Boston University

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Boston University and Algorand

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Hong Kong University of Science and Technology

Clearwater Bay Hong Kong dipapado@cse.ust.hk

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CFRG public key cryptography hashing authenticated denial A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private key can compute the hash, but anyone with public key can verify the correctness of the hash. VRFs are useful for preventing enumeration of hash-based data structures. This document specifies several VRF constructions that are secure in the cryptographic random oracle model. One VRF uses RSA and the other VRF uses Elliptic Curves (EC).

A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private VRF key can compute the hash, but anyone with corresponding public key can verify the correctness of the hash. A key application of the VRF is to provide privacy against offline enumeration (e.g. dictionary attacks) on data stored in a hash-based data structure. In this application, a Prover holds the VRF private key and uses the VRF hashing to construct a hash-based data structure on the input data. Due to the nature of the VRF, only the Prover can answer queries about whether or not some data is stored in the data structure. Anyone who knows the public VRF key can verify that the Prover has answered the queries correctly. However, no offline inferences (i.e. inferences without querying the Prover) can be made about the data stored in the data structure.

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in .

The following terminology is used through this document: The private key for the VRF. The public key for the VRF. The input to be hashed by the VRF. The VRF hash output. The VRF proof. The Prover holds the private VRF key SK and public VRF key PK. The Verifier holds the public VRF key PK.

A VRF comes with a key generation algorithm that generates a public VRF key PK and private VRF key SK. The prover hashes an input alpha using the private VRF key SK to obtain a VRF hash output beta 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 private 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 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 VRF also comes with an algorithm VRF_verify(PK, alpha, pi) that outputs (VALID, beta = VRF_proof_to_hash(pi)) if pi is valid, and INVALID otherwise.

VRFs are designed to ensure the following security properties.

Uniqueness means that, for any fixed public VRF key and for any input alpha, there is a unique VRF output beta that can be proved to be valid. Uniqueness must hold even for an adversarial Prover that knows the VRF private key SK. More precisely, "full uniqueness" states that a computationally-bounded adversary cannot choose a VRF public key PK, a VRF input alpha, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha, pi1) outputs (VALID, beta1), VRF_verify(PK, alpha, pi2) outputs (VALID, beta2), and beta1 is not equal to beta2. A slightly weaker security property called "trusted uniqueness" suffices for many applications. Trusted uniqueness is the same as full uniqueness, but it must hold only if the VRF keys PK and SK were generated in a trustworthy manner. In other words, uniqueness might not hold if keys were generated in an invalid manner or with bad randomness.

Like any cryptographic hash function, VRFs need to be collision resistant. Collison resistance must hold even for an adversarial Prover that knows the VRF private key SK. More precisely, "full collision resistance" states that it should be computationally infeasible for an adversary to find two distinct VRF inputs alpha1 and alpha2 that have the same VRF hash beta, even if that adversary knows the private VRF key SK. For most applications, a slightly weaker security property called "trusted collision resistance" suffices. Trusted collision resistance is the same as collision resistance, but it holds only if PK and SK were generated in a trustworthy manner.

Pseudorandomness ensures that when an adversarial Verifier sees a VRF hash output beta without its corresponding VRF proof pi, then beta is indistinguishable from a random value. More precisely, suppose the public and private VRF keys (PK, SK) were generated in a trustworthy manner. Pseudorandomness ensures that the VRF hash output beta (without its corresponding VRF proof pi) on any adversarially-chosen "target" VRF input alpha looks indistinguishable from random for any computationally bounded adversary who does not know the private VRF key SK. This holds even if the adversary also gets to choose other VRF inputs alpha' and observe their corresponding VRF hash outputs beta' and proofs pi'. With "full pseudorandomness", the adversary is allowed to choose the "target" VRF input alpha at any time, even after it observes VRF outputs beta' and proofs pi' on a variety of chosen inputs alpha'. "Selective pseudorandomness" is a weaker security property which suffices in many applications. Here, the adversary must choose the target VRF input alpha independently of the public VRF key PK, and before it observes VRF outputs beta' and proofs pi' on inputs alpha' of its choice. It is important to remember that the VRF output beta does not look random to the Prover, or to any other party that knows the private VRF key SK! Such a party can easily distinguish beta from a random value by comparing beta to the result of VRF_hash(SK, alpha). Also, the VRF output beta does not look random to any party that knows the valid VRF proof pi corresponding to the VRF input alpha, even if this party does not know the private VRF key SK. Such a party can easily distinguish beta from a random value by checking whether VRF_verify(PK, alpha, pi) returns (VALID, beta). Also, the VRF output beta may not look random if VRF key generation was not done in a trustworthy fashion. (For example, if VRF keys were generated with bad randomness.)

