CFRG S. Goldberg Internet-Draft L. Reyzin Intended status: Standards Track Boston University Expires: December 30, 2018 D. Papadopoulos Hong Kong University of Science and Techology J. Vcelak NS1 June 28, 2018 Verifiable Random Functions (VRFs) draft-irtf-cfrg-vrf-02 Abstract A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private key can compute the hash, but anyone with public key can verify the correctness of the hash. VRFs are useful for preventing enumeration of hash-based data structures. This document specifies several VRF constructions that are secure in the cryptographic random oracle model. One VRF uses RSA and the other VRF uses Eliptic Curves (EC). Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on December 30, 2018. Copyright Notice Copyright (c) 2018 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 Goldberg, et al. Expires December 30, 2018 [Page 1]

Internet-Draft VRF June 2018 publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 3 1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3 2. VRF Algorithms . . . . . . . . . . . . . . . . . . . . . . . 4 3. VRF Security Properties . . . . . . . . . . . . . . . . . . . 4 3.1. Full Uniqueness or Trusted Uniqueness . . . . . . . . . . 5 3.2. Full Collison Resistance or Trusted Collision Resistance 5 3.3. Full Pseudorandomness or Selective Pseudorandomness . . . 5 3.4. An additional pseudorandomness property . . . . . . . . . 6 4. RSA Full Domain Hash VRF (RSA-FDH-VRF) . . . . . . . . . . . 7 4.1. RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . . 8 4.2. RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . . 8 4.3. RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . . 9 5. Elliptic Curve VRF (ECVRF) . . . . . . . . . . . . . . . . . 9 5.1. ECVRF Proving . . . . . . . . . . . . . . . . . . . . . . 11 5.2. ECVRF Proof To Hash . . . . . . . . . . . . . . . . . . . 12 5.3. ECVRF Verifying . . . . . . . . . . . . . . . . . . . . . 13 5.4. ECVRF Auxiliary Functions . . . . . . . . . . . . . . . . 13 5.4.1. ECVRF Hash To Curve . . . . . . . . . . . . . . . . . 13 5.4.2. ECVRF Hash Points . . . . . . . . . . . . . . . . . . 17 5.4.3. ECVRF Decode Proof . . . . . . . . . . . . . . . . . 17 5.5. ECVRF Ciphersuites . . . . . . . . . . . . . . . . . . . 18 5.6. When the ECVRF Keys are Untrusted . . . . . . . . . . . . 19 5.6.1. ECVRF Validate Key . . . . . . . . . . . . . . . . . 20 6. Implementation Status . . . . . . . . . . . . . . . . . . . . 20 7. Security Considerations . . . . . . . . . . . . . . . . . . . 21 7.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 21 7.1.1. Uniqueness and collision resistance with untrusted keys . . . . . . . . . . . . . . . . . . . . . . . . 21 7.1.2. Pseudorandomness with untrusted keys . . . . . . . . 22 7.2. Selective vs Full Pseudorandomness . . . . . . . . . . . 22 7.3. Proper pseudorandom nonce for ECVRF . . . . . . . . . . . 22 7.4. Timing attacks . . . . . . . . . . . . . . . . . . . . . 23 7.5. Proofs Provide No Secrecy for VRF Input . . . . . . . . . 23 7.6. Prehashing . . . . . . . . . . . . . . . . . . . . . . . 23 7.7. Hash function domain separation and future-proofing . . . 24 8. Change Log . . . . . . . . . . . . . . . . . . . . . . . . . 24 9. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 25 Goldberg, et al. Expires December 30, 2018 [Page 2]

Internet-Draft VRF June 2018 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 25 10.1. Normative References . . . . . . . . . . . . . . . . . . 25 10.2. Informative References . . . . . . . . . . . . . . . . . 26 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 27 1. Introduction 1.1. Rationale A Verifiable Random Function (VRF) [MRV99] is the public-key version of a keyed cryptographic hash. Only the holder of the private VRF key can compute the hash, but anyone with corresponding public key can verify the correctness of the hash. A key application of the VRF is to provide privacy against offline enumeration (e.g. dictionary attacks) on data stored in a hash-based data structure. In this application, a Prover holds the VRF private key and uses the VRF hashing to construct a hash-based data structure on the input data. Due to the nature of the VRF, only the Prover can answer queries about whether or not some data is stored in the data structure. Anyone who knows the public VRF key can verify that the Prover has answered the queries correctly. However no offline inferences (i.e. inferences without querying the Prover) can be made about the data stored in the data strucuture. 1.2. Requirements The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119]. 1.3. Terminology The following terminology is used through this document: SK: The private key for the VRF. PK: The public key for the VRF. alpha: The input to be hashed by the VRF. beta: The VRF hash output. pi: The VRF proof. Prover: The Prover holds the private VRF key SK and public VRF key PK. Goldberg, et al. Expires December 30, 2018 [Page 3]

