d6a6656b2956524ed8711898a0c3abae84371d + a0283bc5f433fc384d810a3c118ed302c0b03da16bee70b80ba3480e7acc1eb358 + b3f20fbe90cc4c8a7e2ba9e28b2a3800a5efbaa3c264f79b231f7cdc9577818df1 + bac60ef7a3f78a44f046fd29b0689556da7a7f61eefe67427f3f691aee0a4b1efe + 2ee2e0e6091143ebb7d69254c9d8ab01ff5e0ad7329f566082f9251e64f436c547 + e68de75351ea3a09746ceb7efed2d234121088aaed01696583c172ec88bc173a0d + 4d8ec43f4dcc18ff8379317e83ef9685536283368c9c6deb783075 + e = 010001 + d = d5c5ceab929a841e2a654536de4788f7f0a2a086d44bbb245f8aab3df00db9 + 24e8d644c3b502820f4cce98adacf09e73bc0e9762b50ae2b697aaa24914fa08b5 + 1758f59c07cf827341bb2a0597e126f9c69db031d60692c9cadf62842444696f08 + 223154a1b0be752a325725748644e6d12935b1c66f983379773bcc8c65d06262e9 + 3b5bb774dd2784265c23e9a7fc5e8871eb6bcc9968a6bc360a98874b623ec59f41 + af0a9ecec6af095cb7e5aca11472363950dcbbfcf678fe003358b4ff0060a391da + a45a1bd81c166b6221fb07e4f5da75e27d8d5fdbbf87ecbd7f5a4d804597070faa + ed22f197511b218788816689375245ddf7fa12337f3e7e898fb9d9 + + 3072-bit key: + + p = ee5adea28491084e6635bd73fd95649915a11da410d3f361c8eccc90a4b834 + 25146da7b9e9d3994fd37d5fad7fb759ae451eb99b1102d4671ead23a2925133d1 + 9df49cf9d7e9dcb69fd7555ca095338d0d2a84abb6825050eaf5fffaeff17ccb08 + 33c6079081dfcbd98ced36a593557d29d64b0e0253ce2ee4e07fe2a06269dfe5ca + 230fad221a593a69d9534b2521c1b41d116cafdee02106228ff41433605453e237 + 777626953e79b46a84f50069e25b4f50496a928708abce30559eb183cf + q = fb585bbc12f5695951f70a25e27682dc568acf56115ad749709b2a6e915cdd + 66dfa06db09b390c00b7c7ebeea00845f73c999d8ea9352b1128bdf10113c7500b + 76a03f6b38d0920b5589961549be3d841ccc306f3edd600a53b4b9d4fa1249af87 + af58dfb3ed694289477e853f7d062f58911f7bdb98033b001ee90f11b78f031cff + ac2b5a07e11b01a2a6c1cda059a728f8253a5ff87267623253fc022d3993b27e2f + 344b99eb6072ff7c7ee160724f8fbca562be49247ffae42b55ea79dad5 + n = ea055cef495dec2d8fb3aef519ca87bd1575fa0ae15dd433f4a5f6c40d34ed + 6ba2388172ab7d2183ed970a669d427dc2774ced66a3f082b8e23e94e7de7532f4 + f30bb4a5bbf2e1db2cba0752858a7c7a9bb892c5d6af7e90a7cee8f0097d14498c + 8b482f86348640af61b66640538e834f23ba8f906048db0e57b6fdc162ba2a8a0e + aedd5423f23d8f89413223d89f473029cba11a211eb59e41fb8f0b8ddc651d115d + 9f07ac30296485a9adbd71cc5d9e4a448bd6d70785e838a978b2e66513eb897c96 + 2e85f00a36cc0a3a613183d8bd1572f895901eb8155af9797dbd4aa14726f41571 + 2bf0eb29fa0a9e938cf5325def05d3af7e686227456d903233e316c8cc50341615 + e59b665f0a4a2c32cfccbf9469bdf89564481fb7afc27a7127741f79424e0a35cd + c466dd33ef5a2067f75c86e06af9c03c68c6e78be5f1a4f49ea03569cd9f74c3a0 + ff290ca4ce2c2fa5b770ef8032b26a517c257b7b1c424622c5c04cf20f2290a268 + 939e0cc79dfbac71842f94727b07bfafaded7db6c7f13b + e = 010001 + d = 6e68e957dbfd7c1862dc1b87780b9dcf0ff9016770bc9c09873b66194941d7 + 6218bf2013c1e4df9326dd4402f5df110656d2ec8ea87a28b2a1cb74e590872aeb + 765fe772ea21c57d6ab4ba0fad019189273f05c061719afd14af02277dd28d67c5 + ef50b75b521ca51819b9bcb44cb7c82be66776a45f490050dc0171e77374f1ed00 + d06f8beb09b711a9682107d8840d4a23edf6ac25441fdbf2b584dfa6a67cee21eb + 51c484f09416e11914e774713f1a17600fb9e4e99fbbd83fdcba4b09145dd98094 + 49a1713777161c912d5d595362314b0ea9d1199e97780e8b3293a39af4019fcc74 + 6aaf78dbb7db06852c3358a9ed02ab1d15831a148b27b932c117445a4a6f5114ed + fa3ccc9a9862df714b78a5362aab5e30501b4a729af73e3cdcabe19aac4928b668 + 969780ad33d9df206d904b978a055f4abbc64987744526856e16ef55962453e3ed + 7a8055b0d79d051c50c94584ec7501dbd4856d7a21e43f25d8749e683cca2f53f5 + 75af1d80f39d8e6932ffdf201d179cbf98314c4048c6c1 + + 4096-bit key: + + p = ac803464c8b2082153e15d5a0698d0a2990397fa01c1edd6171a5315e743c9 + 9feb7acd31c37529d4f83405e657c390488d19f7da9ef9d9f9cff4b460d2a26eb1 + 0f90cf4aaf55a19e21dc3bb697723a673e12bbc6580adc7bb72adaddf4682d656f + f5b992e62379bc7b0ac977f2bfbcfac634e04ed597ef302684be72c6bf7db10b80 + f452d412d09e63e017acba378ccc6ea58e683e5641d1e72248f3201a5632f4af75 + 25e91f9e0733731d264fe36802f416cb3e182b21e67a12e3bfba9a9cf40a45ff32 + addfae78063933120238ac61fbb995300a8602aa84f993bed375d6ccba86ad0c8e + fa5f0950aa2c92779febce9d05fa7a1f0d6e5c0d785de93c108297 + q = feb39bb6ee78adfa524e9c0821f60c20d3cff74f8b49731d67ea27d218bcb2 + 0c87498d30dfd398bc23daff7b33dc330db93e6c0e5e6196e035446c6db7cfdb98 + 68b9518d94670b31f9c4d2109cf32c9cc8ac2fc4a6c2e1078510522c81610a81a7 + 07997933ee24030b572a76ee51aa683312ecaa51d8558b3b19cccf65fc867354ae + 193fd5c4f5d5a7180c5ca1e90fcc42f6915dff69a3d1e49046f6c3ef841b262ba8 + 9ddcfde2ed3caeb5bd594181a76f6f1ce01fc65c6f925f6d5b77037c2cbf7b6047 + e19f7b9c846c80238f1c8284c33bfd90c79de91381bb883b0de568aaf4b4a3c3f9 + c98f92e9f6a51f010bcc1dacfd72bfdfda29f527d7f4913153bef7 + n = aba03a8d8527bfc0cbea1cb9a100f4ee7870aedd74a6406f108f7a07f37433 + 6025357e256d655b342d73369102d03c7dcf3c14ed70aac7ebb62498c570068f71 + f1f165e14527f96d946ba839412252eacea604e7d6fd47a0bb9de776679fa9ad64 + 85a076fda04a2015322626dcd2eb91d6b6248802e6d453eb4cbf5e1bfebed02d6a + b36cfe3dd1e8b9749d4853a029940a0bed3aa3128fd8e2e6cd1115db15405bb383 + 7012f56bdc5a6895ec5cc6bca52f7952cfe3c7d5d81d4d3d1c9a29a429eeedfbf5 + 8da0a5b17480875b8071f49eb568fc8d8c023c83b3ed870c3775aaf0578485d757 + b4ab18d8e5fdb30c2b5586047e6203ab1636e376f1c7031f171e2807a2058ece89 + 0cc8fae29ba819df76b45ddb514caee63db1c5e7a3af7468febff82bfe2eb79e3c + 5d1383b7ebee86f02e9cc1853f0f4486f7eb8fee23a2f794317ffd1c39471086df + bfc0e3c0f412f917225f5c551557f38c11f172eca257e4b5908a571e4daa7c7434 + 903701f21937df87d10de9b50ada97e65855d5e786db8f3f86248b55d999ec3153 + 8bd1a409f3e13de46dccc05325774e89016708f8a96240ae1c16641e8b12ab0725 + 7e88aa50d3546e7a91073d85ed601775a3c08e9b7c242d20664dfd4e70a05218d9 + f2c7d760fab3cd772d9362527917cf5b51817e8c2aef51cb3b0dd8cb838097e513 + 537f1d9c3c4708f44ed270db963c7d72cf11b1 + e = 010001 + d = 1efd8dd524282b4deb04592f83cd226d353e53b5156d37d15652321ce16f28 + 1fc258487105b1f9a81054ef937bc89243bd7a01e56624d078d5a9021514c77a7b + 7eceb230dd45fc9a36e4c1b9a4f347b9b29af3e3d14466fcb5242c398b389f70f9 + e7cf33ed54564e38c597720909e513ae8bb149060d1c6612e506e13d78e087c2cb + b39e88c22cf73315c598dbd0ddf1276743ed04a943644c84949ef32d5e4702c805 + 81e54a7fb18879be28b21008dc63182b45f2c190f1b748cd322efc39f2807c64b4 + d06023cb49583418e7b6ac0f447eb2abf48e2ad335583cbc8dff2760c2cce14623 + 46326708336f7e374253ed213e990044927c52d29591f414571e509afc2396a6af + 9843303a19673bcdec1e3fc7c0d6c3f43b4bf88ce83e2bdcfb5e39069fe32800cf + 3f6f6d9917b8083a66ce23a9ab5b0c95bbcc6dfc21d38dadecc20725b13ce2954b + a1bd45ec151a8877fed317cac60b2afaa96c826df6d1c48e7c10649dccc75bdf90 + 5c362c6934da06c3ce30f5befc1cf776d7fda673625147b1108ecb5473f7f58827 + 9533eb184d748230443694b9761b01532ba707563ffa4962321e44fdb710025e8a + 6e00d29bf01ea040618ee111b5d79ac860083f91aa614777cc99d739458f7c53d6 + 3cea7155b118068e0b30b35ed6d0cfc75672f18d075157a3ed31bfa1ce2cea2343 + 57ec76117cc687c274636077abc437cb70a029 + +A.1. RSA-FDH-VRF-SHA256 + + Example 1, using the 2048-bit key above: + + alpha = (the empty string) + EM = 092ea69ca4f5630d4bd1012805ad23528a5f44c040829b4a0208491913ee3 + 9711889bce5347765072efb0b7f8ad9798c830085d9babe10c29f1a649dbb9a64c + 93a8cdaa325d37814faa15a1071ba81c39275f3cd66ce70fd21ee3acc7ac127c5d + e8f2a816b05aff19e4e63451cfe51fef059b2547302387449b4df1ab8eaa5bfc84 + dbbc5edf3b07eb8fe3fe2a93858bd0d55d6f0686f2eb449ed4c609b3083de04b49 + d409a425509d89d282de806a6ce66892edc30337f780b15c7695b26383516f1fc1 + 8f7eab52557c654467600e2e272ef41e7e4a060b42f7533bae603a7fa50f497a64 + a1508b93826d99643a2001d1c958a7a06da0370668634d678a5de + pi = 14234ff8a9487e1b36a23086e258135b8a8a7ff2e23f19c0dfeca0c0a943f + 119ebd336fdc292ef67b56e32ba06f9941893754a8b97c82f68974b2b34c17f6d4 + 3bfd55eb110cd7ea3452d59a24e4ddb8d4cdf040c814e22e3537ca09c2e2dc5dd8 + ea281e6492ad335378f9f437eed30c51eeeee66ef14efb4000c75c802e9c5a6bb8 + 039c0258d4347981159d0ef6990b5e9c8ac2fb03915d7ff1ffa0626e2e11714a63 + 342e59124c1fcea8e2816c1d9a7751feaaa66cf6c82cd3c58ffde66460d98246ab + 358cc33baefae4dfb0d191e9b6d6c0e3f92c35200408925dc8bef39b78d1259f81 + 63a5003a693555f05290ef2e68345f27c6e2a8847c5c919d92e7505 + beta = + 79f0615d4677fb72571889453644013f1a31b08d222e3cee349d64ce1c41045a + + Example 2, using the 3072-bit key above: + + alpha = 74657374 (4 bytes; ASCII "test") + EM = 20a059b7f7034d0d7696c63328cbbd4b40f7c656a632b4129915018fe6c5d + ee8b5bde68ec2a5a78b1ca8483386e3a1a0fa07b4d329ea55facc3145c663ca90d + f5ae46c903211a21bf908dc9a33bd09410cc09c7b4de5fbc79de3413bc80bccf2d + 3aca2fc9c60c776619849ed3e704057ac3d5deacff845d5bc8084ac730c19a1466 + 8e53b5b8b90446b2272eaf59cdf985a7804c7b91cea1ce2582099b7b0f20163b11 + d23110939dd62081b5aa46c62db76b2ac28473d2488970d480bdd8bef8cae9e812 + 74fe3f9925b012c1b55cba8c4291ec7433223cb872e422bb9e0d3775670d587e40 + 3660ff440a9c11a18a488abc716ae36840b2ef5b0db4a90d88f91d79536cef378b + f8e76d173288e26241df522a3cf6bece49c960e43a2d93e7bed10b90580c5b3aff + 056507b4ef27368579832cb4aeecc99c2d8ba402117457df5ae0ed28068ef8b2e0 + d4582f8edacfcad02c83bfab778460b979e9e984827bbefe2b544c0f3ed715dde6 + dc1d7fc7c0f1f87d78aed8e148004b9f62e0321214c7c + pi = 69f6042d400dfad4bdb9974fb73d12ec7823c6632df6b0a97ebc14d8a443f + 74e1eb1a99b37204ba5c7e53bdaf7e3e3fae9efe47cc01d0b061585c8d757ecf00 + 663b3e1bd447d55b6ebd066b814a8d9c4434b224e9cb053a1fd038a58f3bf6b0c7 + 5b6f48f3c8d1ca398a730c133f86f244655f24c445324fdacd291d6d907f93efb2 + 4b59e509f2f370392f5e262fc106292792352d93800f0a1e3a389786619a622f60 + 05cab78ea5f0b5b7ca91ad2a9c6c34fc4a3f9b0332b99e907ffa7f750cdc8342e1 + 2da78f13ad49953bae1751c983ce3cd3335288ac856f85057a7f05acba6465a1c6 + 901ba30bc65b79fb7a847c42a5b4942d600ef316030f2ccafbc6f2e1ff0b46fb5c + 8517cd98c93f81acf370cfdab559bb4270d07db5466e2342d56c476089f4738404 + 