As explained in , pseudorandomness is guaranteed only if the VRF keys were generated in a trustworthy fashion. 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 and ). This property is similar to the unpredictability achieved by an (ordinary, unkeyed) cryptographic hash function: if the input has enough entropy (i.e., cannot be predicted), then the correct output is indistinguishable from uniform. A formal definition of this property appears in Section 3.2 of . The VRF schemes presented in this specification are believed to satisfy this property if the public key was generated in a trustworthy manner. Additionally, the ECVRF is believed to also satisfy this property even if the public key was not generated in a trustworthy manner, as long as the public key satisfies the key validation procedure in .

The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in . Its security follows from the standard RSA assumption in the random oracle model. Formal security proofs are in . The VRF computes the proof pi as a deterministic RSA signature on input alpha using the RSA Full Domain Hash Algorithm parametrized with the selected hash algorithm. RSA signature verification is used to verify the correctness of the proof. The VRF hash output beta is simply obtained by hashing the proof pi with the selected hash algorithm. The key pair for RSA-FDH-VRF MUST be generated in a way that it satisfies the conditions specified in Section 3 of . In this document, the notation from is used. Parameters used: (n, e) - RSA public key K - RSA private key k - length in octets of the RSA modulus n (k must be less than 2^32) Fixed options: Hash - cryptographic hash function hLen - output length in octets of hash function Hash Primitives used: I2OSP - Conversion of a nonnegative integer to an octet string as defined in Section 4.1 of (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 nonnegative integer as defined in Section 4.2 of (given a big-endian encoding of an integer, produces the integer) RSASP1 - RSA signature primitive as defined in Section 5.2.1 of (given a secret key and an input, raises the input to the secret RSA exponent modulo n) RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of (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 Section B.2.1 of || - octet string concatenation

RSAFDHVRF_prove(K, alpha_string) Input: K - RSA private key alpha_string - VRF hash input, an octet string Output: pi_string - proof, an octet string of length k Steps: one_string = 0x01 = I2OSP(1, 1), a single octet with value 1 EM = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) || alpha_string, k - 1) m = OS2IP(EM) s = RSASP1(K, m) pi_string = I2OSP(s, k) Output pi_string

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 pi_string that is known to have been produced by RSAFDHVRF_prove, or from within RSAFDHVRF_verify as specified in . Steps: two_string = 0x02 = I2OSP(2, 1), a single octet with value 2 beta_string = Hash(two_string || pi_string) Output beta_string

RSAFDHVRF_verify((n, e), alpha_string, pi_string) 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 n Output: ("VALID", beta_string), where beta_string is the VRF hash output, an octet string of length hLen; or "INVALID" Steps: s = OS2IP(pi_string) m = RSAVP1((n, e), s) EM = I2OSP(m, k - 1) one_string = 0x01 = I2OSP(1, 1), a single octet with value 1 EM' = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) || alpha_string, k - 1) If EM and EM' are equal, output ("VALID", RSAFDHVRF_proof_to_hash(pi_string)); else output "INVALID".

The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that satisfies the trusted uniqueness, trusted collision resistance, and full pseudorandomness properties defined in . The security of this VRF follows from the decisional Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are in . To additionally satisfy "full uniqueness" and "full collision resistance", the Verifier MUST additionally perform the validation procedure specified in upon receipt of the public VRF key. 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 Fixed options (specified in ): F - finite field 2n - length, in octets, of a field element in F, rounded up to the nearest even integer E - elliptic curve (EC) defined over F ptLen - length, in octets, of an EC point 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., smallest integer such that 2^(8qLen)>q (note that in the typical case, qLen equals 2n or is close to 2n) 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; must be at least 2n ECVRF_hash_to_curve - a function that hashes strings to an EC point. ECVRF_nonce_generation - a function that derives a pseudorandom nonce from SK and the input as part of ECVRF proving. suite_string - a single nonzero octet specifying the ECVRF ciphersuite, which determines the above options as well as type conversions and parameter generation Type conversions (specified in ): int_to_string(a, len) - conversion of nonnegative integer a to octet string of length len string_to_int(a_string) - conversion of an octet string a_string to a nonnegative integer point_to_string - conversion of EC point to an ptLen-octet string string_to_point - conversion of an ptLen-octet string to EC point. string_to_point returns INVALID if the octet string does not convert to a valid EC point. 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. 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 ): SK - VRF private 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, an EC point

ECVRF_prove(SK, alpha_string) Input: SK - VRF private key alpha_string = input alpha, an octet string Output: pi_string - VRF proof, octet string of length ptLen+n+qLen Steps: 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 ; these values can be cached, for example, after key generation, and need not be rederived each time) H = ECVRF_hash_to_curve(Y, alpha_string) h_string = point_to_string(H) Gamma = x*H k = ECVRF_nonce_generation(SK, h_string) c = ECVRF_hash_points(H, Gamma, k*B, k*H) (see ) s = (k + c*x) mod q pi_string = point_to_string(Gamma) || int_to_string(c, n) || int_to_string(s, qLen) Output pi_string