Internet-Draft VRF June 2018 Verifier: The Verifier holds the public VRF key PK. 2. VRF Algorithms 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 a pair of inputs (SK, alpha). The prover also uses the private key SK to construct a proof pi that beta is the correct hash output 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 as beta = VRF_proof2hash(pi) Notice that this means that VRF_hash(SK, alpha) = VRF_proof2hash(VRF_prove(SK, alpha)) and thus this document will specify VRF_prove and VRF_proof2hash 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 if beta=VRF_proof2hash(pi) is the correct VRF hash of alpha under key PK, and outputs INVALID otherwise. 3. VRF Security Properties VRFs are designed to ensure the following security properties. Goldberg, et al. Expires December 30, 2018 [Page 4]

Internet-Draft VRF June 2018 3.1. Full Uniqueness or Trusted Uniqueness 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 percisely, "full uniqueness" states that a computationally- bounded adversary cannot choose a VRF public key PK, a VRF input alpha, two different VRF hash outputs beta1 and beta2, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha, pi1) and VRF_verify(PK, alpha, pi2) both output VALID. A slightly weaker security property called "trusted uniquness" sufficies for many applications. Trusted uniqueness is the same as full uniqueness, but it must hold only if the VRF keys PK and SK were generated in a trustworthy manner. In other words, uniqueness might not hold if keys were generated in an invalid manner or with bad randomness. 3.2. Full Collison Resistance or Trusted Collision Resistance Like any cryprographic hash function, VRFs need to be collision resistant. Collison resistance must hold even for an adversarial Prover that knows the VRF private key SK. More percisely, "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. 3.3. Full Pseudorandomness or Selective Pseudorandomness 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 percisely, 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 Goldberg, et al. Expires December 30, 2018 [Page 5]

Internet-Draft VRF June 2018 gets to choose other VRF inputs alpha' and observe their corresponding VRF hash outputs beta' and proofs pi'. With "full pseudorandomness", the adversary is allowed to choose the "target" VRF input alpha at any time, even after it observes VRF outputs beta' and proofs pi' on a variety of chosen inputs alpha'. "Selective pseudorandomness" is a weaker security property which suffices in many applications. Here, the adversary must choose the target VRF input alpha independently of the public VRF key PK, and before it observes VRF outputs beta' and proofs pi' on inputs alpha' of its choice. It is important to remember that the VRF output beta does not look random to the Prover, or to any other party that knows the private VRF key SK! Such a party can easily distinguish beta from a random value by comparing beta to the result of VRF_hash(SK, alpha). Also, the VRF output beta does not look random to any party that knows valid VRF proof pi corresponding to the VRF input alpha, even if this party does not know the private VRF key SK. Such a party can easily distinguish beta from a random value by checking whether VRF_verify(PK, alpha, pi) returns "VALID" and beta = VRF_proof2hash(pi). 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.) 3.4. An additional pseudorandomness property [TODO: This property is not needed for applications that use VRFs to prevent enumeration of hash-based data structures. However, we noticed that some other applications of VRF (e.g. Algorand, Oroborus) rely on this property. We are waiting on a formal definition of this property in the literature, and a proof that our ECVRF scheme can satisfy this property. Preliminary analysis suggests that acheiving this property requires ECVRF verifiers to run an VRF_validate_key() key function upon receipt of VRF public keys and the proof2hash function to be modified to take in (gamma, beta, pk) rather than just gamma.] Pseudorandomness, as defined in Section 3.3, does not hold if the VRF keys were generated adversarially. There is, however, a different type of pseudorandomness that could hold even if the VRF keys are generated adversarially, as long as the VRF input alpha is unpredictable. This property is similar to the Goldberg, et al. Expires December 30, 2018 [Page 6]

