Hashing to Elliptic Curves
draftirtfcfrghashtocurve04
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Authors  Armando FazHernandez , Sam Scott , Nick Sullivan , Riad S. Wahby , Christopher A. Wood  
Last updated  20190708 (Latest revision 20190311)  
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draftirtfcfrghashtocurve04
Network Working Group A. FazHernandez InternetDraft Cloudflare Intended status: Informational S. Scott Expires: January 9, 2020 Cornell Tech N. Sullivan Cloudflare R. Wahby Stanford University C. Wood Apple Inc. July 08, 2019 Hashing to Elliptic Curves draftirtfcfrghashtocurve04 Abstract This document specifies a number of algorithms that may be used to encode or hash an arbitrary string to a point on an elliptic curve. Status of This Memo This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79. InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current Internet Drafts is at https://datatracker.ietf.org/drafts/current/. InternetDrafts 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 InternetDrafts as reference material or to cite them other than as "work in progress." This InternetDraft will expire on January 9, 2020. Copyright Notice Copyright (c) 2019 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/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect FazHernandez, et al. Expires January 9, 2020 [Page 1] InternetDraft hashtocurve July 2019 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. Requirements . . . . . . . . . . . . . . . . . . . . . . 4 2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1. Elliptic curves . . . . . . . . . . . . . . . . . . . . . 4 2.2. Terminology . . . . . . . . . . . . . . . . . . . . . . . 5 2.2.1. Mappings . . . . . . . . . . . . . . . . . . . . . . 5 2.2.2. Encodings . . . . . . . . . . . . . . . . . . . . . . 6 2.2.3. Random oracle encodings . . . . . . . . . . . . . . . 6 2.2.4. Serialization . . . . . . . . . . . . . . . . . . . . 7 2.2.5. Domain separation . . . . . . . . . . . . . . . . . . 7 3. Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.1. Domain separation requirements . . . . . . . . . . . . . 9 4. Utility Functions . . . . . . . . . . . . . . . . . . . . . . 10 5. Hashing to a Finite Field . . . . . . . . . . . . . . . . . . 13 5.1. Security considerations . . . . . . . . . . . . . . . . . 13 5.2. Performance considerations . . . . . . . . . . . . . . . 14 5.3. Implementation . . . . . . . . . . . . . . . . . . . . . 15 6. Deterministic Mappings . . . . . . . . . . . . . . . . . . . 16 6.1. Interface . . . . . . . . . . . . . . . . . . . . . . . . 16 6.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . 16 6.3. Sign of the resulting point . . . . . . . . . . . . . . . 16 6.4. Exceptional cases . . . . . . . . . . . . . . . . . . . . 17 6.5. Mappings for Weierstrass curves . . . . . . . . . . . . . 17 6.5.1. Icart Method . . . . . . . . . . . . . . . . . . . . 17 6.5.2. Simplified Shalluevan de WoestijneUlas Method . . . 18 6.6. Mappings for Montgomery curves . . . . . . . . . . . . . 20 6.6.1. Elligator 2 Method . . . . . . . . . . . . . . . . . 21 6.7. Mappings for Twisted Edwards curves . . . . . . . . . . . 23 6.7.1. Rational maps from Montgomery to twisted Edwards curves . . . . . . . . . . . . . . . . . . . . . . . 23 6.7.2. Elligator 2 Method . . . . . . . . . . . . . . . . . 25 6.8. Mappings for Supersingular curves . . . . . . . . . . . . 25 6.8.1. BonehFranklin Method . . . . . . . . . . . . . . . . 25 6.8.2. Elligator 2, A == 0 Method . . . . . . . . . . . . . 26 6.9. Mappings for PairingFriendly curves . . . . . . . . . . 27 6.9.1. Shalluevan de Woestijne Method . . . . . . . . . . . 27 6.9.2. Simplified SWU for PairingFriendly Curves . . . . . 30 7. Clearing the cofactor . . . . . . . . . . . . . . . . . . . . 31 8. Suites for Hashing . . . . . . . . . . . . . . . . . . . . . 32 8.1. Suites for NIST P256 . . . . . . . . . . . . . . . . . . 33 8.2. Suites for NIST P384 . . . . . . . . . . . . . . . . . . 34 FazHernandez, et al. Expires January 9, 2020 [Page 2] InternetDraft hashtocurve July 2019 8.3. Suites for NIST P521 . . . . . . . . . . . . . . . . . . 34 8.4. Suites for curve25519 and edwards25519 . . . . . . . . . 35 8.5. Suites for curve448 and edwards448 . . . . . . . . . . . 36 8.6. Suites for SECP256K1 . . . . . . . . . . . . . . . . . . 37 8.7. Suites for BLS12381 . . . . . . . . . . . . . . . . . . 37 9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 39 10. Security Considerations . . . . . . . . . . . . . . . . . . . 39 11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 39 12. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 39 13. References . . . . . . . . . . . . . . . . . . . . . . . . . 39 13.1. Normative References . . . . . . . . . . . . . . . . . . 40 13.2. Informative References . . . . . . . . . . . . . . . . . 40 Appendix A. Related Work . . . . . . . . . . . . . . . . . . . . 45 Appendix B. Rational maps from twisted Edwards to Weierstrass and Montgomery curves . . . . . . . . . . . . . . . 47 Appendix C. Isogenous curves and corresponding maps for BLS12381 . . . . . . . . . . . . . . . . . . . . . 48 C.1. 11isogeny map for G1 . . . . . . . . . . . . . . . . . . 48 C.2. 3isogeny map for G2 . . . . . . . . . . . . . . . . . . 52 Appendix D. Sample Code . . . . . . . . . . . . . . . . . . . . 53 D.1. Interface and projective coordinate systems . . . . . . . 53 D.2. P256 (Simplified SWU) . . . . . . . . . . . . . . . . . 54 D.3. curve25519 (Elligator 2) . . . . . . . . . . . . . . . . 56 D.4. edwards25519 (Elligator 2) . . . . . . . . . . . . . . . 57 D.5. curve448 (Elligator 2) . . . . . . . . . . . . . . . . . 57 D.6. edwards448 (Elligator 2) . . . . . . . . . . . . . . . . 58 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 59 1. Introduction Many cryptographic protocols require a procedure that encodes an arbitrary input, e.g., a password, to a point on an elliptic curve. This procedure is known as hashing to an elliptic curve. Prominent examples of cryptosystems that hash to elliptic curves include Simple Password Exponential Key Exchange [J96], Password Authenticated Key Exchange [BMP00], IdentityBased Encryption [BF01] and BonehLynn Shacham signatures [BLS01]. Unfortunately for implementors, the precise hash function that is suitable for a given scheme is not necessarily included in the description of the protocol. Compounding this problem is the need to pick a suitable curve for the specific protocol. This document aims to bridge this gap by providing a thorough set of recommended algorithms for a range of curve types. Each algorithm conforms to a common interface: it takes as input an arbitrarylength bit string and produces as output a point on an elliptic curve. We provide implementation details for each algorithm, describe the FazHernandez, et al. Expires January 9, 2020 [Page 3] InternetDraft hashtocurve July 2019 security rationale behind each recommendation, and give guidance for elliptic curves that are not explicitly covered. 1.1. 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]. 2. Background 2.1. Elliptic curves The following is a brief definition of elliptic curves, with an emphasis on important parameters and their relation to hashing to curves. For further reference on elliptic curves, consult [CFADLNV05] or [W08]. Let F be the finite field GF(q) of prime characteristic p. In most cases F is a prime field, so q = p. Otherwise, F is a field extension, so q = p^m for an integer m > 1. This document assumes that elements of field extensions are written in a primitive element or polynomial basis, i.e., as of m elements of GF(p) written in ascending order by degree. For example, if q = p^2 and the primitive element basis is {1, i}, then the vector (a, b) corresponds to the element a + b * i. An elliptic curve E is specified by an equation in two variables and a finite field F. An elliptic curve equation takes one of several standard forms, including (but not limited to) Weierstrass, Montgomery, and Edwards. The curve E induces an algebraic group whose elements are those points with coordinates (x, y) satisfying the curve equation, and where x and y are elements of F. This group has order n, meaning that there are n distinct points. This document uses additive notation for the elliptic curve group operation. For security reasons, groups of prime order MUST be used. Elliptic curves induce subgroups of prime order. Let G be a subgroup of the curve of prime order r, where n = h * r. In this equation, h is an integer called the cofactor. An algorithm that takes as input an arbitrary point on the curve E and produces as output a point in the subgroup G of E is said to "clear the cofactor." Such algorithms are discussed in Section 7. Certain hashtocurve algorithms restrict the form of the curve equation, the characteristic of the field, and/or the parameters of FazHernandez, et al. Expires January 9, 2020 [Page 4] InternetDraft hashtocurve July 2019 the curve. For each algorithm presented, this document lists the relevant restrictions. Summary of quantities: ++++  Symbol  Meaning  Relevance  ++++  F,q,p  Finite field F of  For prime fields, q = p;    characteristic p and #F =  otherwise, q = p^m and m>1.    q = p^m.        E  Elliptic curve.  E is specified by an     equation and a field F.       n  Number of points on the  n = h * r, for h and r    elliptic curve E.  defined below.       G  A subgroup of the elliptic  Destination group to which    curve.  bit strings are encoded.       r  Order of G.  This number MUST be prime.       h  Cofactor, h >= 1.  An integer satisfying n = h     * r.  ++++ 2.2. Terminology In this section, we define important terms used in the rest of this document. 2.2.1. Mappings A mapping is a deterministic function from an element of the field F to a point on an elliptic curve E defined over F. In general, the set of all points that a mapping can produce over all possible inputs may be only a subset of the points on an elliptic curve (i.e., the mapping may not be surjective). In addition, a mapping may output the same point for two or more distinct inputs (i.e., the mapping may not be injective). For example, consider a mapping from F to an elliptic curve having n points: if the number of elements of F is not equal to n, then this mapping cannot be bijective (i.e., both injective and surjective) since it is defined to be deterministic. FazHernandez, et al. Expires January 9, 2020 [Page 5] InternetDraft hashtocurve July 2019 Mappings may also be invertible, meaning that there is an efficient algorithm that, for any point P output by the mapping, outputs an x in F such that applying the mapping to x outputs P. Some of the mappings given in Section 6 are invertible, but this document does not discuss inversion algorithms. 2.2.2. Encodings Encodings are closely related to mappings. Like a mapping, an encoding is a function that outputs a point on an elliptic curve. In contrast to a mapping, however, the input to an encoding is an arbitrary bit string. Encodings can be deterministic or probabilistic. Deterministic encodings are preferred for security, because probabilistic ones can leak information through side channels. This document constructs deterministic encodings by composing a hash function H with a deterministic mapping. In particular, H takes as input an arbitrary bit string and outputs an element of F. The deterministic mapping takes that element as input and outputs a point on an elliptic curve E defined over F. Since the hash function H takes arbitrary bit strings as inputs, it cannot be injective: the set of inputs is larger than the set of outputs, so there must be distinct inputs that give the same output (i.e., there must be collisions). Thus, any encoding built from H is also not injective. Like mappings, encodings may be invertible, meaning that there is an efficient algorithm that, for any point P output by the encoding, outputs a bit string s such that applying the encoding to s outputs P. The hash function used by all encodings specified in this document (Section 5) is not invertible; thus, the encodings are also not invertible. 