ECVRF_proof_to_hash(pi_string) Input: pi_string - VRF proof, octet string of length ptLen+n+qLen Output: "INVALID", or beta_string - VRF hash output, octet string of length hLen Important note: ECVRF_proof_to_hash should be run only on pi_string that is known to have been produced by ECVRF_prove, or from within ECVRF_verify as specified in . Steps: D = ECVRF_decode_proof(pi_string) (see ) If D is "INVALID", output "INVALID" and stop (Gamma, c, s) = D three_string = 0x03 = int_to_string(3, 1), a single octet with value 3 zero_string = 0x00 = int_to_string(0, 1), a single octet with value 0 beta_string = Hash(suite_string || three_string || point_to_string(cofactor * Gamma) || zero_string) Output beta_string

ECVRF_verify(Y, pi_string, alpha_string) Input: Y - public key, an EC point pi_string - VRF proof, octet string of length ptLen+n+qLen alpha_string - VRF input, octet string Output: ("VALID", beta_string), where beta_string is the VRF hash output, octet string of length hLen; or "INVALID" Steps: D = ECVRF_decode_proof(pi_string) (see ) If D is "INVALID", output "INVALID" and stop (Gamma, c, s) = D H = ECVRF_hash_to_curve(Y, alpha_string) U = s*B - c*Y V = s*H - c*Gamma c' = ECVRF_hash_points(H, Gamma, U, V) (see ) If c and c' are equal, output ("VALID", ECVRF_proof_to_hash(pi_string)); else output "INVALID"

The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and converts it to H, an EC point in G. This algorithm is the only place the VRF input alpha is used for proving and verifying. See for further discussion. This section specifies a number of such algorithms, which are not compatible with each other. The choice of a particular algorithm from the options specified in this section is made in .

The following ECVRF_hash_to_curve_try_and_increment(Y, alpha_string) algorithm implements ECVRF_hash_to_curve in a simple and generic way that works for any elliptic curve. The running time of this algorithm depends on alpha_string. For the ciphersuites specified in , this algorithm is expected to find a valid curve point after approximately two attempts (i.e., when ctr=1) on average. However, because the running time of algorithm depends on alpha_string, this algorithm SHOULD be avoided in applications where it is important that the VRF input alpha remain secret. ECVRF_hash_to_try_and_increment(Y, alpha_string) Input: Y - public key, an EC point alpha_string - value to be hashed, an octet string Output: H - hashed value, a finite EC point in G Fixed option (specified in ): arbitrary_string_to_point - conversion of an arbitrary octet string to an EC point. Steps: ctr = 0 PK_string = point_to_string(Y) one_string = 0x01 = int_to_string(1, 1), a single octet with value 1 zero_string = 0x00 = int_to_string(0, 1), a single octet with value 0 H = "INVALID" While H is "INVALID" or H is the identity element of the elliptic curve group: ctr_string = int_to_string(ctr, 1) hash_string = Hash(suite_string || one_string || PK_string || alpha_string || ctr_string || zero_string) H = arbitrary_string_to_point(hash_string) If H is not "INVALID" and cofactor > 1, set H = cofactor * H ctr = ctr + 1 Output H

The ECVRF_hash_to_curve_h2c_suite(Y, alpha_string) algorithm implements ECVRF_hash_to_curve using one of the several hash-to-curve options defined in . The specific choice of the hash-to-curve option (called Suite ID in ) is given by the h2c_suite_ID_string parameter. ECVRF_hash_to_curve_h2c_suite(Y, alpha_string) Input: alpha_string - value to be hashed, an octet string Y - public key, an EC point Output: H - hashed value, a finite EC point in G Fixed option (specified in ): h2c_suite_ID_string - a hash-to-curve suite ID, encoded in ASCII (see discussion below) Steps PK_string = point_to_string(Y) string_to_hash = PK_string || alpha_string H = encode(string_to_hash) (the encode function is discussed below) Output H The encode function is provided by the hash-to-curve suite whose ID is h2c_suite_ID_string, as specified in , Section 8. The domain separation tag DST, a parameter to 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).

The following algorithms generate the nonce value k in a deterministic pseudorandom fashion. This section specifies a number of such algorithms, which are not compatible with each other. The choice of a particular algorithm from the options specified in this section is made in Section 5.5.