34cbcdbd1853b487a6df346208d12c17a48fe50b73b96f640a9761f570a516f615 + 7432b83dd18a1d05cc27b6f283a02fcfda147cf1471772e469961004bde7fa1585 + 7e7bf97b5a83c33fddbd9f4b2e2488f4ed5f7463c93f30b + beta = + bfe966f3fabde6f38a2792ad59bc836bbca39de6eff64f15a42886deff6dfcc5 + + Example 3, using the 4096-bit key above: + + alpha = 73616d706c65 (6 bytes; ASCII "sample") + EM = 17524fa1710b2f8a04e55da403b9b287b99a47afe9b81d3421482e3959b73 + b4d4d4f4b52243ff2bfd2d29b1f030b521d0699065faa2b8903cca2b24cff42956 + 1234fcbd7bcccdac61b7dcb7bc61cd857287b4b42357adbd2fc83ecfc0d5bc199e + 1f6e298b5e470bfc540bc85e933b02035792d65d861096dc03f048cae51c9adc6c + 1ec09e7f5e595681b3d3976d94ba1a65c83c7e82503db5478d3d91b2e00a0f24e7 + ffee1faed68aa62ad7ba4b2912ceb636064766f0535d3ca1369760d8edebc3c8d7 + f5b4de784b644b59e44e24e436298cc33a3cd0f676d6fa0b76ca3b9b11aa68e078 + 9e83bd27b3af08518b9a5eb5f34f4953a79dc25c1285b20fa73e558dd99638eb51 + bf89c80d7989f6e925d8ca5ed1d3f29cc1e1065400e4abbdbcf898791be12c5ae2 + 5661bf7de58a4cb6608c9a4dcc18150638068bb6452b25589ae0a943a67f024dd4 + b5d9e7940c01886f798316156e5771c19457f9104618e271ae7863b65fd07f87fc + d7862690115ce2d963eeac60f78b47c037d6ed3000b43d8149cee08df10a97158e + e1daaf0a3963d23fb6ab0615891734e3039417d8ce03bfc18920c832a40385de95 + d99b546b4bd24ecbfb2e75e9158ae1769bcff444990f54aa40e6a14e0aca52df00 + 062afbf81f6ce8193c53f8d26ac71324fc1db878379178abd695cf04a0fae3432d + 1efffa73bba15b4e81fbaf598146a0c3edafc + pi = 745cc4b6cb75b925194374cdf91b498e8d687c5d9cae1eb5352446c554c2c + 43ac4aa3e2db5cf5e366df635ce156a277ebdbe78c5598588c98257069253127e5 + 7c9735b498f2939f14e1d019795cbd74cee2693acda2666624f174e8f666494aa1 + 2641bce0677acd20e5552d2690117bddb38678a18acdc380bd9d93f3b10960f9be + 0c141fc14f5f30da324ff14020cb5b8aed9fbca3fc44b4973d8e5527bd81f5ae5d + a67e5cc995abd1f7f9cdd3fa89b243fd4d5d5086ddb4eed77a2851fda1d4463f5e + e037a4015aa40c420c2e609d5d0da4ef4a1622131022bdd9c9dc26d177b392663e + a42050ef485fe9d53a8d28d84b82a21101bed5b213c82b578ce7c9c6f7c1bf9eca + 3c248ace9f8835f3850158749111ce1a3bdf5766add72a95a47c8866a4817c42c5 + cbd85d7bef52afab567e564f6625be9e04be6f7da012af68e6623ce4f29c692ba0 + b5f7665bb435a2168bd3b88aae0c6168bb87ea6977f35bb5ad833d96dd14d340f2 + a67b241b01fd8caf415842fd0a9dd5f4ccf4e70f15efdb85332e1df2bb186be15f + 7195176435e01bfd00592710023c3a88ac0eea7189b32296f865a310375111a5f1 + 1b74d0c74b98dfe4c41ccbe695ea801ba47f37b878c1ed0fff8302705b63c89120 + 9ea63defa892969e015a86d97945189444524e5fb660f2b9d1dce337a12e0d003e + a6262ca3194515cc3aa10b1a03ac9dd6995b54d + beta = + b663c5f90da1c12cd5d0e6d049679459e6f79f9fe16bc8b8e7e4d64d66500bd9 + +A.2. RSA-FDH-VRF-SHA384 + + Example 4, using the 2048-bit key above: + + alpha = (the empty string) + EM = 1fa5c0079423d46edb63a833abb2e6ecfd5f39d1f2bd68fc666274d9e8ed8 + ea8a13411126861167a4ba1d014d5ae213372de6bb4227b12e68e16e13ce108536 + acb25f7219c49388f757219716fcb74eb0245b826c7e47ca793864885684b7673e + 2f8579f26e78d63a940eacb23bf7619290cb5cd20859482c410fbd6d83a61f8940 + 866f512be7ac041fc23c3ee71d918ec994f3efa62f4f1f44eaa29f5b37a1e93e24 + 73d8677fcbec312838379a3e05899ce44227c0c428fbd7d4f2d0b46cfde7254e39 + 67b220f8661f5dfbce7a3bf19364f522914478cead3eff0f0e02d166c251319bcf + 86701af1c48436f49ceac990f52940f7da6ac6f5fdafa5c55dc77 + pi = cffe6067bd9a1285dc1e8e543e8582c1250407cbfbcb2d01c4ddbc0d4ecb5 + edeb721fb33147cf95f3084f7ce611f9877814770b14b8a671abc7ff085cf5cbe9 + 1e72d17f076d62db478d4758412a4e4b77a5591dc32b764a501d27e34e56189ba7 + 347a96f141ed1290f8ef7c4ce4009a9aba0715cbd0148721ea72bce00a22e59460 + 421a21e4d121fc0b4eda62479d93724afae7556abe66326487be38cfb795ac1968 + c33a3890f2d8c0f7dfbe88bc76f16cbfd2b0f7ee8663abfd7b789caa5f6c77dd1c + a991c9a9cc532f7550ad6184c8ece12ca4bea7e67f32405416a1f83245b09d06e7 + b4214157fb444be12a2eddc4381678f2b862fb240fcedd2da7ffcb3 + beta = dc37e83f8de0e990abada5096a05ca74754cfe7fe8e46b831e241009194 + 15415dcd5a305f5fb8195713cebc78649c8d1 + + Example 5, using the 3072-bit key above: + + alpha = 74657374 (4 bytes; ASCII "test") + EM = fa2fd7c735c961b43b01c005faefc4e39505ede3914076d4dee40d52acf72 + 7de1782386ad6e9e07faf7666c8f45fde93b024d97c40651b957cfcccc42b8596a + 68a5495c02313ed9ecbb705ab0689c38b9e57af035189e377ad50b4704004c2a97 + 3d9f7554204b03e8b925a973d41a9c3432246eb2eab2f729f03d3a63c9c38c0cc2 + baa440ed5e2d61644405e4b5c1acaac85d8de75a4de00419a478e6c44a97b3e898 + 75c318400ce8d75b84c416ffd501ba78dd3203f21c6610fcaa4d8fc94f45e80dc6 + 5b7e48967199e7acdb18d82413b7018192a6fa2da5d6838adb8e6139f8d12abcce + 7d5fd20cfa031c4971e563d4863d498591dc652a937db5e0bfd68535e3c9db9611 + 8874287c2291a5d3b29aa142795e60f1ade2c8c4d627ee678b652f5fded61f9a60 + d2fa9cf5fb5e6a7fd63d81c91ea2269388f0a96fae77da0957695779385c757489 + 56972ab1cb5e19ad3cc6a357b9ffd368ca985dd9c0e53dd42aff5985f7a234af96 + ad9e34e459a958b808e858f6d7be2e964c33cefad9660 + pi = 22c9278e7171183cf6a3ce108f0400e308a9177c39a171f77777c106c966e + b041824ce43fa56c5c77576646dd110e0b5d7f838bd5b1d1bf2c1feb1520397dd5 + 2d3cea6dbb49d786aa3bf3f5235e7692e583d290c7192102a6e0cb64f5229a326d + 4d00267fd75aae9687167ea0d3d450b2d63519ad605e64c77438728a190a129b11 + 63939a5b7b0721b8d81efbf99a96944f63bf80ecc932fe40402d67c3e099a317cd + 1d13ac6947096308050ea6dad18fdb0958ae565d07d29e619673798f52b8d1dfdb + f29b4641324ea6db5b9f35870acde7bf68e0829534d1c1f43ca9a16861efd82fb8 + 83e35d581f613d2dfbc89d01a84fdf081a3a850f2e865188cd995857222160c547 + 80dc310a6ec100b9bac30f3af92e641360cad8dc255b56fa28e88ffcbef8ebe6ba + 8557e4ec44a7d0ebef882ade36db0d89be71ecaa2b35026c9d328d2384b54ae68d + e2ea70160ddde9aced5a8d896590fc185b408732cc04a249eff27501594902bf3a + f4a3743c4da50c5d62a74746007dedb8358ecfef78c75ab + beta = 5bdf742667ad10080f4ca573ec66f751e82e4077d0db1b281df421af68d + 39412e70362dc5101b4b46e1e453eea7e0989 + + Example 6, using the 4096-bit key above: + + alpha = 73616d706c65 (6 bytes; ASCII "sample") + EM = 1b1d2f330ee20b9b1754f5e6ee4126cf03ea2c7f4e8c52d96111da7f99509 + 042428ec2f2eafdf41716c04a9976a26df77b3d4cea8b10b216e7786fb49e923d9 + 84a2ee13ad82b95783b68fcf3444b65d1353619602ae06e392dc030be105d4cebc + 6ff8a647b79115357833bd5312b9d3f0df1a307e782ff4db8de0eb16259d6bff2f + 57b3dd60a57693d607c42013cbcfc140a77d4a651492854afbacca377ed6729d1c + be72999a62a96190fb630e5abc54d5cbe93254426df4e2315dbc777360ffb2401b + 3dedbed1acacf4b3a63b5ff8e5ab6c0f8ffd9e2a34fffd68a8a593c64de2660dce + daaab13cd42ebf5720d49f3120b01f45f29d1f465e995b148c9266aa97793a9da2 + f38831d00f95f9688b1c50b52a4cbcc14f8287db822381cdddf609c9c178286b1b + c2f94d7ef4d5ceb1293dd7b0fac16d1b3a8b2a7fcc454e52efd2de5a799397fd55 + a909641fa775463f4808b520c3ebe0f94e2765f8538d91a4f53bb746e7d5eaf55b + 3876503760f5c015f9e52bc54bdfc9632028db5e88b7dc0b1e9f1661d0a9b3574e + 46311de8ef6278c4c14f68375763e5df0d4cf221a4c3e84493ed0c36984c172d87 + b513857af4b6c10174dea9db6464e2bab210aa492987f0255d2c5588b1c79769da + 03b62f691d5c4e5fac65505c317bf96b4f70e97c002aa0a032b02e48ee3aead570 + 3bde3186ce138f29ba36219fd3558af417945 + pi = 89d801e364fd48c3b8672e7d7abd8a2a1e5bd36bb1e38af5aaefa2f01cde6 + 86fa2e33f88fdcc8eb3babcf1c66cbbf7dcddb614041813990787be5feabe86bbe + c373d2cbf7c080caa0e37a339d5de1d1455de28f9bef76cd72500c669e9cab4599 + b55dc155d9dd5810174c170f646d3b0b459347c17347c0281eecf5055cf887d6bd + 0a2c962c77d5ff9355a53cea64c34ea0888110ec4eb32da69022e293a8843d4c06 + c9d6e020c594335720467a8337c6a939fb2c5d710f7bdab48a52f4e7483dae062c + 1b9f66f7c9038ba9ceef3d61cb4cc004319c94a267a2425b5f042cd7f1a17922d6 + 596a88a6fefaef41fc87742f2badee7d7613179589b4d02611ac8fd7895d926f48 + 4f79542cdf7f034dd536c9596da2f588ac9840f6bb05875bd17107e7458cc5ea36 + 8a7699fd60c35b54253a718c26cf518712be9d86213b2c6bddd0b7dd169f9240e7 + 7bfc44223675454f9c5596ad2e6e607ea65011a721ecbfa993172ae372ae874377 + 9b33278d25e11ced77b14bc481fce60e4fc10a8a211d8b359906509d6830c653d9 + 1c1a86865219db43f62c70ac6780644d2bd73c5c256527a3eaefaebaf1f2207324 + 17e17dbf598636616f70f2088969ac796a853dc8a5f270a1c505797e83d1675e4f + 40b59c150ca06c49bb0967a2e0c7e74eff9e182d0f7bb6f54f68fe788b89d2191c + 87bbf7f3927978449c2174baa581dc64a9c58ed + beta = 8ec4d150788513c85eea3490d1a1ee1b7a397602d3f9c8b467527f09fab + 5252e539f82e8002825608295ebbba19644dd + +A.3. RSA-FDH-VRF-SHA512 + + Example 7, using the 2048-bit key above: + + alpha = (the empty string) + EM = 7b08a7fff4e5d8fd4978ac5a0ddf48537a2bb3f952dcdc00affb25d747b40 + 85c29c68dddaa87378db32396219ce784acebe70699286318f42794927f546de5d + 85bbefd80a02c3aa714fc17090baa0d0f7fb504e1af0b79ea02d41dc0bf576b8f2 + 1472dd4c55f96bd64772d3ebd0347abe74b9fdf35b754d0405e42ceb0e290fdd91 + ef766a3e27ff59cd86572d15274f6fd49400ec4d126145f3cae200d67d5d108999 + 61658ece7dcbf41f1cca63f8b50399955416a1f55e0af116fac2a9fd1f2dc0085e + 6ad6c1c4bc12d9308d9a030c3e2ea7f037d1c98beb23d43d67a97e5bf52382b8e8 + 90c5967ab42f2010cac985d3a52fe726045746d4ffef901127646 + pi = a280db108df5ad6ac1bed67efbc5c6fc6da0d301b9c0b41d26e379cd223c6 + 13c59d52c987e4baaa6de4de2103284ddd56aa0b662dfe8faa8f6a503b83b7c81f + 481e23a08761d49a151ada1d9daa132138bbd6f80204c7fa87716b120df957224f + 92b32a3a0f96c3b209080c408618a92382ab5575f10a57c24ee0ffd01d6b822dc3 + 6b27600bf36aafadf0a01e65aa6a0f2fc1a9dcd207d9bf5181a9ca69120e154108 + 00a26efd3ce619349592eeff7b1851737bd033a83f88744ddd3d3e782efb6d2438 + ffda22ddcaa32c821c6730a05d5bdab88c354809d615884744ff10276496bee70b + 62feb6ed07a3948823e9ee2a453dbcd4450192c9de0128adfc7e147 + beta = 808ca1f8f66a48118aacb011394bd4e5f0011c89ca913943d467b81cc5c + 43086e588abdde061c3ee30f4c15b2a6b51ad0ada42c0737fd7b2206fb43d35c8e + d22 + + Example 8, using the 3072-bit key above: + + alpha = 74657374 (4 bytes; ASCII "test") + EM = 803b6618f0ad47da2db309b1f57807a286500020c9e2b1427ebd9fff1104e + 3aa8a69210441cd58344bd810c4900825c84b1e5e36825f1e397df54c4419f8525 + d9a09a49e7fedc18b8d906cbd9ea831c55f2aaa0461e19ddd6ec9d14dafa1fcf49 + b77458a65427b7f060bc7425538e5d3af1813752cb452d0b098514110399734d1f + 55870c65ea3e799e6d9024a9e2fb95883e580578811a8c7d34b18f8fabc6c05fb9 + 697335fcf2cb1b7576ee7a39dcff129e1f142106c45f30a8ae62370f576d1d1d8c + 6307fccfe25cb431f348dea81b6b7e6307bbefda2a0b23036653a612226392a573 + b7d62e28f9fecc7f4be0bf0a3049ce8ed276b34130faa943aeeedf962b42a3fc6c + 881bbf9a62039e9c0850f1393a2a02c6848d06c3520e086541d8af99ea3ef9f9da + 2e3b2bd3172682a47e5965899bc576b66e29a0b8dcf06871202a1a4e7f2ff19bdd + 9eac2241129a73d7d01303b80372ac62a0d5b6bfd1d7119e561ace229cd53d2c99 + 63d6127b9ade16dce4b07d1cd89247ffc438811dc8b3c + pi = 1aa828e0a751074fed2fa776fd29336a84987c064eeebcd3a8129fb688b47 + eb7109987d01db0c3624ba7cc75e2f1ad60f5e204a250a329048bc34df34d41bfe + ea6651774d249ff9fc29aeabdf524400527aa1c4100b1af86b2dcc2e7aecc77f38 + 6b80f29ccd807cb705b5057431832dafe56733a1e7bfcba1d052a26d1a8512f297 + b5abad5afd64fbcf21b57531a9b2c8217c0d9f1c875c196d998f61e8017f6b6ebe + 7317545ed390e18305bc96abb1514ec271963d02bed91ccf029d022189f84bac8c + fa216da54e39919118348dfea6f4f6532b49da7820ee2a21f42b762e107722ad0a + bf62271e0640d6b1c4d1a39b94ebd74b4283de2d6550cbdb1f29cac51671e9c8fc + 0ea0fdbb082a14a221e0531615f2bcfba0d70e99e4997cb00f81fcab2b95566322 + 0234a5e90f29bd08e6fa50dd92770d9e514e0f9eb27aee634877bcea681ffd7da2 + b5be2f80c1dde1243b17ac726401cf961c5ce06640eb93352402c1ebc59c92188c + 511b375d63124846b46017fe36dc13fc2d34dbd80b312e0 + beta = 9202b6715b7921c5eb35572ed9ebb85848d3345efadd665049ce889be46 + 322586d4177864c9179468473518c6b6ac2e9c85ae5ee5fcd3c0d8e6d4d8f18be6 + 238 + + Example 9, using the 4096-bit key above: + + alpha = 73616d706c65 (6 bytes; ASCII "sample") + EM = 289607894786ccf223b1e758232f3402aa50cd48bca3d2bd64b2ea9b4a69d + c91756a42b2c1feaaa777763b9b7c91888c580433ac85f5fbb360f129ecee739f6 + 9b560657687d38f7d43c84f6605005c38f56c91310eff27bae49b14d8a36542d69 + b70efca8637b845be0f029c085a7b6aad6ca0eb65fffdf8d55d9538d3b54044ebb + c26e092b2f3ddaece7aa5b4b234ec848bfdc72a4ecdc10c66fb845cfc5ad39756e + 7f26007cea0ebf1e878636f4e39308fe7a317a9b7e90051536ac028bc1a2ec200a + 5dad0e3b74717bd9e7ea620919e315799e0fea7b0895fcbc0b95686b2495dc23e3 + cd56a16652b0df0dbd3ca8a6c96b13973a0c31a5541e211229da8a56e588a616c7 + 21baab8e2d30313008c2374887f147598468b378bf8949ab1165b9348245d0a6a5 + f795918fce05f0d072f81f78c7224e7f1c4684877d714f231d5775c88759086121 + 2eae2d174761158ac7d653b18f0d4b71362c0eb8a67bed1a48a4e7dc739b2b4469 + 4514cae7f192d236afb1ab2409f24dfa94a2d705d0087860d844ba04564bb6733c + a20089417d74eaed86d7e68ced681e9d88a9c3d7e6a33927592820cb9a38d45393 + 32e509296489e54cd6b8495f100c36debe1f719578b15e8a99cc8febc3212e8147 + 8aeca616a5230ae84e7079f52aefb2ec2a97157fb5d60e1ddcf03b134be2c93ffb + a41d5d068750adc8df07e5a264640f7e586bd + pi = 17d7635cac33b0b72ea1c0afb1f681d1a96c5073ed9f88ed8bb54eb428d7b + 2db4ee3355eee512ddc7af50694b37fd389f990278e22095b2582c78c4ed6070b0 + c7382b0308b6d546141a9b0d6ebb3af97abd93c16a5d34a2d805d8aa444fed2297 + d017571a693d221fda094d40500ab9b203d397a7543e72b26b06e561d49696e01d + eebfed58b46611dd5a346e227d7519f8ffd1dc76a172c9f7f355c3e7e5ee7773ed + ab00a22af5c39367f3779da68ce6da9f8a594f5f6149012501181653572fe5549a + 9c2bf36148b3bdc94feaedd600727fe5c11b7dcbfd73002ae08061cb4b84ba47f1 + bf8c5d46bc2acb7cb4964a6dca7eedc396e663a64121d93dade8b83cea09d76653 + cca1a8d20d6b7323a890651dc575025ba1be02d08c5946f50cde438339b06e8633 + 198da0d467d2cac7d98ae62dd71353f6fb19aa9daac851d0ce237b21db93b91e51 + 8d5c1ac36cdf874975deb7aab3942acc3980f221f33ad1254eb8ac3138e087d045 + c4746e0b7eedcaf2a1a173559783eba8691555c1b0e468f8efe6501679b760038e + d6fc9ce6aa5ae24b3f1178713793c8e5ee96035a2f0ee02e2d10ac098613358d3c + ff10f4dff3437f2a48252c5d6805288fbd7ee05356f80db12aaeabf6638677abf5 + b8eb2376fb76861cf1b817d5a0b878dae6beac44f078f37d982d941a77582a7778 + 4fabd632e28d664d9f705f31e24d1ca623dfac7 + beta = 6026f6defaf534cc79ce7c1b0370fb53e4825d2d44f549f696e06d693c3 + 9e852e21a5e3b6ff093618dd277b40678957e1b90e8e6ca742efed30dc309b3b24 + 2b8 + +Appendix B. Test Vectors for the ECVRF Ciphersuites + + The test vectors in this section were generated using code provided + at <https://github.com/reyzin/ecvrf>. + +B.1. ECVRF-P256-SHA256-TAI + + The example secret keys and messages in Examples 10 and 11 are taken + from Appendix A.2.5 of [RFC6979]. + + Example 10: + + SK = x = + c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 + PK = + 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 + alpha = 73616d706c65 (6 bytes; ASCII "sample") + try_and_increment succeeded on ctr = 1 + H = + 0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4 + k = + 