2.2.3. Random oracle encodings Two different types of encodings are possible: nonuniform encodings, whose output distribution is not uniformly random, and random oracle encodings, whose output distribution is indistinguishable from uniformly random. Some protocols require a random oracle for security, while others can be securely instantiated with a nonuniform encoding. When the required encoding is not clear, applications SHOULD use a random oracle. Care is required when constructing a random oracle from a mapping function. A simple but insecure approach is to use the output of a cryptographically secure hash function H as the input to the mapping. Because in general the mapping is not surjective, the output of this FazHernandez, et al. Expires January 9, 2020 [Page 6] InternetDraft hashtocurve July 2019 construction is distinguishable from uniformly random, i.e., it does not behave like a random oracle. Brier et al. [BCIMRT10] describe two generic constructions whose outputs are indistinguishable from a random oracle. Farashahi et al. [FFSTV13] and Tibouchi and Kim [TK17] refine the analysis of one of these constructions. That construction is described in Section 3. 2.2.4. Serialization A procedure related to encoding is the conversion of an elliptic curve point to a bit string. This is called serialization, and is typically used for compactly storing or transmitting points. For example, [SECG1] gives a standard method for serializing points. The reverse operation, deserialization, converts a bit string to an elliptic curve point. Deserialization is different from encoding in that only certain strings (namely, those output by the serialization procedure) can be deserialized. In contrast, this document is concerned with encodings from arbitrary bit strings to elliptic curve points. This document does not cover serialization or deserialization. 2.2.5. Domain separation Cryptographic protocols that use random oracles are often analyzed under the assumption that random oracles answer only queries generated by that protocol. In practice, this assumption does not hold if two protocols query the same random oracle. Concretely, consider protocols P1 and P2 that query random oracle R: if P1 and P2 both query R on the same value x, the security analysis of one or both protocols may be invalidated. A common approach to addressing this issue is called domain separation, which allows a single random oracle to simulate multiple, independent oracles. This is effected by ensuring that each simulated oracle sees queries that are distinct from those seen by all other simulated oracles. For example, to simulate two oracles R1 and R2 given a single oracle R, one might define R1(x) := R("R1"  x) R2(x) := R("R2"  x) In this example, "R1" and "R2" are called domain separation tags; they ensure that queries to R1 and R2 cannot result in identical queries to R. Thus, it is safe to treat R1 and R2 as independent oracles. FazHernandez, et al. Expires January 9, 2020 [Page 7] InternetDraft hashtocurve July 2019 3. Roadmap This section presents a general framework for encoding bit strings to points on an elliptic curve. To construct these encodings, we rely on three basic functions: o The function hash_to_base, {0, 1}^* x {0, 1, 2} > F, hashes arbitrarylength bit strings to elements of a finite field; its implementation is defined in Section 5. o The function map_to_curve, F > E, calculates a point on the elliptic curve E from an element of the finite field F over which E is defined. Section 6 describes mappings for a range of curve families. o The function clear_cofactor, E > G, sends any point on the curve E to the subgroup G of E. Section 7 describes methods to perform this operation. We describe two highlevel encoding functions (Section 2.2.2). Although these functions have the same interface, the distributions of their outputs are different. o Nonuniform encoding (encode_to_curve). This function encodes bit strings to points in G. The distribution of the output is not uniformly random in G. encode_to_curve(alpha) Input: alpha, an arbitrarylength bit string. Output: P, a point in G. Steps: 1. u = hash_to_base(alpha, 2) 2. Q = map_to_curve(u) 3. P = clear_cofactor(Q) 4. return P o Random oracle encoding (hash_to_curve). This function encodes bit strings to points in G. The distribution of the output is indistinguishable from uniformly random in G provided that map_to_curve is "well distributed" ([FFSTV13], Def. 1). All of the map_to_curve functions defined in Section 6 meet this requirement. FazHernandez, et al. Expires January 9, 2020 [Page 8] InternetDraft hashtocurve July 2019 hash_to_curve(alpha) Input: alpha, an arbitrarylength bit string. Output: P, a point in G. Steps: 1. u0 = hash_to_base(alpha, 0) 2. u1 = hash_to_base(alpha, 1) 3. Q0 = map_to_curve(u0) 4. Q1 = map_to_curve(u1) 5. R = Q0 + Q1 // point addition 6. P = clear_cofactor(R) 7. return P Instances of these functions are given in Section 8, which defines a list of suites that specify a full set of parameters matching elliptic curves and algorithms. 3.1. Domain separation requirements When invoking hash_to_curve from a higherlevel protocol, implementors MUST use domain separation (Section 2.2.5) to avoid interfering with other protocols that also use the hash_to_curve functionality. Protocols that use encode_to_curve SHOULD use domain separation if possible, though it is not required in this case. Protocols that instantiate multiple, independent random oracles based on hash_to_curve MUST enforce domain separation between those oracles. This requirement applies both in the case of multiple oracles to the same curve and in the case of multiple oracles to different curves. This is because the hash_to_base primitive (Section 5) requires domain separation to guarantee independent outputs. Care is required when choosing a domain separation tag. Implementors SHOULD observe the following guidelines: 1. Tags should be prepended to the value being hashed, as in the example in Section 2.2.5. 2. Tags should have fixed length, or should be encoded in a way that makes the length of a given tag unambiguous. If a variable length tag is used, it should be prefixed with a fixedlength field that encodes the length of the tag. 3. Tags should begin with a fixed protocol identification string. Ideally, this identification string should be unique to the protocol. FazHernandez, et al. Expires January 9, 2020 [Page 9] InternetDraft hashtocurve July 2019 4. Tags should include a protocol version number. 5. For protocols that support multiple ciphersuites, tags should include a ciphersuite identifier. As an example, consider a fictional key exchange protocol named Quux. A reasonable choice of tag is "QUUXV<xx>CS<yy>", where <xx> and <yy> are twodigit numbers indicating the version and ciphersuite, respectively. Alternatively, if a variablelength ciphersuite string must be used, a reasonable choice of tag is "QUUXV<xx> L<zz><csid>", where <csid> is the ciphersuite string, and <xx> and <zz> are twodigit numbers indicating the version and the length of the ciphersuite string, respectively. As another example, consider a fictional protocol named Baz that requires two independent random oracles, where one oracle outputs points on the curve E1 and the other outputs points on the curve E2. To ensure that these two random oracles are independent, each one must be called with a distinct domain separation tag. Reasonable choices of tags for the E1 and E2 oracles are "BAZV<xx>CS<yy>E1" and "BAZV<xx>CS<yy>E2", respectively, where <xx> and <yy> are as defined above. 4. Utility Functions Algorithms in this document make use of utility functions described below. For security reasons, all field operations, comparisons, and assignments MUST be implemented in constant time (i.e., execution time MUST NOT depend on the values of the inputs), and without branching. Guidance on implementing these lowlevel operations in constant time is beyond the scope of this document. o CMOV(a, b, c): If c is False, CMOV returns a, otherwise it returns b. To prevent against timing attacks, this operation must run in constant time, without revealing the value of c. Commonly, implementations assume that the selector c is 1 for True or 0 for False. In this case, given a bit string C, the desired selector c can be computed by ORing all bits of C together. The resulting selector will be either 0 if all bits of C are zero, or 1 if at least one bit of C is 1. o is_square(x): This function returns True whenever the value x is a square in the field F. Due to Euler's criterion, this function can be calculated in constant time as FazHernandez, et al. Expires January 9, 2020 [Page 10] InternetDraft hashtocurve July 2019 is_square(x) := { True, if x^((q  1) / 2) is 0 or 1 in F; { False, otherwise. o sqrt(x): The sqrt operation is a multivalued function, i.e. there exist two roots of x in the field F whenever x is square. To maintain compatibility across implementations while allowing implementors leeway for optimizations, this document does not require sqrt() to return a particular value. Instead, as explained in Section 6.3, any higherlevel function that computes square roots also specifies how to determine the sign of the result. The preferred way of computing square roots is to fix a deterministic algorithm particular to F. We give algorithms for the three most common cases immediately below; other cases are analogous. Note that Case 3 below applies to GF(p^2) when p = 3 mod 8. [AR13] and [S85] describe methods that work for other field extensions. Regardless of the method chosen, the sqrt function MUST be performed in constant time. s = sqrt(x) Parameters:  F, a finite field of characteristic p and order q = p^m, m >= 1. Input: x, an element of F. Output: s, an element of F such that (s^2) == x. ====== Case 1: q = 3 (mod 4) Constants: 1. c1 = (q + 1) / 4 // Integer arithmetic Procedure: 1. return x^c1 ====== Case 2: q = 5 (mod 8) Constants: 1. c1 = sqrt(1) in F, i.e., (c1^2) == 1 in F 2. c2 = (q + 3) / 8 // Integer arithmetic FazHernandez, et al. Expires January 9, 2020 [Page 11] InternetDraft hashtocurve July 2019 Procedure: 1. t1 = x^c2 2. e = (t1^2) == x 3. s = CMOV(t1 * c1, t1, e) 3. return s ====== Case 3: q = 9 (mod 16) Constants: 1. c1 = sqrt(1) in F, i.e., (c1^2) == 1 in F 2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F 3. c3 = sqrt(c1) in F, i.e., (c3^2) == c1 in F 4. c4 = (q + 7) / 16 // Integer arithmetic Procedure: 1. t1 = x^c4 2. t2 = c1 * t1 3. t3 = c2 * t1 4. t4 = c3 * t1 5. e1 = (t2^2) == x 6. e2 = (t3^2) == x 7. t1 = CMOV(t1, t2, e1) // select t2 if (t2^2) == x 8. t2 = CMOV(t4, t3, e2) // select t3 if (t3^2) == x 9. e3 = (t2^2) == x 10. s = CMOV(t1, t2, e3) // select the sqrt from t1 and t2 11. return s o sgn0(x): This function returns either +1 or 1 indicating the "sign" of x, where sgn0(x) == 1 just when x is lexically greater than x. Thus, this function considers 0 to be positive. The following procedure implements sgn0(x) in constant time. See Section 2.1 for a discussion of representing x as a vector. FazHernandez, et al. Expires January 9, 2020 [Page 12] InternetDraft hashtocurve July 2019 sgn0(x) Parameters:  F, a finite field of characteristic p and order q = p^m, m >= 1. Input: x, an element of F. Output: 1 or 1 (an integer). Notation: x_i is the i^th element of the vector representation of x. Steps: 1. sign = 0 2. for i in (m, m  1, ..., 1): 3. sign_i = CMOV(1, 1, x_i > ((p  1) / 2)) 4. sign_i = CMOV(sign_i, 0, x_i == 0) 5. sign = CMOV(sign, sign_i, sign == 0) 6. return CMOV(sign, 1, sign == 0) // regard x == 0 as positive o abs(x): The absolute value of x is defined in terms of sgn0 in the natural way, namely, abs(x) := sgn0(x) * x. o inv0(x): This function returns the multiplicative inverse of x in F, extended to all of F by fixing inv0(0) == 0. To implement inv0 in constant time, compute inv0(x) := x^(q  2). Notice on input 0, the output is 0 as required. o I2OSP and OS2IP: These functions are used to convert an octet string to and from a nonnegative integer as described in [RFC8017]. o a  b: denotes the concatenation of bit strings a and b. 5. Hashing to a Finite Field The hash_to_base function hashes a string msg of any length into an element of a field F. This function is parametrized by the field F (Section 2.1) and by H, a cryptographic hash function that outputs b bits. 5.1. Security considerations For security, hash_to_base should be collision resistant and its output distribution should be uniform over F. To this end, hash_to_base requires a cryptographic hash function H which satisfies the following properties: 1. The number of bits output by H should be b >= 2 * k for sufficient collision resistance, where k is the target security FazHernandez, et al. Expires January 9, 2020 [Page 13] InternetDraft hashtocurve July 2019 level in bits. (This is needed for a birthday bound of approximately 2^(k).) 