ECVRF_nonce_generation_RFC6979(SK, h_string) Input: SK - an ECVRF secret key h_string - an octet string Output: k - an integer between 1 and q-1 The ECVRF_nonce_generation function is as specified in Section 3.2 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 is set as hLen*8 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 All the other values and primitives as defined in

The following is from Steps 2-3 of Section 5.1.6 in . ECVRF_nonce_generation_RFC8032(SK, h_string) Input: SK - an ECVRF secret key h_string - an octet string Output: k - an integer between 0 and q-1 Steps: hashed_sk_string = Hash(SK) truncated_hashed_sk_string = hashed_sk_string[32]...hashed_sk_string[63] k_string = Hash(truncated_hashed_sk_string || h_string) k = string_to_int(k_string) mod q

ECVRF_hash_points(P1, P2, ..., PM) Input: P1...PM - EC points in G Output: c - hash value, integer between 0 and 2^(8n)-1 Steps: two_string = 0x02 = int_to_string(2, 1), a single octet with value 2 Initialize str = suite_string || two_string for PJ in [P1, P2, ... PM]: str = str || point_to_string(PJ) zero_string = 0x00 = int_to_string(0, 1), a single octet with value 0 str = str || zero_string c_string = Hash(str) truncated_c_string = c_string[0]...c_string[n-1] c = string_to_int(truncated_c_string) Output c

ECVRF_decode_proof(pi_string) Input: pi_string - VRF proof, octet string (ptLen+n+qLen octets) Output: "INVALID", or Gamma - EC point c - integer between 0 and 2^(8n)-1 s - integer between 0 and 2^(8qLen)-1 Steps: let gamma_string = pi_string[0]...pi_string[ptLen-1] let c_string = pi_string[ptLen]...pi_string[ptLen+n-1] let s_string =pi_string[ptLen+n]...pi_string[ptLen+n+qLen-1] Gamma = string_to_point(gamma_string) if Gamma = "INVALID" output "INVALID" and stop. c = string_to_int(c_string) s = string_to_int(s_string) Output Gamma, c, and s

This document defines ECVRF-P256-SHA256-TAI as follows: suite_string = 0x01 = int_to_string(1, 1), a single octet with value 1. The EC group G is the NIST P-256 elliptic curve, with curve parameters as specified in (Section D.1.2.3) and (Section 2.6). For this group, 2n = qLen = 32 and cofactor = 1. The key pair generation primitive is specified in Section 3.2.1 of (q, B, SK, and PK in this document correspond to n, G, d, and Q in Section 3.2.1 of ). In this ciphersuite, the secret scalar x is equal to the private key SK. The ECVRF_nonce_generation function is as specified in . The int_to_string function is the I2OSP function specified in Section 4.1 of . (This is big-endian representation.) The string_to_int function is the OS2IP function specified in Section 4.2 of . (This is big-endian representation.) The point_to_string function converts an EC point to an octet string according to the encoding specified in Section 2.3.3 of with point compression on. This implies ptLen = 2n + 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 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 an EC point according to the encoding specified in Section 2.3.4 of . This function MUST output INVALID if the octet string does not decode to an EC point. The hash function Hash is SHA-256 as specified in , with hLen = 32. The ECVRF_hash_to_curve function is as specified in , with arbitrary_string_to_point(s) = string_to_point(0x02 || s) (where 0x02 is a single octet with value 2, 0x02=int_to_string(2, 1)). The input s to arbitrary_string_to_point is a 32-octet string and the output is either an EC point or "INVALID". This document defines ECVRF-P256-SHA256-SSWU as identical to ECVRF-P256-SHA256-TAI, except that: suite_string = 0x02 = int_to_string(2, 1), a single octet with value 2. the ECVRF_hash_to_curve function is as specified in with h2c_suite_ID_string = P256_XMD:SHA-256_SSWU_NU_ (the suite is defined in Section 8.2) This document defines ECVRF-EDWARDS25519-SHA512-TAI as follows: suite_string = 0x03 = int_to_string(3, 1), a single octet with value 3. The EC group G is the edwards25519 elliptic curve with parameters defined in Table 1 of . For this group, 2n = qLen = 32 and cofactor = 8. The private key and generation of the secret scalar and the public key are specified in Section 5.1.5 of The ECVRF_nonce_generation function is as specified in . The int_to_string function as specified in the first paragraph of Section 5.1.2 of . (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 an EC point to an octet string according to the encoding specified in Section 5.1.2 of . This implies ptLen = 2n = 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 above encoding. When using such and implementation, the point_to_string function can be treated as the identity function.) The string_to_point function converts an octet string to an EC point according to the encoding specified in Section 5.1.3 of . This function MUST output INVALID if the octet string does not decode to an EC point. The hash function Hash is SHA-512 as specified in , with hLen = 64. The ECVRF_hash_to_curve function is as specified in , with arbitrary_string_to_point(s) = string_to_point(s[0]...s[31]). This document defines ECVRF-EDWARDS25519-SHA512-ELL2 as identical to ECVRF-EDWARDS25519-SHA512-TAI, except: suite_string = 0x04 = int_to_string(4, 1), a single octet with value 4. the ECVRF_hash_to_curve function is as specified in with h2c_suite_ID_string = edwards25519_XMD:SHA-512_ELL2_NU_ (the suite is defined in Section 8.5.)