Internet-Draft VRF June 2018 pseudorandomness achieved by an (ordinary, unkeyed) cryptographic hash function. [TODO: Formal definition here.] 4. RSA Full Domain Hash VRF (RSA-FDH-VRF) The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. Its security follows from the standard RSA assumption in the random oracle model. Formal security proofs are in [nsec5ecc]. The VRF computes the proof pi as a deterministic RSA signature on input alpha using the RSA Full Domain Hash Algorithm [RFC8017] parametrized with the selected hash algorithm. RSA signature verification is used to verify the correctness of the proof. The VRF hash output beta is simply obtained by hashing the proof pi with the selected hash algorithm. The key pair for RSA-FDH-VRF MUST be generated in a way that it satisfies the conditions specified in Section 3 of [RFC8017]. In this document, the notation from [RFC8017] is used. Parameters used: (n, e) - RSA public key K - RSA private key k - length in octets of the RSA modulus n Fixed options: Hash - cryptographic hash function hLen - output length in octets of hash function Hash Constraints on options: Cryptographic security of Hash is at least as high as the cryptographic security level of the RSA key. Primitives used: I2OSP - Coversion of a nonnegative integer to an octet string as defined in Section 4.1 of [RFC8017] Goldberg, et al. Expires December 30, 2018 [Page 7]

Internet-Draft VRF June 2018 OS2IP - Coversion of an octet string to a nonnegative integer as defined in Section 4.2 of [RFC8017] RSASP1 - RSA signature primitive as defined in Section 5.2.1 of [RFC8017] RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of [RFC8017] MGF1 - Mask Generation Function based on a hash function as defined in Section B.2.1 of [RFC8017] 4.1. RSA-FDH-VRF Proving RSAFDHVRF_prove(K, alpha) Input: K - RSA private key alpha - VRF hash input, an octet string Output: pi - proof, an octet string of length k Steps: 1. EM = MGF1(alpha, k - 1) 2. m = OS2IP(EM) 3. s = RSASP1(K, m) 4. pi = I2OSP(s, k) 5. Output pi 4.2. RSA-FDH-VRF Proof To Hash RSAFDHVRF_proof2hash(pi) Input: pi - proof, an octet string of length k Output: Goldberg, et al. Expires December 30, 2018 [Page 8]

Internet-Draft VRF June 2018 beta - VRF hash output, an octet string of length hLen Steps: 1. beta = Hash(pi) 2. Output beta 4.3. RSA-FDH-VRF Verifying RSAFDHVRF_verify((n, e), alpha, pi) Input: (n, e) - RSA public key alpha - VRF hash input, an octet string pi - proof to be verified, an octet string of length n Output: "VALID" or "INVALID" Steps: 1. s = OS2IP(pi) 2. m = RSAVP1((n, e), s) 3. EM = I2OSP(m, k - 1) 4. EM' = MGF1(alpha, k - 1) 5. If EM and EM' are equal, output "VALID"; else output "INVALID". 5. Elliptic Curve VRF (ECVRF) The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that satisfies the trusted uniqueness, trusted collision resistance, and full pseudorandomness properties defined in Section 3. The security of this VRF follows from the decisional Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are in [nsec5ecc]. To additionally satisfy "full uniqueness" and "full collision resistance", the Verifier MUST additionally perform the validation Goldberg, et al. Expires December 30, 2018 [Page 9]

Internet-Draft VRF June 2018 procedure specified in Section 5.6 upon receipt of the public VRF key. Fixed options (specified in Section 5.5): F - finite field 2n - length, in octets, of a field element in F; must be even E - elliptic curve (EC) defined over F m - 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; must be less than 2^{2n} cofactor - number of points on E divided by q g - generator of group G Hash - cryptographic hash function hLen - output length in octets of Hash; must be at least 2n suite - a single nonzero octet specifying the ECVRF ciphersuite, which determines the above options Notation and primitives used: p^k - When p is an EC point: point multiplication, i.e. k repetitions of the EC group operation applied to the EC point p. When p is an integer: exponentiation || - octet string concatenation I2OSP - nonnegative integer conversion to octet string as defined in Section 4.1 of [RFC8017] OS2IP - Coversion of an octet string to a nonnegative integer as defined in Section 4.2 of [RFC8017] EC2OSP - conversion of EC point to an m-octet string as specified in Section 5.5 OS2ECP - conversion of an m-octet string to EC point as specified in Section 5.5. OS2ECP returns INVALID if the octet string does not convert to a valid EC point. Goldberg, et al. Expires December 30, 2018 [Page 10]