2. H is modeled as a random oracle, so its output must be indistinguishable from a uniformly random bit string. For example, for 128bit security, b >= 256 bits; in this case, SHA256 would be an appropriate choice for H. Ensuring that the hash_to_base output is a uniform random element of F requires care, even when H outputs a uniformly random string. For example, if H is SHA256 and F is a field of characteristic p = 2^255  19, then the result of reducing H(msg) (a 256bit integer) modulo p is slightly more likely to be in [0, 38] than if the value were selected uniformly at random. In this example the bias is negligible, but in general it can be significant. To control bias, the input msg should be hashed to an integer comprising at least ceil(log2(p)) + k bits; reducing this integer modulo p gives bias at most 2^k, which is a safe choice for a cryptosystem with kbit security. To obtain such an integer, HKDF [RFC5869] is used to expand the input msg to a Lbyte string, where L = ceil((ceil(log2(p)) + k) / 8); this string is then interpreted as an integer via OS2IP [RFC8017]. For example, for p a 255bit prime and k = 128bit security, L = ceil((255 + 128) / 8) = 48 bytes. Section 3.1 discusses requirements for domain separation and recommendations for choosing domain separation tags. The hash_to_curve function takes such a tag as a parameter, DST; this is the recommended way of applying domain separation. As an alternative, implementations MAY instead prepend a domain separation tag to the input msg; in this case, DST SHOULD be the empty string. Section 5.3 details the hash_to_base procedure. Note that implementors SHOULD NOT use rejection sampling to generate a uniformly random element of F. The reason is that these procedures are difficult to implement in constant time, and later wellmeaning "optimizations" may silently render an implementation nonconstant time. 5.2. Performance considerations The hash_to_base function uses HKDFExtract to combine the input msg and domain separation tag DST into a short digest, which is then passed to HKDFExpand [RFC5869]. For short messages, this entails at most two extra invocations of H, which is a negligible overhead in the context of hashing to elliptic curves. FazHernandez, et al. Expires January 9, 2020 [Page 14] InternetDraft hashtocurve July 2019 A related issue is that the random oracle construction described in Section 3 requires evaluating two independent hash functions H0 and H1 on msg. A standard way to instantiate independent hashes is to append a counter to the value being hashed, e.g., H(msg  0) and H(msg  1). If msg is long, however, this is either inefficient (because it entails hashing msg twice) or requires nonblackbox use of H (e.g., partial evaluation). To sidestep both of these issues, hash_to_base takes a second argument, ctr, which it passes to HKDFExpand. This means that two invocations of hash_to_base on the same msg with different ctr values both start with identical invocations of HKDFExtract. This is an improvement because it allows sharing one evaluation of HKDFExtract among multiple invocations of hash_to_base, i.e., by factoring out the common computation. 5.3. Implementation The following procedure implements hash_to_base. hash_to_base(msg, ctr) Parameters:  DST, a domain separation tag (see discussion above).  H, a cryptographic hash function.  F, a finite field of characteristic p and order q = p^m.  L = ceil((ceil(log2(p)) + k) / 8), where k is the security parameter of the cryptosystem (e.g., k = 128).  HKDFExtract and HKDFExpand are as defined in RFC5869, instantiated with the hash function H. Inputs:  msg is the message to hash.  ctr is 0, 1, or 2. This is used to efficiently create independent instances of hash_to_base (see discussion above). Output:  u, an element in F. Steps: 1. m' = HKDFExtract(DST, msg) 2. for i in (1, ..., m): 3. info = "H2C"  I2OSP(ctr, 1)  I2OSP(i, 1) 4. t = HKDFExpand(m', info, L) 5. e_i = OS2IP(t) mod p 6. return u = (e_1, ..., e_m) FazHernandez, et al. Expires January 9, 2020 [Page 15] InternetDraft hashtocurve July 2019 6. Deterministic Mappings The mappings in this section are suitable for constructing either nonuniform or random oracle encodings using the constructions of Section 3. 6.1. Interface The generic interface shared by all mappings in this section is as follows: (x, y) = map_to_curve(u) The input u and outputs x and y are elements of the field F. The coordinates (x, y) specify a point on an elliptic curve defined over F. Note that the point (x, y) is not a uniformly random point. If uniformity is required for security, the random oracle construction of Section 3 MUST be used instead. 6.2. Notation As a rough style guide the following convention is used: o All arithmetic operations are performed over a field F, unless explicitly stated otherwise. o u: the input to the mapping function. This is an element of F produced by the hash_to_base function. o (x, y): are the affine coordinates of the point output by the mapping. Indexed values are used when the algorithm calculates some candidate values. o t1, t2, ...: are reusable temporary variables. For notable variables, distinct names are used easing the debugging process when correlating with test vectors. o c1, c2, ...: are constant values, which can be computed in advance. 6.3. Sign of the resulting point In general, elliptic curves have equations of the form y^2 = g(x). Most of the mappings in this section first identify an x such that g(x) is square, then take a square root to find y. Since there are two square roots when g(x) != 0, this results in an ambiguity regarding the sign of y. FazHernandez, et al. Expires January 9, 2020 [Page 16] InternetDraft hashtocurve July 2019 To resolve this ambiguity, the mappings in this section specify the sign of the ycoordinate in terms of the input to the mapping function. Two main reasons support this approach. First, this covers elliptic curves over any field in a uniform way, and second, it gives implementors leeway to optimize their squareroot implementations. 6.4. Exceptional cases Mappings may have have exceptional cases, i.e., inputs u on which the mapping is undefined. These cases must be handled carefully, especially for constanttime implementations. For each mapping in this section, we discuss the exceptional cases and show how to handle them in constant time. Note that all implementations SHOULD use inv0 (Section 4) to compute multiplicative inverses, to avoid exceptional cases that result from attempting to compute the inverse of 0. 6.5. Mappings for Weierstrass curves The following mappings apply to elliptic curves defined by the equation E: y^2 = g(x) = x^3 + A * x + B, where 4 * A^3 + 27 * B^2 != 0. 6.5.1. Icart Method The function map_to_curve_icart(u) implements the Icart method from [Icart09]. Preconditions: An elliptic curve over F, such that p > 3 and q = p^m = 2 (mod 3), or p = 2 (mod 3) and odd m. Constants: A and B, the parameters of the Weierstrass curve. Sign of y: this mapping does not compute a square root, so there is no ambiguity regarding the sign of y. Exceptions: The only exceptional case is u == 0. Implementations MUST detect this case by testing whether u == 0 and setting u = 1 if so. Operations: FazHernandez, et al. Expires January 9, 2020 [Page 17] InternetDraft hashtocurve July 2019 1. If u == 0, set u = 1 2. v = (3 * A  u^4) / (6 * u) 3. w = (2 * p  1) / 3 // Integer arithmetic 4. x = (v^2  B  (u^6 / 27))^w + (u^2 / 3) 5. y = u * x + v 6. return (x, y) 6.5.1.1. Implementation The following procedure implements Icart's algorithm in a straight line fashion. map_to_curve_icart(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = (2 * p  1) / 3 // Integer arithmetic 2. c2 = 1 / 3 3. c3 = c2^3 4. c4 = 3 * A Steps: 1. e = u == 0 2. u = CMOV(u, 1, e) // handle exceptional case u == 0 3. u2 = u^2 // u^2 4. u4 = u2^2 // u^4 5. v = c4  u4 // 3 * A  u^4 6. t1 = 6 * u // 6 * u 7. t1 = inv0(t1) // 1 / (6 * u) 8. v = v * t1 // v = (3 * A  u^4) / (6 * u) 9. x = v^2 // v^2 10. x = x  B // v^2  B 11. u6 = u4 * c3 // u^4 / 27 12. u6 = u6 * u2 // u^6 / 27 13. x = x  u6 // v^2  B  u^6 / 27 14. x = x^c1 // (v^2  B  u^6 / 27)^(1 / 3) 15. t1 = u2 * c2 // u^2 / 3 16. x = x + t1 // x = (v^2  B  u^6 / 27)^(1 / 3) + (u^2 / 3) 17. y = u * x // u * x 18. y = y + v // y = u * x + v 19. return (x, y) 6.5.2. Simplified Shalluevan de WoestijneUlas Method The function map_to_curve_simple_swu(u) implements a simplification of the Shalluevan de WoestijneUlas mapping [U07] described by Brier et al. [BCIMRT10], which they call the "simplified SWU" map. Wahby FazHernandez, et al. Expires January 9, 2020 [Page 18] InternetDraft hashtocurve July 2019 and Boneh [WB19] generalize this mapping to curves over fields of odd characteristic p > 3. Preconditions: A Weierstrass curve over F such that A != 0 and B != 0. Constants: o A and B, the parameters of the Weierstrass curve. o Z, the unique element of F meeting all of the following criteria: 1. Z is nonsquare in F, 2. g(B / (Z * A)) is square in F, 3. there is no other Z' meeting criteria (1) and (2) for which abs(Z') < abs(Z) (Section 4), and 4. if Z and Z both meet the above criteria, Z is the element such that sgn0(Z) == 1. Sign of y: Inputs u and u give the same xcoordinate. Thus, we set sgn0(y) == sgn0(u). Exceptions: The exceptional cases are values of u such that Z^2 * u^4 + Z * u^2 == 0. This includes u == 0, and may include other values depending on Z. Implementations must detect this case and set x1 = B / (Z * A), which guarantees that g(x1) is square by the condition on Z given above. Operations: 1. t1 = inv0(Z^2 * u^4 + Z * u^2) 2. x1 = (B / A) * (1 + t1) 3. If t1 == 0, set x1 = B / (Z * A) 4. gx1 = x1^3 + A * x1 + B 5. x2 = Z * u^2 * x1 6. gx2 = x2^3 + A * x2 + B 7. If is_square(gx1), set x = x1 and y = sqrt(gx1) 8. Else set x = x2 and y = sqrt(gx2) 9. If sgn0(u) != sgn0(y), set y = y 10. return (x, y) FazHernandez, et al. Expires January 9, 2020 [Page 19] InternetDraft hashtocurve July 2019 6.5.2.1. Implementation The following procedure implements the simplified SWU mapping in a straightline fashion. Appendix D gives an optimized straightline procedure for P256 [FIPS1864]. For discussion of how to generalize to q = 1 (mod 4), see [WB19] (Section 4) or the example code found at [hash2curverepo]. map_to_curve_simple_swu(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = B / A 2. c2 = 1 / Z Steps: 1. t1 = Z * u^2 2. t2 = t1^2 3. x1 = t1 + t2 4. x1 = inv0(x1) 5. e1 = x1 == 0 6. x1 = x1 + 1 7. x1 = CMOV(x1, c2, e1) // if (t1 + t2) == 0, set x1 = 1 / Z 8. x1 = x1 * c1 // x1 = (B / A) * (1 + (1 / (Z^2 * u^4 + Z * u^2))) 9. gx1 = x1^2 10. gx1 = gx1 + A 11. gx1 = gx1 * x1 12. gx1 = gx1 + B // gx1 = g(x1) = x1^3 + A * x1 + B 13. x2 = t1 * x1 // x2 = Z * u^2 * x1 14. t2 = t1 * t2 15. gx2 = gx1 * t2 // gx2 = (Z * u^2)^3 * gx1 16. e2 = is_square(gx1) 17. x = CMOV(x2, x1, e2) // If is_square(gx1), x = x1, else x = x2 18. y2 = CMOV(gx2, gx1, e2) // If is_square(gx1), y2 = gx1, else y2 = gx2 19. y = sqrt(y2) 20. e3 = sgn0(u) == sgn0(y) // fix sign of y 21. y = CMOV(y, y, e3) 22. return (x, y) 6.6. Mappings for Montgomery curves The mapping defined in Section 6.6.1 implements Elligator 2 [BHKL13] for curves defined by the Weierstrass equation y^2 = x^3 + A * x^2 + B * x, where A * B * (A^2  4 * B) != 0 and A^2  4 * B is nonsquare in F. FazHernandez, et al. Expires January 9, 2020 [Page 20] InternetDraft hashtocurve July 2019 Such a Weierstrass curve is related to the Montgomery curve B' * y'^2 = x'^3 + A' * x'^2 + x' by the following change of variables: o A = A' / B' o B = 1 / B'^2 o x = x' / B' o y = y' / B' The Elligator 2 mapping given below returns a point (x, y) on the Weierstrass curve defined above. This point can be converted to a point (x', y') on the original Montgomery curve by computing o x' = B' * x o y' = B' * y Note that when B and B' are equal to 1, the above two curve equations are identical and no conversion is necessary. This is the case, for example, for Curve25519 and Curve448 [RFC7748]. 