The ECVRF as specified above is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in . In order to obtain "full uniqueness" and "full collision resistance" (which provide protection against a malicious VRF public key), the Verifier MUST perform the following additional validation procedure upon receipt of the public VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID". Note that this procedure is not sufficient if the elliptic curve E or the point B, the generator of group G, is untrusted. If the prover is untrusted, the Verifier MUST obtain E and B from a trusted source, such as a ciphersuite specification, rather than from the prover. This procedure supposes that the public key provided to the Verifier is an octet string. The procedure returns "INVALID" if the public key in invalid. Otherwise, it returns Y, the public key as an EC point.

ECVRF_validate_key(PK_string) Input: PK_string - public key, an octet string Output: "INVALID", or Y - public key, an EC point Steps: Y = string_to_point(PK_string) If Y is "INVALID", output "INVALID" and stop If cofactor*Y is the identity element of the elliptic curve group, output "INVALID" and stop Output Y Note that if the cofactor = 1, then Step 3 need not multiply Y by the cofactor; instead, it suffices to output "INVALID" if Y is the identity element of the elliptic curve group. Moreover, when cofactor>1, it is not necessary to verify that Y is in the subgroup G; Step 3 suffices. Therefore, if the cofactor is small, the total number of points that could cause Step 3 to output "INVALID" may be small, and it may be more efficient to simply check Y against a fixed list of such points. For example, the following algorithm can be used for the edwards25519 curve: Y = string_to_point(PK_string) If Y is "INVALID", output "INVALID" and stop y_string = PK_string oneTwentySeven_string = 0x7F = int_to_string(127, 1) (a single octet with value 127) y_string[31] = y_string[31] & oneTwentySeven_string (this step clears the high-order bit of octet 31) bad_pk[0] = int_to_string(0, 32) bad_pk[1] = int_to_string(1, 32) bad_y2 = 2707385501144840649318225287225658788936804267575313519463743609750303402022 bad_pk[2] = int_to_string(bad_y2, 32) bad_pk[3] = int_to_string(p-bad_y2, 32) bad_pk[4] = int_to_string(p-1, 32) bad_pk[5] = int_to_string(p, 32) bad_pk[6] = int_to_string(p+1, 32) If y_string is in bad_pk[0]...bad_pk[6], output "INVALID" and stop Output Y (bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys, depending on the sign of the x-coordinate, which was cleared in step 5, in order to make sure that it does not affect the comparison. bad_pk[1] and bad_pk[4] each match one bad public key, because x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6] are simply bad_pk[0] and bad_pk[1] shifted by p, in case the y-coordinate had not been modular reduced by p. There is no need to shift the other bad_pk values by p, because they will exceed 2^255. These bad keys, which represent all points of order 1, 2, 4, and 8, have been obtained by converting the points specified in to Edwards coordinates.)

Note to RFC editor: Remove before publication A reference C++ implementation of ECVRF-P256-SHA256-TAI, ECVRF-P256-SHA256-SSWU, ECVRF-EDWARDS25519-SHA512-TAI, and ECVRF-EDWARDS25519-SHA512-ELL2 is available at . This implementation is neither secure nor especially efficient, but can be used to generate test vectors. A Python implementation of an older version of ECVRF-EDWARDS25519-SHA512-ELL2 from the -05 version of this draft is available at . A C implementation of an older version of ECVRF-EDWARDS25519-SHA512-ELL2 from the -03 version of this draft is available at . A Rust implementation of an older version of ECVRF-P256-SHA256-TAI from the -05 version of this draft, as well as variants for the sect163k1 and secp256k1 curves, is available at . A C implementation of a variant of ECVRF-P256-SHA256-TAI from the -05 version of this draft adapted for the secp256k1 curve is available at . An implementation of an earlier version of RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256-TAI was first developed as a part of the NSEC5 project and is available at . The Key Transparency project at Google uses a VRF implementation that is similar to the ECVRF-P256-SHA256-TAI, with a few changes including the use of SHA-512 instead of SHA-256. Its implementation is available at An implementation by Ryuji Ishiguro following an older version of ECVRF-EDWARDS25519-SHA512-TAI from the -00 version of this draft is available at . An implementation similar to ECVRF-EDWARDS25519-SHA512-ELL2 (with some changes, including the use of SHA-3) is available as part of the CONIKS implementation in Golang at . Open Whisper Systems also uses a VRF similar to ECVRF-EDWARDS25519-SHA512-ELL2, called VXEdDSA, and specified here and here . Implementations in C and Java are available at and .

Applications that use the VRFs defined in this document MUST ensure that that the VRF key is generated correctly, using good randomness.