Internet-Draft VRF June 2018 RS2ECP - conversion of a random 2n-octet string to an EC point as specified in Section 5.5 ECVRF_hash_to_curve - collision resistant hash of strings to an EC point; options described in Section 5.4.1 and specified in Section 5.5. ECVRF_nonce_generation - derives a pseudorandom nonce from SK and the input as part of ECVRF proving. Specified in Section 5.5 ECVRF_hash_points - collision resistant hash of EC points to an n-octet string. Specified in Section 5.4.2. Parameters used (the generation of these parameters is specified in Section 5.5): 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 = g^x - VRF public key, an EC point 5.1. ECVRF Proving Note: this function must have the VRF private key SK as input. Below we make it more efficient by supplying it also with the secret scalar x and the public key y as additional inputs; however, each of these can be computed from SK if desired. ECVRF_prove(y, x, alpha) Input: SK - VRF private key x - VRF secret scalar y = g^x - VRF public key Output: pi - VRF proof, octet string of length m+3n Steps: Goldberg, et al. Expires December 30, 2018 [Page 11]

Internet-Draft VRF June 2018 1. h = ECVRF_hash_to_curve(suite, y, alpha) 2. h1 = EC2OSP(h) 3. gamma = h^x 4. k = ECVRF_nonce_generation(SK, h1) 5. c = ECVRF_hash_points(h, gamma, g^k, h^k) 6. s = (k + c*x) mod q (where * denotes integer multiplication) 7. pi = EC2OSP(gamma) || I2OSP(c, n) || I2OSP(s, 2n) 8. Output pi 5.2. ECVRF Proof To Hash ECVRF_proof2hash(pi) Input: pi - VRF proof, octet string of length m+3n Output: "INVALID", or beta - VRF hash output, octet string of length 2n Steps: 1. D = ECVRF_decode_proof(pi) 2. If D is "INVALID", output "INVALID" and stop 3. (gamma, c, s) = D 4. three = 0x03 = I2OSP(3, 1), a single octet with value 3 5. preBeta = Hash(suite || three || EC2OSP(gamma^cofactor)) 6. beta = first 2n octets of preBeta 7. Output beta Goldberg, et al. Expires December 30, 2018 [Page 12]

Internet-Draft VRF June 2018 5.3. ECVRF Verifying ECVRF_verify(y, pi, alpha) Input: y - public key, an EC point pi - VRF proof, octet string of length 5n+1 alpha - VRF input, octet string Output: "VALID" or "INVALID" Steps: 1. D = ECVRF_decode_proof(pi) 2. If D is "INVALID", output "INVALID" and stop 3. (gamma, c, s) = D 4. u = g^s / y^c (where / denotes EC point subtraction, i.e. the group operation applied to g^s and the inverse of y^c) 5. h = ECVRF_hash_to_curve(suite, y, alpha) 6. v = h^s / gamma^c (where / again denotes EC point subtraction) 7. c' = ECVRF_hash_points(h, gamma, u, v) 8. If c and c' are equal, output "VALID"; else output "INVALID" 5.4. ECVRF Auxiliary Functions 5.4.1. ECVRF Hash To Curve 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 alpha is used in the proving and verfying. See Section 7.6 for further discussion. The algorithms in this section are not compatible with each other; the choice of algorithm is made in Section 5.5. Goldberg, et al. Expires December 30, 2018 [Page 13]