6.6.1. Elligator 2 Method Preconditions: A Weierstrass curve y^2 = x^3 + A * x^2 + B * x where A != 0, B != 0, and A^2  4 * B is nonzero and nonsquare in F. Constants: o A and B, the parameters of the elliptic curve. o Z, the unique element of F meeting all of the following criteria: 1. Z is nonsquare in F, 2. there is no other nonsquare Z' for which abs(Z') < abs(Z) (Section 4), and 3. if Z and Z both met the above criteria, Z is the element such that sgn0(Z) == 1. Sign of y: Inputs u and u give the same xcoordinate. Thus, we set sgn0(y) == sgn0(u). Exceptions: The exceptional case is Z * u^2 == 1, i.e., 1 + Z * u^2 == 0. Implementations must detect this case and set x1 = A. Note that this can only happen when q = 3 (mod 4). FazHernandez, et al. Expires January 9, 2020 [Page 21] InternetDraft hashtocurve July 2019 Operations: 1. x1 = A * inv0(1 + Z * u^2) 2. If x1 == 0, set x1 = A. 3. gx1 = x1^3 + A * x1^2 + B * x1 4. x2 = x1  A 5. gx2 = x2^3 + A * x2^2 + B * x2 6. If is_square(gx1), set x = x1 and y = sqrt(gx1) 7. Else if is_square(gx2), set x = x2 and y = sqrt(gx2) 8. If sgn0(u) != sgn0(y), set y = y 9. return (x, y) 6.6.1.1. Implementation The following procedure implements Elligator 2 in a straightline fashion. Appendix D gives optimized straightline procedures for curve25519 and curve448 [RFC7748]. map_to_curve_elligator2(u) Input: u, an element of F. Output: (x, y), a point on E. Steps: 1. t1 = u^2 2. t1 = Z * t1 // Z * u^2 3. x1 = t1 + 1 4. x1 = inv0(x1) 5. e1 = x1 == 0 6. x1 = CMOV(x1, 1, e1) // if x1 == 0, set x1 = 1 7. x1 = A * x1 // x1 = A / (1 + Z * u^2) 8. gx1 = x1 + A 9. gx1 = gx1 * x1 10. gx1 = gx1 + B 11. gx1 = gx1 * x1 // gx1 = x1^3 + A * x1^2 + B * x1 12. x2 = x1  A 13. gx2 = t1 * gx1 14. e2 = is_square(gx1) 15. x = CMOV(x2, x1, e2) // If is_square(gx1), x = x1, else x = x2 16. y2 = CMOV(gx2, gx1, e2) // If is_square(gx1), y2 = gx1, else y2 = gx2 17. y = sqrt(y2) 18. e3 = sgn0(u) == sgn0(y) // fix sign of y 19. y = CMOV(y, y, e3) 20. return (x, y) FazHernandez, et al. Expires January 9, 2020 [Page 22] InternetDraft hashtocurve July 2019 6.7. Mappings for Twisted Edwards curves Twisted Edwards curves (a class of curves that includes Edwards curves) are closely related to Montgomery curves (Section 6.6): every twisted Edwards curve is birationally equivalent to a Montgomery curve ([BBJLP08], Theorem 3.2). This equivalence yields an efficient way of hashing to a twisted Edwards curve: first, hash to the equivalent Montgomery curve, then transform the result into a point on the twisted Edwards curve via a rational map. This method of hashing to a twisted Edwards curve thus requires identifying a corresponding Montgomery curve and rational map. We describe how to identify such a curve and map immediately below. 6.7.1. Rational maps from Montgomery to twisted Edwards curves There are two ways to identify the correct Montgomery curve and rational map for use when hashing to a given twisted Edwards curve. When hashing to a standardized twisted Edwards curve for which a corresponding Montgomery form and rational map are also standardized, the standard Montgomery form and rational map MUST be used to ensure compatibility with existing software. Two such standardized curves are the edwards25519 and edwards448 curves, which correspond to the Montgomery curves curve25519 and curve448, respectively. For both of these curves, [RFC7748] lists both the Montgomery and twisted Edwards forms and gives the corresponding rational maps. The rational map for edwards25519 ([RFC7748], Section 4.1) uses the constant sqrt_neg_486664 = sqrt(486664) mod 2^255  19. To ensure compatibility, this constant MUST be chosen such that sgn0(sqrt_neg_486664) == 1. Analogous ambiguities in other standardized rational maps MUST be resolved in the same way: for any constant k whose sign is ambiguous, k MUST be chosen such that sgn0(k) == 1. The 4isogeny map from curve448 to edwards448 ([RFC7748], Section 4.2) is unambiguous with respect to sign. When defining new twisted Edwards curves, a Montgomery equivalent and rational map SHOULD be specified, and the sign of the rational map SHOULD be stated unambiguously. When hashing to a twisted Edwards curve that does not have a standardized Montgomery form or rational map, the following procedure MUST be used to derive them. For a twisted Edwards curve given by a * x^2 + y^2 = 1 + d * x^2 * y^2, first compute A and B, the parameters of the equivalent curve given by y'^2 = x'^3 + A * x'^2 + B * x', as follows: FazHernandez, et al. Expires January 9, 2020 [Page 23] InternetDraft hashtocurve July 2019 o A = (a + d) / 2 o B = (a  d)^2 / 16 Note that the above curve is given in the Weierstrass form required by the Elligator 2 mapping of Section 6.6.1. The rational map from the point (x', y') on this Weierstrass curve to the point (x, y) on the twisted Edwards curve is given by o x = x' / y' o y = (B' * x'  1) / (B' * x' + 1), where B' = 1 / sqrt(B) = 4 / (a  d) For completeness, we give the inverse map in Appendix B. Note that the inverse map is not used when hashing to a twisted Edwards curve. Rational maps may be undefined, for example, when the denominator of one of the rational functions is zero. For example, in the map described above, the exceptional cases are y' == 0 or B' * x' == 1. Implementations MUST detect exceptional cases and return the value (x, y) = (0, 1), which is a valid point on all twisted Edwards curves given by the equation above. The following straightline implementation of the above rational map handles the exceptional cases. Implementations of other rational maps (e.g., the ones give in [RFC7748]) are analogous. rational_map(x', y') Input: (x', y'), a point on the curve y'^2 = x'^3 + A * x'^2 + B * x'. Output: (x, y), a point on the equivalent twisted Edwards curve. 1. t1 = y' * B' 2. t2 = x' + 1 3. t3 = t1 * t2 4. t3 = inv0(t3) 5. x = t2 * t3 6. x = x * x' 7. y = x'  1 8. y = y * t3 9. y = y * t1 10. e = y == 0 11. y = CMOV(y, 1, e) 12. return (x, y) FazHernandez, et al. Expires January 9, 2020 [Page 24] InternetDraft hashtocurve July 2019 6.7.2. Elligator 2 Method Preconditions: A twisted Edwards curve E and an equivalent curve M meeting the requirements in Section 6.7.1. Helper functions: o map_to_curve_elligator2 is the mapping of Section 6.6.1 to the curve M. o rational_map is a function that takes a point (x', y') on M and returns a point (x, y) on E, as defined in Section 6.7.1. Sign of y: for this map, the sign is determined by map_to_curve_elligator2. No further sign adjustments are required. Exceptions: The exceptions for the Elligator 2 mapping are as given in Section 6.6.1. The exceptions for the rational map are as given in Section 6.7.1. No other exceptions are possible. The following procedure implements the Elligator 2 mapping for a twisted Edwards curve. map_to_curve_elligator2_edwards(u) Input: u, an element of F. Output: (x, y), a point on E. 1. (x', y') = map_to_curve_elligator2(u) // (x', y') is on M 2. (x, y) = rational_map(x', y') // (x, y) is on E 3. return (x, y) 6.8. Mappings for Supersingular curves 6.8.1. BonehFranklin Method The function map_to_curve_bf(u) implements the BonehFranklin method [BF01] which covers the supersingular curves defined by y^2 = x^3 + B over a field F such that q = 2 (mod 3). Preconditions: A supersingular curve over F such that q = 2 (mod 3). Constants: B, the parameter of the supersingular curve. Sign of y: determined by sign of u. No adjustments are necessary. Exceptions: none. Operations: FazHernandez, et al. Expires January 9, 2020 [Page 25] InternetDraft hashtocurve July 2019 1. w = (2 * q  1) / 3 // Integer arithmetic 2. x = (u^2  B)^w 3. y = u 4. return (x, y) 6.8.1.1. Implementation The following procedure implements the BonehFranklin's algorithm in a straightline fashion. map_to_curve_bf(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = (2 * q  1) / 3 // Integer arithmetic Steps: 1. t1 = u^2 2. t1 = t1  B 3. x = t1^c1 // x = (u^2  B)^((2 * q  1) / 3) 4. y = u 5. return (x, y) 6.8.2. Elligator 2, A == 0 Method The function map_to_curve_ell2A0(u) implements an adaptation of Elligator 2 [BLMP19] targeting curves given by y^2 = x^3 + B * x over F such that q = 3 (mod 4). Preconditions: An elliptic curve over F such that q = 3 (mod 4). Constants: B, the parameter of the elliptic curve. Sign of y: Inputs u and u give the same xcoordinate. Thus, we set sgn0(y) == sgn0(u). Exceptions: none. Operations: FazHernandez, et al. Expires January 9, 2020 [Page 26] InternetDraft hashtocurve July 2019 1. x1 = u 2. gx1 = x1^3 + B * x1 3. x2 = x1 4. gx2 = gx1 5. If gx1 is square, x = x1 and y = sqrt(gx1) 6. Else x = x2 and y = sqrt(gx2) 7. If sgn0(u) != sgn0(y), set y = y. 8. return (x, y) 6.8.2.1. Implementation The following procedure implements the Elligator 2 mapping for supersingular curves in a straightline fashion. map_to_curve_ell2A0(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = (p + 1) / 4 // Integer arithmetic Steps: 1. x1 = u 2. x2 = x1 3. gx1 = x1^2 4. gx1 = gx1 + B 5. gx1 = gx1 * x1 // gx1 = x1^3 + B * x1 6. y = gx1^c1 // this is either sqrt(gx1) or sqrt(gx2) 7. e1 = (y^2) == gx1 8. x = CMOV(x2, x1, e1) 9. e2 = sgn0(u) == sgn0(y) 10. y = CMOV(y, y, e2) 11. return (x, y) 6.9. Mappings for PairingFriendly curves 6.9.1. Shalluevan de Woestijne Method Shallue and van de Woestijne [SW06] describe a mapping that applies to essentially any elliptic curve. Fouque and Tibouchi [FT12] give a concrete set of parameters for this mapping geared toward Barreto Naehrig pairingfriendly curves [BN05], i.e., curves y^2 = x^3 + B over fields of characteristic q = 1 (mod 3). Wahby and Boneh [WB19] suggest a small generalization of the FouqueTibouchi parameters that results in a uniform method for handling exceptional cases. The Shalluevan de Woestijne mapping method covers curves not handled by other methods, e.g., SECP256K1 [SEC2]. It also covers pairing FazHernandez, et al. Expires January 9, 2020 [Page 27] InternetDraft hashtocurve July 2019 friendly curves in the BN [BN05], KSS [KSS08], and BLS [BLS03] families. (Note, however, that the mapping described in Section 6.9.2 is faster, when it applies.) Preconditions: An elliptic curve y^2 = g(x) = x^3 + B over F such that q = 1 (mod 3) and B != 0. Constants: o B, the parameter of the Weierstrass curve. o Z, the unique element of F meeting all of the following criteria: 1. g((sqrt(3 * Z^2)  Z) / 2) is square in F, 2. there is no other Z' meeting criterion (1) for which abs(Z') < abs(Z) (Section 4), and 3. if Z and Z both meet the above criteria, Z is the element such that sgn0(Z) == 1. Sign of y: Inputs u and u give the same xcoordinate. Thus, we set sgn0(y) == sgn0(u). Exceptions: The exceptional cases for u occur when u^2 * (u^2 + g(Z)) == 0. The restriction on Z given above ensures that implementations that use inv0 to invert this product are exception free. Operations: 1. t1 = u^2 + g(Z) 2. t2 = inv0(u^2 * t1) 3. t3 = u^4 * t2 * sqrt(3 * Z^2) 4. x1 = ((sqrt(3 * Z^2)  Z) / 2)  t3 5. x2 = t3  ((sqrt(3 * Z^2) + Z) / 2) 6. x3 = Z  (t1^3 * t2 / (3 * Z^2)) 7. If is_square(g(x1)), set x = x1 and y = sqrt(g(x1)) 8. Else If is_square(g(x2)), set x = x2 and y = sqrt(g(x2)) 9. Else set x = x3 and y = sqrt(g(x3)) 10. If sgn0(u) != sgn0(y), set y = y 11. return (x, y) 6.9.1.1. Implementation The following procedure implements the Shallue and van de Woestijne method in a straightline