0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77 + U = k*B = + 02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672 + V = k*H = + 02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394 + pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5 + 071b4a53f0a46f018bc2c56e58d383f2305e0975972c26feea0eb122fe7893c15a + f376b33edf7de17c6ea056d4d82de6bc02f + beta = + a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e + + Example 11: + + SK = x = + c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 + PK = + 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 + alpha = 74657374 (4 bytes; ASCII "test") + try_and_increment succeeded on ctr = 3 + H = + 02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974 + k = + 5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961 + U = k*B = + 022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48e + V = k*H = + 033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119 + pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080 + ddb56c19e067b15a8a8174905b13617804534214f935b94c2287f797e393eb0816 + 969d864f37625b443f30f1a5a33f2b3c854 + beta = + a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d + + The example secret key in Example 12 is taken from Appendix L.4.2 of + [ANSI.X9-62-2005]. + + Example 12: + + SK = x = + 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8 + PK = + 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d + alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20 + 417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035 + (62 bytes; ASCII "Example using ECDSA key from Appendix L.4.2 of + ANSI.X9-62-2005") + try_and_increment succeeded on ctr = 1 + H = + 0258055c26c4b01d01c00fb57567955f7d39cd6f6e85fd37c58f696cc6b7aa761d + k = + 5689e2e08e1110b4dda293ac21667eac6db5de4a46a519c73d533f69be2f4da3 + U = k*B = + 020f465cd0ec74d2e23af0abde4c07e866ae4e5138bded5dd1196b8843f380db84 + V = k*H = + 036cb6f811428fc4904370b86c488f60c280fa5b496d2f34ff8772f60ed24b2d1d + pi = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b + 8c667d091c00b0f5c3619d10ecea44363b5a599cadc5b2957e223fec62e81f7b48 + 25fc799a771a3d7334b9186bdbee87316b1 + beta = + 90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19 + +B.2. ECVRF-P256-SHA256-SSWU + + The example secret keys and messages in Examples 13 and 14 are taken + from Appendix A.2.5 of [RFC6979]. + + Example 13: + + SK = x = + c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 + PK = + 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 + alpha = 73616d706c65 (6 bytes; ASCII "sample") + In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c + 2a4b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7e + In SSWU: u = + df565615a2372e8b31b8771f7503bafc144e48b05688b97958cc27ce29a8d810 + In SSWU: x1 = + e7e39eb8a4c982426fcff629e55a3e13516cfeb62c02c369b1e750316f5e94eb + In SSWU: gx1 is a nonsquare + H = + 02b31973e872d4a097e2cfae9f37af9f9d73428fde74ac537dda93b5f18dbc5842 + k = + e92820035a0a8afe132826c6312662b6ea733fc1a0d33737945016de54d02dd8 + U = k*B = + 031490f49d0355ffcdf66e40df788bee93861917ee713acff79be40d20cc91a30a + V = k*H = + 03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3 + pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394 + c7888782fffde7b842c38c20c08de6ec6c2e7027a97000f2c9fa4425d5c03e639f + b48fde58114d755985498d7eb234cf4aed9 + beta = + 21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b + + Example 14: + + SK = x = + c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 + PK = + 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 + alpha = 74657374 (4 bytes; ASCII "test") + In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109af + b243b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3 + In SSWU: u = + d8b0107f7e7aa36390240d834852f8703a6dc407019d6196bda5861b8fc00181 + In SSWU: x1 = + ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7 + In SSWU: gx1 is a square + H = + 03ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7 + k = + febc3451ea7639fde2cf41ffd03f463124ecb3b5a79913db1ed069147c8a7dea + U = k*B = + 031200f9900e96f811d1247d353573f47e0d9da601fc992566234fc1a5b37749ae + V = k*H = + 02d3715dcfee136c7ae50e95ffca76f4ca6c29ddfb92a39c31a0d48e75c6605cd1 + pi = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa + 8db2743aad348a3006dc1caad7da28687320740bf7dd78fe13c298867321ce3b36 + b79ec3093b7083ac5e4daf3465f9f43c627 + beta = + 8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd + + The example secret key in Example 15 is taken from Appendix L.4.2 of + [ANSI.X9-62-2005]. + + Example 15: + + SK = x = + 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8 + PK = + 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d + alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20 + 