The ECVRF as specified in - satisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. If the Verifier trusts the VRF keys are generated correctly, it MAY use the public key Y as is. However, if the ECVRF uses keys that could be generated adversarially, then the Verifier MUST first perform the validation procedure ECVRF_validate_key(PK) (specified in ) upon receipt of the public key PK as an octet string. If the validation procedure outputs "INVALID", then the public key MUST not be used. Otherwise, the procedure will output a valid public key Y, and the ECVRF with public key Y satisfies the "full uniqueness" and "full collision resistance" properties. The RSA-FDH-VRF satisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. These properties may not hold if the keys are generated adversarially (e.g., if the RSA function specified in the public key is not bijective). Meanwhile, the "full uniqueness" and "full collision resistance" are properties that hold even if VRF keys are generated by an adversary. The RSA-FDH-VRF defined in this document does not have these properties. However, if adversarial key generation is a concern, the RSA-FDH-VRF may be modified to have these properties by adding additional cryptographic checks that its public key has the right form. These modifications are left for future specification.

Without good randomness, the "pseudorandomness" properties of the VRF may not hold. Note that it is not possible to guarantee pseudorandomness in the face of adversarially generated VRF keys. This is because an adversary can always use bad randomness to generate the VRF keys, and thus, the VRF output may not be pseudorandom.

presents cryptographic reductions to an underlying hard problem (e.g. Decisional Diffie-Hellman for the ECVRF, or the standard RSA assumption for RSA-FDH-VRF) that prove the VRFs specified in this document possess full pseudorandomness as well as selective pseudorandomness. However, the cryptographic reductions are tighter for selective pseudorandomness than for full pseudorandomness. This means that the VRFs have quantitively stronger security guarantees for selective pseudorandomness. Applications that are concerned about tightness of cryptographic reductions therefore have two options. They may choose to ensure that selective pseudorandomness is sufficient for the application. That is, that pseudorandomness of outputs matters only for inputs that are chosen independently of the VRF key. If full pseudorandomness is required for the application, the application may increase security parameters to make up for the loose security reduction. For RSA-FDH-VRF, this means increasing the RSA key length. For ECVRF, this means increasing the cryptographic strength of the EC group G. For both RSA-FDH-VRF and ECVRF, the cryptographic strength of the hash function Hash may also potentially need to be increased.

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 private VRF key x (and thus break pseudorandomness of the VRF) after observing several valid VRF proofs pi. The nonce generation methods specified in the ECVRF ciphersuites of are designed with this requirement in mind.

Side channel attacks on cryptographic primitives are an important issue. Here we discuss only one such side channel: timing attacks that can be used to leak information about the VRF input alpha. Implementers should take care to avoid side-channel attacks that leak information about the VRF private key SK (and the nonce k used in the ECVRF). The ECVRF_hash_to_curve_try_and_increment algorithm defined in SHOULD NOT be used in applications where the VRF input alpha is secret and is hashed by the VRF on-the-fly. This is because the algorithm's running time depends on the VRF input alpha, and thus creates a timing channel that can be used to learn information about alpha. That said, for most inputs the amount of information obtained from such a timing attack is likely to be small (1 bit, on average), since the algorithm is expected to find a valid curve point after only two attempts. However, there might be inputs which cause the algorithm to make many attempts before it finds a valid curve point; for such inputs, the information leaked in a timing attack will be more than 1 bit. ECVRF-P256-SHA256-SSWU and ECVRF-EDWARDS25519-SHA512-ELL2 can be made to run in time independent of alpha, following recommendations in .

The VRF proof pi is not designed to provide secrecy and, in general, may reveal the VRF input alpha. Anyone who knows PK and pi is able to perform an offline dictionary attack to search for alpha, by verifying guesses for alpha using VRF_verify. This is in contrast to the VRF hash output beta which, without the proof, is pseudorandom and thus is designed to reveal no information about alpha.

The VRFs specified in this document allow for read-once access to the input alpha for both signing and verifying. Thus, additional prehashing of alpha (as specified, for example, in for EdDSA signatures) is not needed, even for applications that need to handle long alpha or to support the Initialized-Update-Finalize (IUF) interface (in such an interface, alpha is not supplied all at once, but rather in pieces by a sequence of calls to Update). The ECVRF, in particular, uses alpha only in ECVRF_hash_to_curve. The curve point H becomes the representative of alpha thereafter. Note that the suite_string octet and the public key are hashed together with alpha in ECVRF_hash_to_curve, which ensures that the curve (including the generator B) and the public key are included indirectly into subsequent hashes.