fashion. FazHernandez, et al. Expires January 9, 2020 [Page 28] InternetDraft hashtocurve July 2019 map_to_curve_svdw(u) Input: u, an element of F. Output: (x, y), a point on E. Constants: 1. c1 = g(Z) 2. c2 = sqrt(3 * Z^2) 3. c3 = (sqrt(3 * Z^2)  Z) / 2 4. c4 = (sqrt(3 * Z^2) + Z) / 2 5. c5 = 1 / (3 * Z^2) Steps: 1. t1 = u^2 2. t2 = t1 + c1 // t2 = u^2 + g(Z) 3. t3 = t1 * t2 4. t4 = inv0(t3) // t4 = 1 / (u^2 * (u^2 + g(Z))) 5. t3 = t1^2 6. t3 = t3 * t4 7. t3 = t3 * c2 // t3 = u^2 * sqrt(3 * Z^2) / (u^2 + g(Z)) 8. x1 = c3  t3 9. gx1 = x1^2 10. gx1 = gx1 * x1 11. gx1 = gx1 + B // gx1 = x1^3 + B 12. e1 = is_square(gx1) 13. x2 = t3  c4 14. gx2 = x2^2 15. gx2 = gx2 * x2 16. gx2 = gx2 + B // gx2 = x2^3 + B 17. e2 = is_square(gx2) 18. e3 = e1 OR e2 // logical OR 19. x3 = t2^2 20. x3 = x3 * t2 21. x3 = x3 * t4 22. x3 = x3 * c5 23. x3 = Z  x3 // Z  (u^2 + g(Z))^2 / (3 Z^2 u^2) 24. gx3 = x3^2 25. gx3 = gx3 * x3 26. gx3 = gx3 + B // gx3 = x3^3 + B 27. x = CMOV(x2, x1, e1) // select x1 if gx1 is square 28. gx = CMOV(gx2, gx1, e1) 29. x = CMOV(x3, x, e3) // select x3 if gx1 and gx2 are not square 30. gx = CMOV(gx3, gx, e3) 31. y = sqrt(gx) 32. e4 = sgn0(u) == sgn0(y) 33. y = CMOV(y, y, e4) // select correct sign of y 34. return (x, y) FazHernandez, et al. Expires January 9, 2020 [Page 29] InternetDraft hashtocurve July 2019 6.9.2. Simplified SWU for PairingFriendly Curves Wahby and Boneh [WB19] show how to adapt the simplified SWU mapping to certain Weierstrass curves having either A = 0 or B = 0, one of which is almost always true for pairingfriendly curves. Note that neither case is supported by the mapping of Section 6.5.2. This method requires finding another elliptic curve E': y^2 = g'(x) = x^3 + A' * x + B' that is isogenous to E and has A' != 0 and B' != 0. (One might do this, for example, using [SAGE]; details are beyond the scope of this document.) This isogeny defines a map iso_map(x', y') that takes as input a point on E' and produces as output a point on E. Once E' and iso_map are identified, this mapping works as follows: on input u, first apply the simplified SWU mapping to get a point on E', then apply the isogeny map to that point to get a point on E. Note that iso_map is a group homomorphism, meaning that point addition commutes with iso_map. Thus, when using this mapping in the hash_to_curve construction of Section 3, one can effect a small optimization by first mapping u0 and u1 to E', adding the resulting points on E', and then applying iso_map to the sum. This gives the same result while requiring only one evaluation of iso_map. Preconditions: An elliptic curve E' with A' != 0 and B' != 0 that is isogenous to the target curve E with isogeny map iso_map(x, y) from E' to E. Helper functions: o map_to_curve_simple_swu is the mapping of Section 6.5.2 to E' o iso_map is the isogeny map from E' to E Sign of y: for this map, the sign is determined by map_to_curve_elligator2. No further sign adjustments are necessary. Exceptions: map_to_curve_simple_swu handles its exceptional cases. Exceptional cases of iso_map should return the identity point on E. Operations: 1. (x', y') = map_to_curve_simple_swu(u) // (x', y') is on E' 2. (x, y) = iso_map(x', y') // (x, y) is on E 3. return (x, y) FazHernandez, et al. Expires January 9, 2020 [Page 30] InternetDraft hashtocurve July 2019 We do not repeat the sample implementation of Section 6.5.2 here. See [hash2curverepo] or [WB19] for details on implementing the isogeny map. 7. Clearing the cofactor The mappings of Section 6 always output a point on the elliptic curve, i.e., a point in a group of order h * r (Section 2.1). Obtaining a point in G may require a final operation commonly called "clearing the cofactor," which takes as input any point on the curve. The cofactor can always be cleared via scalar multiplication by h. For elliptic curves where h = 1, i.e., the curves with a prime number of points, no operation is required. This applies, for example, to the NIST curves P256, P384, and P521 [FIPS1864]. In some cases, it is possible to clear the cofactor via a faster method than scalar multiplication by h. These methods are equivalent to (but usually faster than) multiplication by some scalar h_eff whose value is determined by the method and the curve. Examples of fast cofactor clearing methods include the following: o For certain pairingfriendly curves having subgroup G2 over an extension field, Scott et al. [SBCDBK09] describe a method for fast cofactor clearing that exploits an efficientlycomputable endomorphism. FuentesCastaneda et al. [FKR11] propose an alternative method that is sometimes more efficient. Budroni and Pintore [BP18] give concrete instantiations of these methods for BarretoLynnScott pairingfriendly curves [BLS03]. o Wahby and Boneh ([WB19], Section 5) describe a trick due to Scott for fast cofactor clearing on any elliptic curve for which the prime factorization of h and the structure of the elliptic curve group meet certain conditions. The clear_cofactor function is parameterized by a scalar h_eff. Specifically, clear_cofactor(P) := h_eff * P where * represents scalar multiplication. When a curve does not support a fast cofactor clearing method, h_eff = h and the cofactor MUST be cleared via scalar multiplication. When a curve admits a fast cofactor clearing method, clear_cofactor MAY be evaluated either via that method or via scalar multiplication by the equivalent h_eff; these two methods give the same result. Note that in this case scalar multiplication by the cofactor h does FazHernandez, et al. Expires January 9, 2020 [Page 31] InternetDraft hashtocurve July 2019 not generally give the same result as the fast method, and SHOULD NOT be used. 8. Suites for Hashing This section lists recommended suites for hashing to standard elliptic curves. A suite fully specifies the procedure for hashing bit strings to points on a specific elliptic curve group. Each suite comprises the following parameters: o Suite ID, a short name used to refer to a given suite. o E, the target elliptic curve over a field F. o p, the characteristic of the field F. o m, the extension degree of the field F. o H, the hash function used by hash_to_base (Section 5). o W, the number of evaluations of H in hash_to_base. o f, a mapping function from Section 6. o h_eff, the scalar parameter for clear_cofactor (Section 7). In addition to the above parameters, the mapping f may require additional parameters Z, M, rational_map, E', and/or iso_map. These are specified when applicable. Suites whose ID includes "RO" use the hash_to_curve procedure of Section 3; suites whose ID includes "NU" use the encode_to_curve procedure from that section. Applications whose security requires a random oracle MUST use a "RO" suite. When standardizing a new elliptic curve, corresponding hashtocurve suites SHOULD be specified. The below table lists the curves for which suites are defined and the subsection that gives the corresponding parameters. FazHernandez, et al. Expires January 9, 2020 [Page 32] InternetDraft hashtocurve July 2019 +++  E  Section  +++  NIST P256  Section 8.1      NIST P384  Section 8.2      NIST P521  Section 8.3      curve25519 / edwards25519  Section 8.4      curve448 / edwards448  Section 8.5      SECP256k1  Section 8.6      BLS12381  Section 8.7  +++ 8.1. Suites for NIST P256 The suites P256SHA256SSWURO and P256SHA256SSWUNU are defined for the NIST P256 elliptic curve [FIPS1864]. These suites share the following parameters: o E: y^2 = x^3 + A * x + B, where * A = 3 * B = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e2 7d2604b o p: 2^256  2^224 + 2^192 + 2^96  1 o m: 1 o H: SHA256 o W: 2 o f: Simplified SWU method, Section 6.5.2 o Z: 2 o h_eff: 1 FazHernandez, et al. Expires January 9, 2020 [Page 33] InternetDraft hashtocurve July 2019 8.2. Suites for NIST P384 The suites P384SHA512ICARTRO and P384SHA512ICARTNU are defined for the NIST P384 elliptic curve [FIPS1864]. These suites share the following parameters: o E: y^2 = x^3 + A * x + B, where * A = 3 * B = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5 013875ac656398d8a2ed19d2a85c8edd3ec2aef o p: 2^384  2^128  2^96 + 2^32  1 o m: 1 o H: SHA512 o W: 2 o f: Icart's method, Section 6.5.1 o h_eff: 1 8.3. Suites for NIST P521 The suites P521SHA512SSWURO and P521SHA512SSWUNU are defined for the NIST P384 elliptic curve [FIPS1864]. These suites share the following parameters: o E: y^2 = x^3 + A * x + B, where * A = 3 * B = 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b4899 18ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451f d46b503f00 o p: 2^521  1 o m: 1 o H: SHA512 o W: 2 o f: Simplified SWU method, Section 6.5.2 FazHernandez, et al. Expires January 9, 2020 [Page 34] InternetDraft hashtocurve July 2019 o Z: 2 o h_eff: 1 An optimized example implementation of the above mapping is given in Appendix D.2. 8.4. Suites for curve25519 and edwards25519 This section defines ciphersuites for curve25519 and edwards25519 [RFC7748]. The suites curve25519SHA256ELL2RO and curve25519SHA256ELL2NU share the following parameters, in addition to the common parameters below. o E: B * y^2 = x^3 + A * x^2 + x, where * A = 486662 * B = 1 o f: Elligator 2 method, Section 6.6.1 The suites edwards25519SHA256EDELL2RO and edwards25519SHA256EDELL2NU share the following parameters, in addition to the common parameters below. o E: a * x^2 + y^2 = 1 + d * x^2 * y^2, where * a = 1 * d = 0x52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca1 35978a3 o f: Twisted Edwards Elligator 2 method, Section 6.7.2 o M: curve25519 defined in [RFC7748], Section 4.1 o rational_map: the birational map defined in [RFC7748], Section 4.1 The common parameters for all of the above suites are: o p: 2^255  19 o m: 1 o H: SHA256 FazHernandez, et al. Expires January 9, 2020 [Page 35] InternetDraft hashtocurve July 2019 o W: 2 o Z: 2 o h_eff: 8 Optimized example implementations of the above mappings are given in Appendix D.3 and Appendix D.4. 8.5. Suites for curve448 and edwards448 This section defines ciphersuites for curve448 and edwards448 [RFC7748]. The suites curve448SHA512ELL2RO and curve448SHA512ELL2NU share the following parameters, in addition to the common parameters below. o E: B * y^2 = x^3 + A * x^2 + x, where * A = 156326 * B = 1 o f: Elligator 2 method, Section 6.6.1 The suites edwards448SHA512EDELL2RO and edwards448SHA512EDELL2NU share the following parameters, in addition to the common parameters below. o E: a * x^2 + y^2 = 1 + d * x^2 * y^2, where * a = 1 * d = 39081 o f: Twisted Edwards Elligator 2 method, Section 6.7.2 o M: curve448, defined in [RFC7748], Section 4.2 o rational_map: the 4isogeny map defined in [RFC7748], Section 4.2 The common parameters for all of the above suites are: o p: 2^448  2^224  1 o m: 1 o H: SHA512 FazHernandez, et al. Expires January 9, 2020 [Page 36] InternetDraft hashtocurve July 2019 o W: 2 o Z: 1 o h_eff: 4 Optimized example implementations of the above mappings are given in Appendix D.5 and Appendix D.6. 8.6. Suites for SECP256K1 The suites SECP256K1SHA256SVDWRO and SECP256K1SHA256SVDWNU are defined for the SECP256K1 elliptic curve [SEC2]. These suites share the following parameters: o E: y^2 = x^3 + 7 o p: 2^256  2^32  2^9  2^8  2^7  2^6  2^4  1 o m: 1 o H: SHA256 o W: 2 o f: Shalluevan de Woestijne method, Section 6.9.1 o Z: 1 o h_eff: 1 8.7. Suites for BLS12381 This section defines ciphersuites for groups G1 and G2 of the BLS12381 elliptic curve [draftyonezawapfc01]. The suites BLS12381G1SHA256SSWURO and BLS12381G1SHA256SSWUNU share the following parameters, in addition to the common parameters below. o E: y^2 = x^3 + 4 o m: 1 o Z: 1 o E': y'^2 = x'^3 + A * x' + B, where FazHernandez, et al. Expires January 9, 2020 [Page 37] InternetDraft hashtocurve July 2019 * A = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aefd 881ac98936f8da0e0f97f5cf428082d584c1d * B = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14fc ef35ef55a23215a316ceaa5d1cc48e98e172be0 o iso_map: the 11isogeny map from E' to E given in Appendix C.1 o h_eff: 0xd201000000010001 The suites BLS12381G2SHA256SSWURO and BLS12381G2SHA256SSWUNU share the following parameters, in addition to the common parameters below. o F: GF(p^m), where * p: defined below * m: 2 * (1, i) is the basis for F, where i^2 + 1 == 0 in F o E: y^2 = x^3 + 4 * (1 + i) o Z: 1 + i o E': y'^2 = x'^3 + A * x' + B, where * A = 240 * i * B = 1012 * (1 + i) o iso_map: the isogeny map from E' to E given in Appendix C.2 o h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff0315 08ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689 f6a359894c0adebbf6b4e8020005aaa95551 The common parameters for the above suites are: o p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f 6241eabfffeb153ffffb9feffffffffaaab o H: SHA256 o W: 2 o f: Simplified SWU for pairingfriendly curves, Section 6.9.2 FazHernandez, et al. Expires January 9, 2020 [Page 38] InternetDraft hashtocurve July 2019 Note that the h_eff parameters for all of the above suites are chosen for compatibility with the fast cofactor clearing methods described by Scott for G1 ([WB19] Section 5) and by Budroni and Pintore for G2 ([BP18], Section 4.1). 