Internet-Draft VRF June 2018 5.4.1.1. ECVRF_hash_to_curve_try_and_increment The following ECVRF_hash_to_curve_try_and_increment(suite, y, alpha) 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. 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. See also [Icart09]. However, because the running time of algorithm depends on alpha, this algorithm SHOULD be avoided in applications where it is important that the VRF input alpha remain secret. ECVRF_hash_to_try_and_increment(suite, y, alpha) Input: suite - a single octet specifying ECVRF ciphersuite. y - public key, an EC point alpha - value to be hashed, an octet string Output: h - hashed value, a finite EC point in G Steps: 1. ctr = 0 2. pk = EC2OSP(y) 3. one = 0x01 = I2OSP(1, 1), a single octet with value 1 4. h = "INVALID" 5. While h is "INVALID" or h is EC point at infinity: A. CTR = I2OSP(ctr, 4) B. ctr = ctr + 1 C. hash = Hash(suite || one || pk || alpha || CTR) D. attempted_hash = first 2n bits of hash Goldberg, et al. Expires December 30, 2018 [Page 14]

Internet-Draft VRF June 2018 E. h = RS2ECP(attempted_hash) F. If h is not "INVALID" and cofactor > 1, set h = h^cofactor 6. Output h 5.4.1.2. ECVRF_hash_to_curve_elligator2_25519 The following ECVRF__hash_to_curve_elligator2_25519(suite, y, alpha) algorithm implements ECVRF_hash_to_curve using the elligator2 algorithm exclusively for Curve25519. ECVRF_hash_to_curve_elligator2_25519(y, alpha) Input: alpha - value to be hashed, an octet string y - public key, an EC point suite = a single octet specifying ECVRF ciphersuite. Output: h - hashed value, a finite EC point in G per the encoding in RFC8032 Section 5.1.2 [RFC8032] Fixed options: p = 2^255-19, the size of the finite field F, a prime, for Curve25519 A = 486662, Montgomery curve constant for Curve25519 cofactor = 8 , the cofactor for Curve25519 * denotes integer multiplication Constraints on options: output length of Hash is at least 16n + 1 bits (i.e. greater than 2n octets) Steps: 1. pk = EC2OSP(y) 2. one = 0x01 = I2OSP(1, 1), a single octet with value 1 Goldberg, et al. Expires December 30, 2018 [Page 15]

Internet-Draft VRF June 2018 3. hash = Hash(suite || one || pk || alpha ) 4. r = first 2n octets of hash 5. x_0 = next unused bit of hash 6. u = - A / (1 + 2*(r^2) ) mod p 7. v = u * (u^2 + A*u + 1) mod p 8. Let e equal the Legendre symbol of v modulo p 9. If e is equal to 1 then finalu = u; else finalu = (-A - u) mod p 10. y = (finalu - 1) / (finalu + 1) mod p 11. let h = (x_0,y), an EC point per the encoding in Section 5.1.2 of [RFC8032] 12. h = h^cofactor In order to make this algorithm run in time that is (almost) independent of the input (so-called "constant-time"), implementers should pay particular attention to Steps 7 and 8 above. These steps can be implemented using the following approach: e = v ^ ((p-1)/2) mod p finalu = (e*u + (e-1) * (A/2)) mod p The first step will produce a value e that is either 1 or p-1. The second step is so fast compared to the first that that its dependence on e is likely to not matter; implementers can also ensure that it runs in the same amount of time regardless of e. If having this algorithm run in constant time is not important, then there are much faster algorithms to compute the Legendre symbol (which is the same as the Jacobi symbol because p is a prime). See, for example, Section 12.3 of [ntb]. 5.4.1.3. ECVRF_hash_to_curve_other? [TODO: In addition to Elligator, it would be nice to have a generic ECVRF_hash_to_curve algorithm that runs in constant time and provides a generic way to hash an octet string onto any elliptic curve. There is an an upcoming draft that deals with this [SW18]. Some other useful algorithms include [Icart09], and Shallue-Woestijne-Ulas algorithm from [BCIMRT10].] Goldberg, et al. Expires December 30, 2018 [Page 16]

Internet-Draft VRF June 2018 5.4.2. ECVRF Hash Points ECVRF_hash_points(p_1, p_2, ..., p_j) Input: p_i - EC point in G Output: c - hash value, integer between 0 and 2^(8n)-1 Steps: 1. two = 0x02 = I2OSP(2, 1), a single octet with value 2 2. Initialize str = suite || two 3. for p_i in [p_1, p_2, ... p_j]: str = str || EC2OSP(p_i) 4. c1 = Hash(str) 5. c2 = first n octets of c1 6. c = OS2IP(c2) 7. Output c 5.4.3. ECVRF Decode Proof ECVRF_decode_proof(pi) Input: pi - VRF proof, octet string (m+3n octets) Output: "INVALID", or gamma - EC point c - integer between 0 and 2^(8n)-1 s - integer between 0 and 2^(16n)-1 Steps: Goldberg, et al. Expires December 30, 2018 [Page 17]