Hashing is used for different purposes in the two VRFs (namely, in the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in hash_to_curve, nonce_generation, hash_points, and proof_to_hash). The theoretical analysis assumes each of these functions is a separate random oracle. This analysis still holds even if the same hash function is used, as long as the four queries made to the hash function for a given SK and alpha are overwhelmingly unlikely to equal each other or to any queries made to the hash function for the same SK and different alpha. This is indeed the case for the RSA-FDH-VRF defined in this document, because the first octets of the input to the hash function used in MGF1 and in proof_to_hash are different. This is also the case for the ECVRF ciphersuites defined in this document, because: inputs to the hash function used during nonce_generation are unlikely to equal inputs used in hash_to_curve, proof_to_hash, and hash_points. 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, hash_points, and ECVRF_hash_to_curve_try_and_increment are all different. the last octet of the input to the hash function used in proof_to_hash, hash_points, and ECVRF_hash_to_curve_try_and_increment is always zero, and therefore different from the last octet of the input to the hash function used in ECVRF_hash_to_curve_h2c_suite, which is set equal to the nonzero length of the domain separation tag by . For the RSA VRF, if future designs need to specify variants of the design in this document, such variants should use different first octets in inputs to MGF1 and to the hash function used in proof_to_hash, in order to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant For the elliptic curve VRF, if future designs need to specify variants (e.g., additional ciphersuites) of the design in this document, then, to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant, they should specify a different suite_string constant. This way, the inputs to the hash_to_curve hash function used in producing H are guaranteed to be different; since all the other hashing done by the prover depends on H, inputs all the hash functions used by the prover will also be different as long as hash_to_curve is collision resistant.

Note to RFC Editor: if this document does not obsolete an existing RFC, please remove this appendix before publication as an RFC. 00 - Forked this document from draft-goldbe-vrf-01. 01 - Minor updates, mostly highlighting TODO items. 02 - Added specification of elligator2 for Curve25519, along with ciphersuites for ECVRF-ED25519-SHA512-Elligator. Changed ECVRF-ED25519-SHA256 suite_string to ECVRF-ED25519-SHA512. (This change made because Ed25519 in signatures use SHA512 and not SHA256.) Made ECVRF nonce generation a separate component, so that nonces are deterministic. In ECVRF proving, changed + to - (and made corresponding verification changes) in order to be consistent with EdDSA and ECDSA. Highlighted that ECVRF_hash_to_curve acts like a prehash. Added "suites" variable to ECVRF for futureproofing. Ensured domain separation for hash functions by modifying hash_points and added discussion about domain separation. Updated todos in the "additional pseudorandomness property" section. Added a discussion of secrecy into security considerations. Removed B and PK=Y from ECVRF_hash_points because they are already present via H, which is computed via hash_to_curve using the suite_string (which identifies B) and Y. 03 - Changed Ed25519 conversions to little-endian, to match RFC 8032; added simple key validation for Ed25519; added Simple SWU cipher suite; clarified Elligator and removed the extra x0 bit, to make Montgomery and Edwards Elligator the same; added domain separation for RSA VRF; improved notation throughout; added nonce generation as a section; changed counter in try-and-increment from four bytes to one, to avoid endian issues; renamed try-and-increment ciphersuites to -TAI; added qLen as a separate parameter; changed output length to hLen for ECVRF, to match RSAVRF; made Verify return beta so unverified proofs don't end up in proof_to_hash; added test vectors. 04 - Clarified handling of optional arguments x and PK in ECVRF_prove. Edited implementation status to bring it up to date. 05 - Renamed ed25519 into the more commonly used edwards25519. Corrected ECVRF_nonce_generation_RFC6979 (thanks to Gorka Irazoqui Apecechea and Mario Cao Cueto for finding the problem) and corresponding test vectors for the P256 suites. Added a reference to the Rust implementation. 06 - Made some variable names more descriptive. Added a few implementation references. 07 - Incorporated hash-to-curve draft by reference to replace our own Elligator2 and Simple SWU. Clarified discussion of EC parameters and functions. Added a 0 octet to all hashing to enforce domain separation from hashing done inside hash-to-curve. 08 - Incorporated suggestions from crypto panel review by Chloe Martindale. Changed Reyzin's affiliation. Updated references. 09 - Added a note to remove the implementation page before publication.

This document also would not be possible without the work of Moni Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf Ziv (Facebook). Shumon Huque, David C. Lawerence, Trevor Perrin, Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova, Tony Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood, Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, Gary Belvinm, Piotr Nojszewski, Gorka Irazoqui Apecechea, and Mario Cao Cueto provided valuable input to this draft. Riad Wahby was very helpful with the integration of the hash-to-curve draft.

National Institute for Standards and Technology Standards for Efficient Cryptography Group (SECG)

The test vectors in this section were generated using the reference implementation at .

These two example secret keys and messages are taken from Appendix A.2.5 of . SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 alpha = 73616d706c65 (ASCII "sample") try_and_increment succeeded on ctr = 1 H = 0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4 k = 0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77 U = k*B = 02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672 V = k*H = 02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394 pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5071b498e7c291a16dafb9ccff8c2ae1f039fa92a328d5f7e0d483ee18353067a13f699944a78892ff24939bcd044827eef884 beta = a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 alpha = 74657374 (ASCII "test") try_and_increment succeeded on ctr = 3 H = 02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974 k = 5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961 U = k*B = 022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48e V = k*H = 033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119 pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080ddb56c874cc95b7d29a6a65cb518fe6f4418256385f12b1eccbad023c901bb983ff707b109b3a3b526ca3a1e8661f7b8481a2 beta = a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d This example secret key is taken from Appendix L.4.2 of . SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8 PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035 (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 = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b8c6679d92353c8a4fdfddb2a8540094b686cb5fb50f730d833a098a0399ccad32f3fec4da2299891fc75ebda42baeb65e8c11 beta = 90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19