9. IANA Considerations This document has no IANA actions. 10. Security Considerations When constanttime implementations are required, all basic operations and utility functions must be implemented in constant time, as discussed in Section 4. Each encoding function accepts arbitrary input and maps it to a pseudorandom point on the curve. Directly evaluating the mappings of Section 6 produces an output that is distinguishable from random. Section 3 shows how to use these mappings to construct a function approximating a random oracle. Section 3.1 describes considerations related to domain separation for random oracle encodings. Section 5 describes considerations for uniformly hashing to field elements. 11. Acknowledgements The authors would like to thank Adam Langley for his detailed writeup up Elligator 2 with Curve25519 [L13]. We also thank Sean Devlin and Thomas Icart for feedback on earlier versions of this document. 12. Contributors o Sharon Goldberg Boston University goldbe@cs.bu.edu o Ela Lee Royal Holloway, University of London Ela.Lee.2010@live.rhul.ac.uk 13. References FazHernandez, et al. Expires January 9, 2020 [Page 39] InternetDraft hashtocurve July 2019 13.1. Normative References [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.rfceditor.org/info/rfc2119>. [RFC5869] Krawczyk, H. and P. Eronen, "HMACbased ExtractandExpand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010, <https://www.rfceditor.org/info/rfc5869>. [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, <https://www.rfceditor.org/info/rfc7748>. [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.rfceditor.org/info/rfc8017>. 13.2. Informative References [AFQTZ14] Aranha, D., Fouque, P., Qian, C., Tibouchi, M., and J. Zapalowicz, "Binary Elligator squared", In Selected Areas in Cryptography  SAC 2014, pages 2037, DOI 10.1007/9783319130514_2, 2014, <https://doi.org/10.1007/9783319130514_2>. [AR13] Adj, G. and F. RodriguezHenriquez, "Square Root Computation over Even Extension Fields", In IEEE Transactions on Computers. vol 63 issue 11, pages 28292841, DOI 10.1109/TC.2013.145, November 2014, <https://doi.org/10.1109/TC.2013.145>. [BBJLP08] Bernstein, D., Birkner, P., Joye, M., Lange, T., and C. Peters, "Twisted Edwards curves", In AFRICACRYPT 2008, pages 389405, DOI 10.1007/9783540681649_26, 2008, <https://doi.org/10.1007/9783540681649_26>. [BCIMRT10] Brier, E., Coron, J., Icart, T., Madore, D., Randriam, H., and M. Tibouchi, "Efficient Indifferentiable Hashing into Ordinary Elliptic Curves", In Advances in Cryptology  CRYPTO 2010, pages 237254, DOI 10.1007/9783642146237_13, 2010, <https://doi.org/10.1007/9783642146237_13>. FazHernandez, et al. Expires January 9, 2020 [Page 40] InternetDraft hashtocurve July 2019 [BF01] Boneh, D. and M. Franklin, "Identitybased encryption from the Weil pairing", In Advances in Cryptology  CRYPTO 2001, pages 213229, DOI 10.1007/3540446478_13, August 2001, <https://doi.org/10.1007/3540446478_13>. [BHKL13] Bernstein, D., Hamburg, M., Krasnova, A., and T. Lange, "Elligator: ellipticcurve points indistinguishable from uniform random strings", In Proceedings of the 2013 ACM SIGSAC conference on computer and communications security., pages 967980, DOI 10.1145/2508859.2516734, November 2013, <https://doi.org/10.1145/2508859.2516734>. [BLMP19] Bernstein, D., Lange, T., Martindale, C., and L. Panny, "Quantum circuits for the CSIDH: optimizing quantum evaluation of isogenies", In Advances in Cryptology  EUROCRYPT 2019, DOI 10.1007/9783030176563, 2019, <https://doi.org/10.1007/9783030176563>. [BLS01] Boneh, D., Lynn, B., and H. Shacham, "Short signatures from the Weil pairing", In Journal of Cryptology, vol 17, pages 297319, DOI 10.1007/s0014500403149, July 2004, <https://doi.org/10.1007/s0014500403149>. [BLS03] Barreto, P., Lynn, B., and M. Scott, "Constructing Elliptic Curves with Prescribed Embedding Degrees", In Security in Communication Networks, pages 257267, DOI 10.1007/3540364137_19, 2003, <https://doi.org/10.1007/3540364137_19>. [BMP00] Boyko, V., MacKenzie, P., and S. Patel, "Provably secure passwordauthenticated key exchange using DiffieHellman", In Advances in Cryptology  EUROCRYPT 2000, pages 156171, DOI 10.1007/3540455396_12, May 2000, <https://doi.org/10.1007/3540455396_12>. [BN05] Barreto, P. and M. Naehrig, "PairingFriendly Elliptic Curves of Prime Order", In Selected Areas in Cryptography 2005, pages 319331, DOI 10.1007/11693383_22, 2006, <https://doi.org/10.1007/11693383_22>. [BP18] Budroni, A. and F. Pintore, "Hashing to G2 on BLS pairing friendly curves", In ACM Communications in Computer Algebra, pages 6366, DOI 10.1145/3313880.3313884, September 2018, <https://doi.org/10.1145/3313880.3313884>. FazHernandez, et al. Expires January 9, 2020 [Page 41] InternetDraft hashtocurve July 2019 [CFADLNV05] Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., and F. Vercauteren, "Handbook of Elliptic and Hyperelliptic Curve Cryptography", publisher Chapman and Hall / CRC, ISBN 9781584885184, 2005, <https://www.crcpress.com/9781584885184>. [CK11] Couveignes, J. and J. Kammerer, "The geometry of flex tangents to a cubic curve and its parameterizations", In Journal of Symbolic Computation, vol 47 issue 3, pages 266281, DOI 10.1016/j.jsc.2011.11.003, 2012, <https://doi.org/10.1016/j.jsc.2011.11.003>. [draftyonezawapfc01] Yonezawa, S., Chikara, S., Kobayashi, T., and T. Saito, "Pairingfriendly Curves", March 2019, <https://datatracker.ietf.org/doc/ draftyonezawapairingfriendlycurves/>. [F11] Farashahi, R., "Hashing into Hessian curves", In AFRICACRYPT 2011, pages 278289, DOI 10.1007/9783642219696_17, 2011, <https://doi.org/10.1007/9783642219696_17>. [FFSTV13] Farashahi, R., Fouque, P., Shparlinski, I., Tibouch, M., and J. Voloch, "Indifferentiable deterministic hashing to elliptic and hyperelliptic curves", In Math. Comp. vol 82, pages 491512, DOI 10.1090/S002557182012026068, 2013, <https://doi.org/10.1090/S002557182012026068>. [FIPS1864] National Institute of Standards and Technology (NIST), "FIPS Publication 1864: Digital Signature Standard", July 2013, <https://nvlpubs.nist.gov/nistpubs/FIPS/ NIST.FIPS.1864.pdf>. [FJT13] Fouque, P., Joux, A., and M. Tibouchi, "Injective encodings to elliptic curves", In ACISP 2013, pages 203218, DOI 10.1007/9783642390593_14, 2013, <https://doi.org/10.1007/9783642390593_14>. [FKR11] FuentesCastaneda, L., Knapp, E., and F. Rodriguez Henriquez, "Fast Hashing to G2 on PairingFriendly Curves", In Selected Areas in Cryptography, pages 412430, DOI 10.1007/9783642284960_25, 2011, <https://doi.org/10.1007/9783642284960_25>. FazHernandez, et al. Expires January 9, 2020 [Page 42] InternetDraft hashtocurve July 2019 [FSV09] Farashahi, R., Shparlinski, I., and J. Voloch, "On hashing into elliptic curves", In Journal of Mathematical Cryptology, vol 3 no 4, pages 353360, DOI 10.1515/JMC.2009.022, 2009, <https://doi.org/10.1515/JMC.2009.022>. [FT10] Fouque, P. and M. Tibouchi, "Estimating the size of the image of deterministic hash functions to elliptic curves.", In Progress in Cryptology  LATINCRYPT 2010, pages 8191, DOI 10.1007/9783642147128_5, 2010, <https://doi.org/10.1007/9783642147128_5>. [FT12] Fouque, P. and M. Tibouchi, "Indifferentiable Hashing to BarretoNaehrig Curves", In Progress in Cryptology  LATINCRYPT 2012, pages 17, DOI 10.1007/9783642334818_1, 2012, <https://doi.org/10.1007/9783642334818_1>. [hash2curverepo] "Hashing to Elliptic Curves  GitHub repository", 2019, <https://github.com/cfrg/draftirtfcfrghashtocurve>. [Icart09] Icart, T., "How to Hash into Elliptic Curves", In Advances in Cryptology  CRYPTO 2009, pages 303316, DOI 10.1007/9783642033568_18, 2009, <https://doi.org/10.1007/9783642033568_18>. [J96] Jablon, D., "Strong passwordonly authenticated key exchange", In SIGCOMM Computer Communication Review, vol 26 issue 5, pages 526, DOI 10.1145/242896.242897, 1996, <https://doi.org/10.1145/242896.242897>. [KLR10] Kammerer, J., Lercier, R., and G. Renault, "Encoding points on hyperelliptic curves over finite fields in deterministic polynomial time", In PAIRING 2010, pages 278297, DOI 10.1007/9783642174551_18, 2010, <https://doi.org/10.1007/9783642174551_18>. [KSS08] Kachisa, E., Schaefer, E., and M. Scott, "Constructing BrezingWeng PairingFriendly Elliptic Curves Using Elements in the Cyclotomic Field", In PairingBased Cryptography  Pairing 2008, pages 126135, DOI 10.1007/9783540855385_9, 2008, <https://doi.org/10.1007/9783540855385_9>. [L13] Langley, A., "Implementing Elligator for Curve25519", 2013, <https://www.imperialviolet.org/2013/12/25/ elligator.html>. FazHernandez, et al. Expires January 9, 2020 [Page 43] InternetDraft hashtocurve July 2019 [S05] Skalba, M., "Points on elliptic curves over finite fields", In Acta Arithmetica, vol 117 no 3, pages 293301, DOI 10.4064/aa11737, 2005, <https://doi.org/10.4064/aa11737>. [S85] Schoof, R., "Elliptic Curves Over Finite Fields and the Computation of Square Roots mod p", In Mathematics of Computation vol 44 issue 170, pages 483494, DOI 10.1090/S00255718198507772806, April 1985, <https://doi.org/10.1090/S00255718198507772806>. [SAGE] The Sage Developers, "SageMath, the Sage Mathematics Software System", 2019, <https://www.sagemath.org>. [SBCDBK09] Scott, M., Benger, N., Charlemagne, M., Dominguez Perez, L., Benger, N., and E. Kachisa, "Fast Hashing to G2 on PairingFriendly Curves", In PairingBased Cryptography  Pairing 2009, pages 102113, DOI 10.1007/9783642032981_8, 2009, <https://doi.org/10.1007/9783642032981_8>. [SEC2] Standards for Efficient Cryptography Group (SECG), "SEC 2: Recommended Elliptic Curve Domain Parameters", January 2010, <http://www.secg.org/sec2v2.pdf>. [SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", May 2009, <http://www.secg.org/sec1v2.pdf>. [SS04] Schinzel, A. and M. Skalba, "On equations y^2 = x^n + k in a finite field.", In Bulletin Polish Acad. Sci. Math. vol 52, no 3, pages 223226, DOI 10.4064/ba5231, 2004, <https://doi.org/10.4064/ba5231>. [SW06] Shallue, A. and C. van de Woestijne, "Construction of rational points on elliptic curves over finite fields", In Algorithmic Number Theory. ANTS 2006., pages 510524, DOI 10.1007/11792086_36, 2006, <https://doi.org/10.1007/11792086_36>. [T14] Tibouchi, M., "Elligator squared: Uniform points on elliptic curves of prime order as uniform random strings", In Financial Cryptography and Data Security  FC 2014, pages 139156, DOI 10.1007/9783662454725_10, 2014, <https://doi.org/10.1007/9783662454725_10>. FazHernandez, et al. Expires January 9, 2020 [Page 44] InternetDraft hashtocurve July 2019 [TK17] Tibouchi, M. and T. Kim, "Improved elliptic curve hashing and point representation", In Designs, Codes, and Cryptography, vol 82, pages 161177, DOI 10.1007/s1062301602882, 2017, <https://doi.org/10.1007/s1062301602882>. [U07] Ulas, M., "Rational points on certain hyperelliptic curves over finite fields", In Bulletin Polish Acad. Sci. Math. vol 55, no 2, pages 97104, DOI 10.4064/ba5521, 2007, <https://doi.org/10.4064/ba5521>. [W08] Washington, L., "Elliptic curves: Number theory and cryptography", edition 2nd, publisher Chapman and Hall / CRC, ISBN 