Internet-Draft VRF June 2018 1. let gamma', c', s' be pi split after m-th and m+n-th octet 2. gamma = OS2ECP(gamma') 3. if gamma = "INVALID" output "INVALID" and stop. 4. c = OS2IP(c') 5. s = OS2IP(s') 6. Output gamma, c, and s 5.5. ECVRF Ciphersuites This document defines ECVRF-P256-SHA256 as follows: o suite = 0x01 = I2OSP(1, 1). o The EC group G is the NIST-P256 elliptic curve, with curve parameters as specified in [FIPS-186-3] (Section D.1.2.3) and [RFC5114] (Section 2.6). For this group, 2n = 32 and cofactor = 1. o The key pair generation primitive is specified in Section 3.2.1 of [SECG1] (q, g, SK, and PK in this document correspond to in n, G, d, and Q in Section 3.2.1 of [SECG1]). In this ciphersuite, the secret scalar x is equal to the private key SK. o The ECVRF_nonce_generation function is as specified in [RFC6979] Section 3.2 Steps b-h (omitting Step a and using h1 that is provided as input to ECVRF_nonce_generation), with the hash function SHA-256 as specified in [RFC6234] o EC2OSP is specified in Section 2.3.3 of [SECG1] with point compression on. This implies m = 2n + 1 = 33. o OS2ECP is specified in Section 2.3.4 of [SECG1]. o RS2ECP(h) = OS2ECP(0x02 || h) (where 0x02 is a single octet with value 2, 0x02=I2OSP(2, 1)). The input h is a 32-octet string and the output is either an EC point or "INVALID". o The hash function Hash is SHA-256 as specified in [RFC6234]. o The ECVRF_hash_to_curve function is as specified in Section 5.4.1.1. This document defines ECVRF-ED25519-SHA512 as follows: Goldberg, et al. Expires December 30, 2018 [Page 18]

Internet-Draft VRF June 2018 o suite = 0x02 = I2OSP(2, 1). o The EC group G is the Ed25519 elliptic curve with parameters defined in Table 1 of [RFC8032]. For this group, 2n = 32 and cofactor = 8. o The private key and generation of the secret scalar and the public key are specified in Section 5.1.5 of [RFC8032] o The ECVRF_nonce_generation function is as specified in [RFC8032] Section 5.1.6 Steps 1-2, with dom2(F, C) equal to the empty string and PH(M) equal to h1. o EC2OSP is specified in Section 5.1.2 of [RFC8032]. This implies m = 2n = 32. o OS2ECP is specified in Section 5.1.3 of [RFC8032]. o RS2ECP is equivalent to OS2ECP. o The hash function Hash is SHA-512 as specified in [RFC6234]. o The ECVRF_hash_to_curve function is as specified in Section 5.4.1.1. This document defines ECVRF-ED25519-SHA512-Elligator2 as follows: o This ciphersuite is identical to ECVRF-ED25519-SHA512 except that the ECVRF_hash_to_curve function is as specified in Section 5.4.1.2 and suite = 0x03 = I2OSP(3, 1). 5.6. When the ECVRF Keys are Untrusted The ECVRF as specified above is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. In order to obtain "full uniqueness" and "full collision resistance" (which provide protection against a malicious VRF public key), the Verifier MUST perform the following additional validation procedure upon receipt of the public VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID". Note that this procedure is not sufficient if the elliptic curve E or the point g, the generator of group G, is untrusted. If the prover is untrusted, the Verifier MUST obtain E and g from a trusted source, such as a ciphersuite specification, rather than from the prover. Goldberg, et al. Expires December 30, 2018 [Page 19]