417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035 + (62 bytes; ASCII "Example using ECDSA key from Appendix L.4.2 of + ANSI.X9-62-2005") + In SSWU: uniform_bytes = 9b81d55a242d3e8438d3bcfb1bee985a87fd14480 + 2c9268cf9adeee160e6e9ff765569797a0f701cb4316018de2e7dd4 + In SSWU: u = + e43c98c2ae06d13839fedb0303e5ee815896beda39be83fb11325b97976efdce + In SSWU: x1 = + be9e195a50f175d3563aed8dc2d9f513a5536c1e9aee1757d86c08d32d582a86 + In SSWU: gx1 is a nonsquare + H = + 022dd5150e5a2a24c66feab2f68532be1486e28e07f1b9a055cf38ccc16f6595ff + k = + 8e29221f33564f3f66f858ba2b0c14766e1057adbd422c3e7d0d99d5e142b613 + U = k*B = + 03a8823ff9fd16bf879261c740b9c7792b77fee0830f21314117e441784667958d + V = k*H = + 02d48fbb45921c755b73b25be2f23379e3ce69294f6cee9279815f57f4b422659d + pi = 039f8d9cdc162c89be2871cbcb1435144739431db7fab437ab7bc4e2651a9 + e99d5488405a11a6c7fc8defddd9e1573a563b7333aab4effe73ae9803274174c6 + 59269fd39b53e133dcd9e0d24f01288de9a + beta = + 4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a + +B.3. ECVRF-EDWARDS25519-SHA512-TAI + + The example secret keys and messages in Examples 16, 17, and 18 are + taken from Section 7.1 of [RFC8032]. + + Example 16: + + SK = + 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60 + PK = + d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a + alpha = (the empty string) + x = + 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f + try_and_increment succeeded on ctr = 0 + H = + 91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318 + k_string = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106 + f345b8594b6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2 + a6afae4 + k = + 8a49edbd1492a8ee09766befe50a7d563051bf3406cbffc20a88def030730f0f + U = k*B = + aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071 + V = k*H = + 5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f + pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f7 + 23f26f8a57ccaed74ee1b190bed1f479d9727d2d0f9b005a6e456a35d4fb0daab1 + 268a1b0db10836d9826a528ca76567805 + beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757 + 876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd + 8ae + + Example 17: + + SK = + 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb + PK = + 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c + alpha = 72 (1 byte) + x = + 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51 + try_and_increment succeeded on ctr = 1 + H = + 5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551 + k_string = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b234 + 2acb9c167cab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc + 16cc5ce + k = + d8c3a66921444cb3427d5d989f9b315aa8ca3375e9ec4d52207711a1fdb44107 + U = k*B = + 1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3 + V = k*H = + fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423 + pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed + 5933bf0864a62558b3ed7f2fea45c92a465301b3bbf5e3e54ddf2d935be3b67926 + da3ef39226bbc355bdc9850112c8f4b02 + beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496 + 310eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5 + 031 + + Example 18: + + SK = + c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7 + PK = + fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025 + alpha = af82 (2 bytes) + x = + 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c + try_and_increment succeeded on ctr = 0 + H = + bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a + k_string = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59f + c31a0febc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6b + b6d6c3a + k = + 5ffdbc72135d936014e8ab708585fda379405542b07e3bd2c0bd48437fbac60a + U = k*B = + 2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5 + V = k*H = + 88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506 + pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccb + f8096bb474e53895c362d8628ee9f9ea3c0e52c7a5c691b6c18c9979866568add7 + a2d41b00b05081ed0f58ee5e31b3a970e + beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c + 61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df + 19f + +B.4. ECVRF-EDWARDS25519-SHA512-ELL2 + + The example secret keys and messages in Examples 19, 20, and 21 are + taken from Section 7.1 of [RFC8032]. + + Example 19: + + SK = + 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60 + PK = + d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a + alpha = (the empty string) + x = + 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f + In Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca + 5298b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269ae + In Elligator2: u = + 30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55c + In Elligator2: gx1 = + 8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08 + In Elligator2: gx1 is a square + H = + b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9 + k_string = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a4245 + 8f817bed7a386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88d + ab8d41c + k = + 55cbb247af9b8372259a97b2cfec656d78868deb33b203d51b9961c364522400 + U = k*B = + 762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9 + V = k*H = + 4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af + pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab + 46f14adf9a3cd8b8412d9038531e865c341cafa73589b023d14311c331a9ad15ff + 2fb37831e00f0acaa6d73bc9997b06501 + beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80 + c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6 + c54 + + Example 20: + + SK = + 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb + PK = + 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c + alpha = 72 (1 byte) + x = + 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51 + In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd7 + 7bb99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6d + In Elligator2: u = + 3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773d + In Elligator2: gx1 = + 25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464d + In Elligator2: gx1 is a nonsquare + H = + 76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57 + k_string = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37 + c4c37f1d41e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab75 + 44e444d + k = + 9565956daeedf376cad61b829b2a4d21ba1b52e9b3e2457477a64630a9711003 + U = k*B = + 8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061 + V = k*H = + fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8b + pi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c50907 + 2ef055b48372bb82efbdce8e10c8cb9a2f9d60e93908f93df1623ad78a86a028d6 + bc064dbfc75a6a57379ef855dc6733801 + beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4 + 635987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc + 735 + + Example 21: + + SK = + c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7 + PK = + fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025 + alpha = af82 (2 bytes) + x = + 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c + In Elligator2: uniform_bytes = be0aed556e36cdfddf8f1eeddbb7356a24f + ad64cf95a922a098038f215588b216beabbfe6acf20256188e883292b7a3a + In Elligator2: u = + f6675dc6d17fc790d4b3f1c6acf689a13d8b5815f23880092a925af94cd6fa24 + In Elligator2: gx1 = + a63d48e3247c903e22fdfb88fd9295e396712a5fe576af335dbe16f99f0af26c + In Elligator2: gx1 is a square + H = 13d2a8b5ca32db7e98094a61f656a08c6c964344e058879a386a947a4e189ed1 + k_string = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5 + aaf79cf576087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861 + b3d683b + k = + 1fda4077f737098b3f361c33a36cccafd7e9e9b720e1f84011254e25f37eed02 + U = k*B = + a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60 + V = k*H = + fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25a + pi = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57f + dce35b46edfc655bc828d44ad09d1150f31374e7ef73027e14760d42e77341fe05 + 467bb286cc2c9d7fde29120a0b2320d04 + beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5c + d090a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314 + a58 + +Contributors + + This document would not be possible without the work of Moni Naor, + Sachin Vasant, and Asaf Ziv. Chloe Martindale provided a thorough + cryptographer's review. Liliya Akhmetzyanova, Tony Arcieri, Gary + Belvin, Mario Cao Cueto, Brian Chen, Sergey Gorbunov, Shumon Huque, + Gorka Irazoqui Apecechea, Marek Jankowski, Burt Kaliski, Mallory + Knodel, David C. Lawrence, Derek Ting-Haye Leung, Antonio Marcedone, + Piotr Nojszewski, Chris Peikert, Colin Perkins, Trevor Perrin, Sam + Scott, Stanislav Smyshlyaev, Adam Suhl, Nick Sullivan, Christopher + Wood, Jiayu Xu, and Annie Yousar provided valuable input to this + document. Christopher Wood, Malte Thomsen, Marcus Rasmussen, and + Tobias Vestergaard provided independent verification of the test + vectors. Riad Wahby helped this document align with [RFC9380]. + +Authors' Addresses + + Sharon Goldberg + Boston University + 665 Commonwealth Avenue + Boston, MA 02215 + United States of America + Email: goldbe@cs.bu.edu + + + Leonid Reyzin + Boston University and Algorand + 665 Commonwealth Avenue + Boston, MA 02215 + United States of America + Email: reyzin@bu.edu + + + Dimitrios Papadopoulos + Hong Kong University of Science and Technology + Clearwater Bay + Hong Kong + Email: dipapado@cse.ust.hk + + + Jan Včelák + NS1 + Email: jvcelak@ns1.com |