9781420071467, 2008, <https://www.crcpress.com/9781420071467>. [WB19] Wahby, R. and D. Boneh, "Fast and simple constanttime hashing to the BLS12381 elliptic curve", Technical report ePrint 2019/403, 2019, <https://eprint.iacr.org/2019/403>. Appendix A. Related Work The problem of mapping arbitrary bit strings to elliptic curve points has been the subject of both practical and theoretical research. This section briefly describes the background and research results that underly the recommendations in this document. This section is provided for informational purposes only. A naive but generally insecure method of mapping a string alpha to a point on an elliptic curve E having n points is to first fix a point P that generates the elliptic curve group, and a hash function Hn from bit strings to integers less than n; then compute Hn(alpha) * P, where the * operator represents scalar multiplication. The reason this approach is insecure is that the resulting point has a known discrete log relationship to P. Thus, except in cases where this method is specified by the protocol, it must not be used; doing so risks catastrophic security failures. Boneh et al. [BLS01] describe an encoding method they call MapToGroup, which works roughly as follows: first, use the input string to initialize a pseudorandom number generator, then use the generator to produce a pseudorandom value x in F. If x is the xcoordinate of a point on the elliptic curve, output that point. Otherwise, generate a new pseudorandom value x in F and try again. Since a random value x in F has probability about 1/2 of corresponding to a point on the curve, the expected number of tries is just two. However, the running time of this method depends on the FazHernandez, et al. Expires January 9, 2020 [Page 45] InternetDraft hashtocurve July 2019 input string, which means that it is not safe to use in protocols sensitive to timing side channels. Schinzel and Skalba [SS04] introduce the first method of constructing elliptic curve points deterministically, for a restricted class of curves and a very small number of points. Skalba [S05] generalizes this construction to more curves and more points on those curves. Shallue and van de Woestijne [SW06] further generalize and simplify Skalba's construction, yielding concretely efficient maps to a constant fraction of the points on almost any curve. Ulas [U07] describes a simpler version of this map, and Brier et al. [BCIMRT10] give a further simplification, which the authors call the "simplified SWU" map. The simplified map applies only to fields of characteristic p = 3 mod 4; Wahby and Boneh [WB19] generalize to fields of any characteristic. Icart gives another deterministic algorithm which maps to any curve over a field of characteristic p = 2 mod 3 [Icart09]. Several extensions and generalizations follow this work, including [FSV09], [FT10], [KLR10], [F11], and [CK11]. Following the work of Farashahi [F11], Fouque et al. [FJT13] describe a mapping to curves of characteristic p = 3 mod 4 having a number of points divisible by 4. Bernstein et al. [BHKL13] optimize this mapping and describe a related mapping that they call "Elligator 2," which applies to any curve over a field of odd characteristic having a point of order 2. This includes Curve25519 and Curve448, both of which are CFRGrecommended curves [RFC7748]. An important caveat regarding all of the above deterministic mapping functions is that none of them map to the entire curve, but rather to some fraction of the points. This means that they cannot be used directly to construct a random oracle that outputs points on the curve. Brier et al. [BCIMRT10] give two solutions to this problem. The first, which Brier et al. prove applies to Icart's method, computes f(H0(msg)) + f(H1(msg)) for two distinct hash functions H0 and H1 from bit strings to F and a mapping f from F to the elliptic curve E. The second, which applies to essentially all deterministic mappings but is more costly, computes f(H0(msg)) + H2(msg) * P, for P a generator of the elliptic curve group and H2 a hash from bit strings to integers modulo n, the order of the elliptic curve group. Farashahi et al. [FFSTV13] improve the analysis of the first method, showing that it applies to essentially all deterministic mappings. Tibouchi and Kim [TK17] further refine the analysis and describe additional optimizations. FazHernandez, et al. Expires January 9, 2020 [Page 46] InternetDraft hashtocurve July 2019 Complementary to the problem of mapping from bit strings to elliptic curve points, Bernstein et al. [BHKL13] study the problem of mapping from elliptic curve points to uniformly random bit strings, giving solutions for a class of curves including Montgomery and twisted Edwards curves. Tibouchi [T14] and Aranha et al. [AFQTZ14] generalize these results. This document does not deal with this complementary problem. Appendix B. Rational maps from twisted Edwards to Weierstrass and Montgomery curves The inverse of the rational map specified in Section 6.7.1, i.e., the map from the point (x', y') on the Weierstrass curve y'^2 = x'^3 + A * x'^2 + B * x' to the point (x, y) on the twisted Edwards curve a * x^2 + y^2 = 1 + d * x^2 * y^2 is given by: o x' = (1 + y) / (B' * (1  y)) o y' = (1 + y) / (B' * x * (1  y)) where o A = (a + d) / 2 o B = (a  d)^2 / 16 o B' = 1 / sqrt(B) = 4 / (a  d) This map is undefined when y == 1 or x == 0. In this case, return the point (0, 0). It may also be useful to map to a Montgomery curve of the form B' * y''^2 = x''^3 + A' * x''^2 + x''. This curve is equivalent to the twisted Edwards curve above via the following rational map ([BBJLP08], Theorem 3.2): o A' = 2 * (a + d) / (a  d) o B' = 4 / (a  d) o x'' = (1 + y) / (1  y) o y'' = (1 + y) / (x * (1  y)) FazHernandez, et al. Expires January 9, 2020 [Page 47] InternetDraft hashtocurve July 2019 Appendix C. Isogenous curves and corresponding maps for BLS12381 This section specifies the isogeny maps for the BLS12381 suites listed in Section 8.7. C.1. 11isogeny map for G1 The 11isogeny map from E' to E is given by the following rational functions: o x = x_num / x_den, where * x_num = k_(1,11) * x'^11 + k_(1,10) * x'^10 + ... + k_(1,0) * x_den = x'^10 + k_(2,9) * x'^9 + ... + k_(2,0) o y = y' * y_num / y_den, where * y_num = k_(3,15) * x'^15 + k_(3,14) * x'^14 + ... + k_(3,0) * y_den = x'^15 + k_(4,14) * x'^14 + ... + k_(4,0) The constants used to compute x_num are as follows: o k_(1,0) = 0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b 56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7 o k_(1,1) = 0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e3 3c70d1e86b4838f2a6f318c356e834eef1b3cb83bb o k_(1,2) = 0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e50 1ec68e25c958c3e3d2a09729fe0179f9dac9edcb0 o k_(1,3) = 0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b332 89f1b330835336e25ce3107193c5b388641d9b6861 o k_(1,4) = 0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb6 5982fac18985a286f301e77c451154ce9ac8895d9 o k_(1,5) = 0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab 9097e68f90a0870d2dcae73d19cd13c1c66f652983 o k_(1,6) = 0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e21 8f9c86b2a8da25128c1052ecaddd7f225a139ed84 o k_(1,7) = 0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c6 2ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e FazHernandez, et al. Expires January 9, 2020 [Page 48] InternetDraft hashtocurve July 2019 o k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c 928c5d1de4fa295f296b74e956d71986a8497e317 o k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314 baf4bb1b7fa3190b2edc0327797f241067be390c9e o k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af 36003b14866f69b771f8c285decca67df3f1605fb7b o k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf 1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229 The constants used to compute x_den are as follows: o k_(2,0) = 0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c95886 17fc8ac62b558d681be343df8993cf9fa40d21b1c o k_(2,1) = 0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2 588c48bf5713daa8846cb026e9e5c8276ec82b3bff o k_(2,2) = 0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e0 0b11aceacd6a3d0967c94fedcfcc239ba5cb83e19 o k_(2,3) = 0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8a bc28d6fd04976d5243eecf5c4130de8938dc62cd8 o k_(2,4) = 0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44 e833b306da9bd29ba81f35781d539d395b3532a21e o k_(2,5) = 0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bce f24b8982f7400d24bc4228f11c02df9a29f6304a5 o k_(2,6) = 0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073 062aede9cea73b3538f0de06cec2574496ee84a3a o k_(2,7) = 0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1 cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e o k_(2,8) = 0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da 67f39883503826692abba43704776ec3a79a1d641 o k_(2,9) = 0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533 978f31c1593174e4b4b7865002d6384d168ecdd0a The constants used to compute y_num are as follows: o k_(3,0) = 0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6 c956543d3cd0c7aee9b3ba3c2be9845719707bb33 FazHernandez, et al. Expires January 9, 2020 [Page 49] InternetDraft hashtocurve July 2019 o k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f9063 43eb67ad34d6c56711962fa8bfe097e75a2e41c696 o k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b 91400da7d26d521628b00523b8dfe240c72de1f6 o k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3e c03251cf9de405aba9ec61deca6355c77b0e5f4cb o k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd 221351adc2ee7f8dc099040a841b6daecf2e8fedb o k_(3,5) = 0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c 3d3ad5544e203f6326c95a807299b23ab13633a5f0 o k_(3,6) = 0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413 c4d634f3747a87ac2460f415ec961f8855fe9d6f2 o k_(3,7) = 0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da 6c26c842642f64550fedfe935a15e4ca31870fb29 o k_(3,8) = 0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6 a1f20cabe69d65201c78607a360370e577bdba587 o k_(3,9) = 0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7 ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30 o k_(3,10) = 0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183 e416389e61031bf3a5cce3fbafce813711ad011c132 o k_(3,11) = 0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bf e7f911f643249d9cdf41b44d606ce07c8a4d0074d8e o k_(3,12) = 0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710 e807b4633f06c851c1919211f20d4c04f00b971ef8 o k_(3,13) = 0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e663 2d3c40659cc6cf90ad1c232a6442d9d3f5db980133 o k_(3,14) = 0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f 5d396a7ce46ba1049b6579afb7866b1e715475224b o k_(3,15) = 0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add 4fa95af01b2b665027efec01c7704b456be69c8b604 The constants used to compute y_den are as follows: FazHernandez, et al. Expires January 9, 2020 [Page 50] InternetDraft hashtocurve July 2019 o k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec323 2b5be72e7a07f3688ef60c206d01479253b03663c1 o k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a103 56f453e01f78a4260763529e3532f6102c2e49a03d o k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5d d279cd2eca6757cd636f96f891e2538b53dbf67f2 o k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da4 89e726af41727364f2c28297ada8d26d98445f5416 o k_(4,4) = 0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0 fc9dec916a20b15dc0fd2ededda39142311a5001d o k_(4,5) = 0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87 b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac o k_(4,6) = 0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365 bc400a0051d5fa9c01a58b1fb93d1a1399126a775c o k_(4,7) = 0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34f d206357132b920f5b00801dee460ee415a15812ed9 o k_(4,8) = 0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc575 0c4bf39b4852cfe2f7bb9248836b233d9d55535d4a o k_(4,9) = 0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb530859 2e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55 o k_(4,10) = 0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b 36e636a5c871a5c29f4f83060400f8b49cba8f6aa8 o k_(4,11) = 0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a 88cea7913516f968986f7ebbea9684b529e2561092 o k_(4,12) = 0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5cee f9a00d9b8693000763e3b90ac11e99b138573345cc o k_(4,13) = 0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748d f27942480e420517bd8714cc80d1fadc1326ed06f7 o k_(4,14) = 0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efc d6356caa205ca2f570f13497804415473a1d634b8f FazHernandez, et al. Expires January 9, 2020 [Page 51] InternetDraft hashtocurve July 2019 C.2. 