These two example secret keys and messages are taken from Appendix A.2.5 of . SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 alpha = 73616d706c65 (ASCII "sample") In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c2a4b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7e 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 = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394c7888a39dfe7432f119228473f37db3f87ca470c63b0237432a791f18f823c1215e276b7ac0962725ba8daec2bf90c0ccc91a beta = 21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b SK = x = c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721 PK = 0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6 alpha = 74657374 (ASCII "test") In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109afb243b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3 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 = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa8db2721e0499b7cecd68027a82f6095da076625a5f2f62908f1c283d5ee9b9e852d85bedf64f2452a4e5094729e101824443e beta = 8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd This example secret key is taken from Appendix L.4.2 of . SK = x = 2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8 PK = 03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035 (ASCII "Example using ECDSA key from Appendix L.4.2 of ANSI.X9-62-2005") In SSWU: uniform_bytes = 9b81d55a242d3e8438d3bcfb1bee985a87fd144802c9268cf9adeee160e6e9ff765569797a0f701cb4316018de2e7dd4 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 = 039f8d9cdc162c89be2871cbcb1435144739431db7fab437ab7bc4e2651a9e99d5288aac70a5e4bd07df303c1d460eb6336bb5fa95436a07c2f6b7aec6fef7cc4846ea901ee1e238dee12bf752029b0b2e beta = 4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a

These three example secret keys and messages are taken from Section 7.1 of . SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60 PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a alpha = (the empty string) x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f try_and_increment succeeded on ctr = 0 H = 91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318 k = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106f345b8594b6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2a6afae4 U = k*B = aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071 V = k*H = 5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f723f5e8bd1839b414219e8626d393787a192241fc442e6569e96c462f62b8079b9ed83ff2ee21c90c7c398802fdeebea4001 beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd8ae SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c alpha = 72 (1 byte) x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51 try_and_increment succeeded on ctr = 1 H = 5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551 k = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b2342acb9c167cab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc16cc5ce U = k*B = 1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3 V = k*H = fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423 pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed593f7eaf3eb2f1a968cba3f6e23b386aeeaab7b1ea44a256e811892e13eeae7c9f6ea8992557453eac11c4d5476b1f35a08 beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496310eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5031 SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7 PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025 alpha = af82 (2 bytes) x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c try_and_increment succeeded on ctr = 0 H = bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a k = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59fc31a0febc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6bb6d6c3a U = k*B = 2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5 V = k*H = 88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506 pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccbf80e29dc513c01c3a980e0e545bcd848222d08a6c3e3665ff5a4cab13a643bef812e284c6b2ee063a2cb4f456794723ad0a beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df19f

These three example secret keys and messages are taken from Section 7.1 of . SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60 PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a alpha = (the empty string) x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f In Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca5298b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269ae In Elligator2: u = 30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55c In Elligator2: gx1 = 8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08 In Elligator2: gx1 is a square H = b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9 k = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a42458f817bed7a386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88dab8d41c U = k*B = 762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9 V = k*H = 4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab46f25898f6bd7d4ed4c75f0282b0f7bb9d0e61b387b76db60b3cbf34bf09109ccb33fab742a8bddc0c8ba3caf5c0b75bb04 beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6c54 SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c alpha = 72 (1 byte) x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51 In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd77bb99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6d In Elligator2: u = 3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773d In Elligator2: gx1 = 25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464d In Elligator2: gx1 is a nonsquare H = 76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57 k = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37c4c37f1d41e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab7544e444d U = k*B = 8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061 V = k*H = fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8b pi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c509072ef9bf1a234f833f72d8fff36075fd9b836da28b5569e74caa418bae7ef521f2ddd35f5727d271ecc70b4a83c1fc8ebc40c beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4635987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc735 SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7 PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025 alpha = af82 (2 bytes) x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c In Elligator2: uniform_bytes = be0aed556e36cdfddf8f1eeddbb7356a24fad64cf95a922a098038f215588b216beabbfe6acf20256188e883292b7a3a In Elligator2: u = f6675dc6d17fc790d4b3f1c6acf689a13d8b5815f23880092a925af94cd6fa24 In Elligator2: gx1 = a63d48e3247c903e22fdfb88fd9295e396712a5fe576af335dbe16f99f0af26c In Elligator2: gx1 is a square H = 13d2a8b5ca32db7e98094a61f656a08c6c964344e058879a386a947a4e189ed1 k = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5aaf79cf576087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861b3d683b U = k*B = a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60 V = k*H = fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25a pi = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57fdce6187befa109606682503b3a1424f0f729ca0418099fbd86a48093e6a8de26307b8d93e02da927e6dd5b73c8f119aee0f beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5cd090a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314a58