Internet-Draft VRF June 2018 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. 5.6.1. ECVRF Validate Key ECVRF_validate_key(PK) Input: PK - public key, an octet string Output: "INVALID", or y - public key, an EC point Steps: 1. y = OS2ECP(PK) 2. If y is "INVALID", output "INVALID" and stop 3. If y^cofactor is the EC point at infinty, output "INVALID" and stop 4. Output y 6. Implementation Status An implementation of the RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256 was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5] and is available at <http://github.com/fcelda/nsec5-crypto>. The ECVRF implementation may be out of date as this spec has evolved. The Key Transparency project at Google uses a VRF implemention that is similar to the ECVRF-P256-SHA256, with a few minor changes including the use of SHA-512 instead of SHA-256. Its implementation is available <https://github.com/google/keytransparency/blob/master/core/vrf/ vrf.go> An implementation by Yahoo! similar to the ECVRF is available at <https://github.com/r2ishiguro/vrf>. Goldberg, et al. Expires December 30, 2018 [Page 20]

Internet-Draft VRF June 2018 An implementation similar to ECVRF is available as part of the CONIKS implementation in Golang at <https://github.com/coniks-sys/coniks- go/tree/master/crypto/vrf>. Open Whisper Systems also uses a VRF very similar to ECVRF- ED25519-SHA512-Elligator, called VXEdDSA, and specified here: <https://whispersystems.org/docs/specifications/xeddsa/> 7. Security Considerations 7.1. Key Generation Applications that use the VRFs defined in this document MUST ensure that that the VRF key is generated correctly, using good randomness. 7.1.1. Uniqueness and collision resistance with untrusted keys The ECVRF as specified in Section 5.1-Section 5.5 statisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. If the Verifier trusts the VRF keys are generated correctly, it MAY use the public key y as is. However, if the ECVRF uses keys that could be generated adversarially, then the the Verfier MUST first perform the validation procedure ECVRF_validate_key(PK) (specified in Section 5.6) upon receipt of the public key PK as an octet string. If the validation procedure outputs "INVALID", then the public key MUST not be used. Otherwise, the procedure will output a valid public key y, and the ECVRF with public key y satisfies the "full uniqueness" and "full collision resistance" properties. The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. These properties may not hold if the keys are generated adversarially (e.g., if RSA is not permutation). Meanwhile, the "full uniqueness" and "full collision resistance" are properties that hold even if VRF keys are generated by an adversary. The RSA-FDH-VRF defined in this document does not have these properties. However, if adversarial key generation is a concern, the RSA-FDH-VRF may be modifed to have these properties by adding additional cryptographic checks that its public key has the right form. These modifications are left for future specification. Goldberg, et al. Expires December 30, 2018 [Page 21]

Internet-Draft VRF June 2018 7.1.2. Pseudorandomness with untrusted keys 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. 7.2. Selective vs Full Pseudorandomness [nsec5ecc] presents cryptographic reductions to an underlying hard problem (e.g. Decisional Diffie Hellman for the ECVRF, or the standard RSA assumption for RSA-FDH-VRF) that prove the VRFs specificied in this document possess full pseudorandomness as well as selective pseudorandomness. However, the cryptographic reductions are tighter for selective pseudorandomness than for full pseudorandomness. This means the the VRFs have quantitavely stronger security guarentees for selective pseudorandomness. Applications that are concerned about tightness of cryptographic reductions therefore have two options. o 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. o 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. 7.3. Proper pseudorandom nonce for ECVRF The security of the ECVRF defined in this document relies on the fact that nonce k used in the ECVRF_prove algorithm is chosen uniformly and pseudorandomly modulo q, and is unknown to the advesrary. Otherwise, an adversary may be able to recover the private VRF key x (and thus break pseudorandomness of the VRF) after observing several valid VRF proofs pi. The nonce generation methods specified in the ECVRF ciphersuites of Section 5.5 are designed with this requirement in mind. Goldberg, et al. Expires December 30, 2018 [Page 22]