3isogeny map for G2 The 3isogeny map from E' to E is given by the following rational functions: o x = x_num / x_den, where * x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + ... + k_(1,0) * x_den = x'^2 + k_(2,1) * x' + k_(2,0) o y = y' * y_num / y_den, where * y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + ... + k_(3,0) * y_den = x'^3 + k_(4,2) * x'^2 + ... + k_(4,0) The constants used to compute x_num are as follows: o k_(1,0) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842 3c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 + 0x5c759507e8e333ebb5b7 a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaa aaaaa97d6 * i o k_(1,1) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71a * i o k_(1,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71e + 0x8ab05f8bdd54cde1909 37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff ffffffe38d * i o k_(1,3) = 0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d610 8f142b85757098e38d0f671c7188e2aaaaaaaa5ed1 The constants used to compute x_den are as follows: o k_(2,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffaa63 * i o k_(2,1) = 0xc + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf 6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f * i The constants used to compute y_num are as follows: o k_(3,0) = 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439 d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 + 0x1530477c7ab4113b59a FazHernandez, et al. Expires January 9, 2020 [Page 52] InternetDraft hashtocurve July 2019 4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc 71c71c6d706 * i o k_(3,1) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842 3c50ae15d5c2638e343d9c71c6238aaaaaaaa97be * i o k_(3,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c 6b4f20a4181472aaa9cb8d555526a9ffffffffc71c + 0x8ab05f8bdd54cde1909 37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff ffffffe38f * i o k_(3,3) = 0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977 c69aa274524e79097a56dc4bd9e1b371c71c718b10 The constants used to compute y_den are as follows: o k_(4,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb + 0x1a0111ea397fe69a4b1 ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9fef fffffffa8fb * i o k_(4,1) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2 a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3 * i o k_(4,2) = 0x12 + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512b f6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99 * i Appendix D. Sample Code This section gives sample implementations optimized for some of the elliptic curves listed in Section 8. A future version of this document will include all listed curves, plus accompanying test vectors. Sample Sage [SAGE] code for each algorithm can also be found in the draft repository [hash2curverepo]. D.1. Interface and projective coordinate systems The sample code in this section uses a different interface than the mappings of Section 6. Specifically, each mapping function in this section has the following signature: (xn, xd, yn, nd) = map_to_curve(u) The resulting point (x, y) is given by (xn / xd, yn / yd). The reason for this modified interface is that it enables further optimizations when working with points in a projective coordinate system. This is desirable, for example, when the resulting point FazHernandez, et al. Expires January 9, 2020 [Page 53] InternetDraft hashtocurve July 2019 will be immediately multiplied by a scalar, since most scalar multiplication algorithms operate on projective points. The following are two commonly used projective coordinate systems and the corresponding conversions: o A point (X, Y, Z) in homogeneous projective coordinates corresponds to the affine point (x, y) = (X / Z, Y / Z); the inverse conversion is given by (X, Y, Z) = (x, y, 1). To convert (xn, xd, yn, yd) to homogeneous projective coordinates, compute (X, Y, Z) = (xn * yd, yn * xd, xd * yd). o A point (X', Y', Z') in Jacobian projective coordinates corresponds to the affine point (x, y) = (X' / Z'^2, Y' / Z'^3); the inverse conversion is given by (X', Y', Z') = (x, y, 1). To convert (xn, xd, yn, yd) to Jacobian projective coordinates, compute (X', Y', Z') = (xn * xd * yd^2, yn * yd^2 * xd^3, xd * yd). D.2. P256 (Simplified SWU) The following is a straightline implementation of the Simplified SWU mapping for P256 [FIPS1864] as specified in Section 8.1. FazHernandez, et al. Expires January 9, 2020 [Page 54] InternetDraft hashtocurve July 2019 map_to_curve_simple_swu_p256(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on P256. Constants: 1. B = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b 2. c1 = B / 3 3. c2 = (p  3) / 4 // Integer arithmetic 4. c3 = sqrt(8) Steps: 1. t1 = u^2 2. t3 = 2 * t1 3. t2 = t3^2 4. xd = t2 + t3 5. x1n = xd + 1 6. x1n = x1n * B 7. xd = xd * 3 8. e1 = xd == 0 9. xd = CMOV(xd, 6, e1) // If xd == 0, set xd = Z * A == 6 10. t2 = xd^2 11. gxd = t2 * xd // gxd == xd^3 12. t2 = 3 * t2 13. gx1 = x1n^2 14. gx1 = gx1 + t2 // x1n^2 + A * xd^2 15. gx1 = gx1 * x1n // x1n^3 + A * x1n * xd^2 16. t2 = B * gxd 17. gx1 = gx1 + t2 // x1n^3 + A * x1n * xd^2 + B * xd^3 18. t4 = gxd^2 19. t2 = gx1 * gxd 20. t4 = t4 * t2 // gx1 * gxd^3 21. y1 = t4^c2 // (gx1 * gxd^3)^((p  3) / 4) 22. y1 = y1 * t2 // gx1 * gxd * (gx1 * gxd^3)^((p  3) / 4) 23. x2n = t3 * x1n // x2 = x2n / xd = 2 * u^2 * x1n / xd 24. y2 = y1 * c3 25. y2 = y2 * t1 26. y2 = y2 * u 27. t2 = y1^2 28. t2 = t2 * gxd 29. e2 = t2 == gx1 30. xn = CMOV(x2n, x1n, e2) // If e2, x = x1, else x = x2 31. y = CMOV(y2, y1, e2) // If e2, y = y1, else y = y2 32. e3 = sgn0(u) == sgn0(y) // fix sign of y 33. y = CMOV(y, y, e3) 34. return (xn, xd, y, 1) FazHernandez, et al. Expires January 9, 2020 [Page 55] InternetDraft hashtocurve July 2019 D.3. curve25519 (Elligator 2) The following is a straightline implementation of Elligator 2 for curve25519 [RFC7748] as specified in Section 8.4. map_to_curve_elligator2_curve25519(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on curve25519. Constants: 1. c1 = (p + 3) / 8 // Integer arithmetic 2. c2 = 2^c1 3. c3 = sqrt(1) 4. c4 = (p  5) / 8 // Integer arithmetic Steps: 1. t1 = u^2 2. t1 = 2 * t1 3. xd = t1 + 1 // nonzero: 1 is square mod p, xd is not 4. x1n = 486662 // x1 = x1n / xd = 486662 / (1 + 2 * u^2) 5. t2 = xd^2 6. gxd = t2 * xd // gxd = xd^3 7. gx1 = 486662 * xd // 486662 * xd 8. gx1 = gx1 + x1n // x1n + 486662 * xd 9. gx1 = gx1 * x1n // x1n^2 + 486662 * x1n * xd 10. gx1 = gx1 + t2 // x1n^2 + 486662 * x1n * xd + xd^2 11. gx1 = gx1 * x1n // x1n^3 + 486662 * x1n^2 * xd + x1n * xd^2 12. t3 = gxd^2 13. t2 = t3^2 // gxd^4 14. t3 = t3 * gxd // gxd^3 15. t3 = t3 * gx1 // gx1 * gxd^3 16. t2 = t2 * t3 // gx1 * gxd^7 17. y11 = t2^c4 // (gx1 * gxd^7)^((p  5) / 8) 18. y11 = y11 * t3 // gx1 * gxd^3 * (gx1 * gxd^7)^((p  5) / 8) 19. y12 = y11 * c3 20. t2 = y11^2 21. t2 = t2 * gxd 22. e1 = t2 == gx1 23. y1 = CMOV(y12, y11, e1) // If g(x1) is square, this is its sqrt 24. x2n = x1n * t1 // x2 = x2n / xd = 2 * u^2 * x1n / xd 25. y21 = y11 * u 26. y21 = y21 * c2 27. y22 = y21 * c3 28. gx2 = gx1 * t1 // g(x2) = gx2 / gxd = 2 * u^2 * g(x1) 29. t2 = y21^2 30. t2 = t2 * gxd 31. e2 = t2 == gx2 FazHernandez, et al. Expires January 9, 2020 [Page 56] InternetDraft hashtocurve July 2019 32. y2 = CMOV(y22, y21, e2) // If g(x2) is square, this is its sqrt 33. t2 = y1^2 34. t2 = t2 * gxd 35. e3 = t2 == gx1 36. xn = CMOV(x2n, x1n, e3) // if e3, x = x1, else x = x2 37. y = CMOV(y2, y1, e3) // if e3, y = y1, else y = y2 38. e4 = sgn0(u) == sgn0(y) // fix sign of y 39. y = CMOV(y, y, e4) 40. return (xn, xd, y, 1) D.4. edwards25519 (Elligator 2) The following is a straightline implementation of Elligator 2 for edwards25519 [RFC7748] as specified in Section 8.4. The subroutine map_to_curve_elligator2_curve25519 is defined in Appendix D.3. map_to_curve_elligator2_edwards25519(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on edwards25519. Constants: 1. c1 = sqrt(486664) // sign MUST be chosen such that sgn0(c1) == 1 Steps: 1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u) 2. xn = xMn * yMd 3. xn = xn * c1 4. xd = xMd * yMn // xn / xd = c1 * xM / yM 5. yn = xMn  xMd 6. yd = xMn + xMd // (n / d  1) / (n / d + 1) = (n  d) / (n + d) 7. return (xn, xd, yn, yd) D.5. curve448 (Elligator 2) The following is a straightline implementation of Elligator 2 for curve448 [RFC7748] as specified in Section 8.5. FazHernandez, et al. Expires January 9, 2020 [Page 57] InternetDraft hashtocurve July 2019 map_to_curve_elligator2_curve448(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on curve448. Constants: 1. c1 = (p  3) / 4 // Integer arithmetic Steps: 1. t1 = u^2 2. xd = 1  t1 3. e1 = xd == 0 4. xd = CMOV(xd, 1, e1) // If xd == 0, set xd = 1 5. x1n = CMOV(156326, 1, e1) // If xd == 0, x1n = 1, else x1n = A 6. t2 = xd^2 7. gxd = t2 * xd // gxd = xd^3 8. gx1 = 156326 * xd // 156326 * xd 9. gx1 = gx1 + x1n // x1n + 156326 * xd 10. gx1 = gx1 * x1n // x1n^2 + 156326 * x1n * xd 11. gx1 = gx1 + t2 // x1n^2 + 156326 * x1n * xd + xd^2 12. gx1 = gx1 * x1n // x1n^3 + 156326 * x1n^2 * xd + x1n * xd^2 13. t3 = gxd^2 14. t2 = gx1 * gxd // gx1 * gxd 15. t3 = t3 * t2 // gx1 * gxd^3 16. y1 = t3^c1 // (gx1 * gxd^3)^((p  3) / 4) 17. y1 = y1 * t2 // gx1 * gxd * (gx1 * gxd^3)^((p  3) / 4) 18. x2n = t1 * x1n // x2 = x2n / xd = 1 * u^2 * x1n / xd 19. y2 = y1 * u 20. t2 = y1^2 21. t2 = t2 * gxd 22. e2 = t2 == gx1 23. xn = CMOV(x2n, x1n, e2) // If e2, x = x1, else x = x2 24. y = CMOV(y2, y1, e2) // If e2, y = y1, else y = y2 25. e3 = sgn0(u) == sgn0(y) // fix sign of y 26. y = CMOV(y, y, e3) 27. return (xn, xd, y, 1) D.6. edwards448 (Elligator 2) The following is a straightline implementation of Elligator 2 for edwards448 [RFC7748] as specified in Section 8.5. The subroutine map_to_curve_elligator2_curve448 is defined in Appendix D.5. FazHernandez, et al. Expires January 9, 2020 [Page 58] InternetDraft hashtocurve July 2019 map_to_curve_elligator2_edwards448(u) Input: u, an element of F. Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a point on edwards448. Steps: 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u) 2. xn2 = xn^2 3. xd2 = xd^2 4. xd4 = xd2^2 5. yn2 = yn^2 6. yd2 = yd^2 7. xEn = xn2  xd2 8. t2 = xEn  xd2 9. xEn = xEn * xd2 10. xEn = xEn * yd 11. xEn = xEn * yn 12. xEn = xEn * 4 13. t2 = t2 * xn2 14. t2 = t2 * yd2 15. t3 = 4 * yn2 16. t1 = t3 + yd2 17. t1 = t1 * xd4 18. xEd = t1 + t2 19. t2 = t2 * xn 20. t4 = xn * xd4 21. yEn = t3  yd2 22. yEn = yEn * t4 23. yEn = yEn  t2 24. t1 = xn2 + xd2 25. t1 = t1 * xd2 26. t1 = t1 * xd 27. t1 = t1 * yn2 28. t1 = 2 * t1 29. yEd = t2 + t1 30. t4 = t4 * yd2 31. yEd = yEd + t4 32. return (xEn, xEd, yEn, yEd) Authors' Addresses Armando FazHernandez Cloudflare 101 Townsend St San Francisco United States of America Email: armfazh@cloudflare.com FazHernandez, et al. Expires January 9, 2020 [Page 59] InternetDraft hashtocurve July 2019 Sam Scott Cornell Tech 2 West Loop Rd New York, New York 10044 United States of America Email: sam.scott@cornell.edu Nick Sullivan Cloudflare 101 Townsend St San Francisco United States of America Email: nick@cloudflare.com Riad S. Wahby Stanford University Email: rsw@cs.stanford.edu Christopher A. Wood Apple Inc. One Apple Park Way Cupertino, California 95014 United States of America Email: cawood@apple.com FazHernandez, et al. Expires January 9, 2020 [Page 60]