Internet-Draft VRF June 2018 7.4. Timing attacks The ECVRF_hash_to_curve_try_and_increment algorithm defined in Section 5.4.1.1 SHOULD NOT be used in applications where the VRF input alpha is secret and is hashed by the VRF on-the-fly. This is because the algorithm's running time depends on the VRF input alpha, and thus creates a timing channel that can be used to learn information about alpha. That said, for most inputs the amount of information obtained from such a timing attack is likely to be small (1 bit, on average), since the algorithm is expected to find a valid curve point after only two attempts. However, there might be inputs which cause the algorithm to make many attempts before it finds a valid curve point; for such inputs, the information leaked in a timing attack will be more than 1 bit. Meanwhile, ECVRF-ED25519-SHA512-Elligator2 runs in constant time if the implementation of the ECVRF_hash_to_curve function specified in Section 5.4.1.2 also runs in constant time. 7.5. Proofs Provide No Secrecy for 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. 7.6. 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 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 octet and the public key are hashed together with alpha in ECVRF_hash_to_curve, which ensures that the curve (including the generator g) and the public key are included indirectly into subsequent hashes. Goldberg, et al. Expires December 30, 2018 [Page 23]

Internet-Draft VRF June 2018 7.7. Hash function domain separation and future-proofing Hashing is used for four different purposes in ECVRF (namely, in hash_to_curve, nonce_generation, hash_points, and proof2hash). The theoretical analysis assumes each of these four 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 ECVRF ciphersuites defined in this document, because: o inputs to the hash function used during nonce_generation are unlikely to equal to inputs given to hash_to_curve, proof2hash, 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 o the second octet of the input to the hash function used in hash_to_curve, proof2hash, and hash_points are all different 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 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. 8. Change Log 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 to ECVRF-ED25519-SHA512. (This change made because Ed25519 in [RFC8032] signatures use SHA512 and not SHA256.) Made ECVRF nonce generation a separate component, so that nonces are determinsitic. In ECVRF proving, changed + to - (and made corresponding verification changes) in order to be Goldberg, et al. Expires December 30, 2018 [Page 24]

Internet-Draft VRF June 2018 consistent with EdDSA and ECDSA. Highlighted that ECVRF_hash_to_curve acts like a prehash. Added "suites" variable to ECVRF for future-proofing. 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 an discussion of secrecy into security considerations. Removed g and PK=y from ECVRF_hash_points because they are already present via h, which is computed via hash_to_curve using the suite (which identifies g) and y. 9. Contributors 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 (Salesforce), David C. Lawerence (Akamai), and Trevor Perrin provided valuable input to this draft. 10. References 10.1. Normative References [FIPS-186-3] National Institute for Standards and Technology, "Digital Signature Standard (DSS)", FIPS PUB 186-3, June 2009. [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>. Goldberg, et al. Expires December 30, 2018 [Page 25]

Internet-Draft VRF June 2018 [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>. [SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", Version 2.0, May 2009, <http://www.secg.org/sec1-v2.pdf>. 10.2. Informative References [BCIMRT10] Brier, E., Coron, J., Icart, T., Madore, D., Randriam, H., and M. Tibouchi, "Efficient Indifferentiable Hashing into Ordinary Elliptic Curves", in CRYPTO, 2010. [I-D.vcelak-nsec5] Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of Existence", draft-vcelak-nsec5-07 (work in progress), June 2018. [Icart09] Icart, T., "How to Hash into Elliptic Curves", in CRYPTO, 2009. [MRV99] Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random Functions", in FOCS, 1999. [nsec5ecc] Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J., Naor, M., Reyzin, L., and S. Goldberg, "Making NSEC5 Practical for DNSSEC", in ePrint Cryptology Archive 2017/099, February 2017, <https://eprint.iacr.org/2017/099.pdf>. [ntb] Shoup, V., "A Computational Introduction to Number Theory and Algebra", 2008, <http://www.shoup.net/ntb/ntb-v2.pdf>. [SW18] Sullivan, E. and C. Wood, "Hashing to Elliptic Curves", 2018. Goldberg, et al. Expires December 30, 2018 [Page 26]

Internet-Draft VRF June 2018 Authors' Addresses Sharon Goldberg Boston University 111 Cummington St, MCS135 Boston, MA 02215 USA EMail: goldbe@cs.bu.edu Leonid Reyzin Boston University 111 Cummington St, MCS135 Boston, MA 02215 USA EMail: reyzin@bu.edu Dimitrios Papadopoulos Hong Kong University of Science and Techology Clearwater Bay Hong Kong EMail: dipapado@cse.ust.hkbu.edu Jan Vcelak NS1 16 Beaver St New York, NY 10004 USA EMail: jvcelak@ns1.com Goldberg, et al. Expires December 30, 2018 [Page 27]