Network Working Group A. Faz-Hernandez
Internet-Draft Cloudflare
Intended status: Informational S. Scott
Expires: May 5, 2020 Cornell Tech
N. Sullivan
Cloudflare
R. Wahby
Stanford University
C. Wood
Apple Inc.
November 02, 2019
Hashing to Elliptic Curves
draft-irtf-cfrg-hash-to-curve-05
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.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1. How to use this document . . . . . . . . . . . . . . . . 4
1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 5
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1. Elliptic curves . . . . . . . . . . . . . . . . . . . . . 5
2.2. Terminology . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1. Mappings . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2. Encodings . . . . . . . . . . . . . . . . . . . . . . 7
2.2.3. Random oracle encodings . . . . . . . . . . . . . . . 7
2.2.4. Serialization . . . . . . . . . . . . . . . . . . . . 8
2.2.5. Domain separation . . . . . . . . . . . . . . . . . . 8
3. Roadmap . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1. Domain separation requirements . . . . . . . . . . . . . 10
4. Utility Functions . . . . . . . . . . . . . . . . . . . . . . 11
4.1. sgn0 variants . . . . . . . . . . . . . . . . . . . . . . 12
4.1.1. Big endian variant . . . . . . . . . . . . . . . . . 13
4.1.2. Little endian variant . . . . . . . . . . . . . . . . 14
5. Hashing to a Finite Field . . . . . . . . . . . . . . . . . . 14
5.1. Security considerations . . . . . . . . . . . . . . . . . 15
5.2. Performance considerations . . . . . . . . . . . . . . . 16
5.3. Implementation . . . . . . . . . . . . . . . . . . . . . 16
5.4. Alternative hash_to_base functions . . . . . . . . . . . 17
6. Deterministic Mappings . . . . . . . . . . . . . . . . . . . 18
6.1. Choosing a mapping function . . . . . . . . . . . . . . . 18
6.2. Interface . . . . . . . . . . . . . . . . . . . . . . . . 19
6.3. Notation . . . . . . . . . . . . . . . . . . . . . . . . 19
6.4. Sign of the resulting point . . . . . . . . . . . . . . . 20
6.5. Exceptional cases . . . . . . . . . . . . . . . . . . . . 20
6.6. Mappings for Weierstrass curves . . . . . . . . . . . . . 20
6.6.1. Shallue-van de Woestijne Method . . . . . . . . . . . 20
6.6.2. Simplified Shallue-van de Woestijne-Ulas Method . . . 24
6.6.3. Simplified SWU for AB == 0 . . . . . . . . . . . . . 25
6.7. Mappings for Montgomery curves . . . . . . . . . . . . . 27
6.7.1. Elligator 2 Method . . . . . . . . . . . . . . . . . 27
6.8. Mappings for Twisted Edwards curves . . . . . . . . . . . 29
6.8.1. Rational maps from Montgomery to twisted Edwards
curves . . . . . . . . . . . . . . . . . . . . . . . 29
6.8.2. Elligator 2 Method . . . . . . . . . . . . . . . . . 31
6.9. Mappings for Supersingular curves . . . . . . . . . . . . 32
6.9.1. Boneh-Franklin Method . . . . . . . . . . . . . . . . 32
6.9.2. Elligator 2, A == 0 Method . . . . . . . . . . . . . 32
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7. Clearing the cofactor . . . . . . . . . . . . . . . . . . . . 34
8. Suites for Hashing . . . . . . . . . . . . . . . . . . . . . 35
8.1. Defining a new hash-to-curve suite . . . . . . . . . . . 36
8.2. Suite ID naming conventions . . . . . . . . . . . . . . . 36
8.3. Suites for NIST P-256 . . . . . . . . . . . . . . . . . . 38
8.4. Suites for NIST P-384 . . . . . . . . . . . . . . . . . . 38
8.5. Suites for NIST P-521 . . . . . . . . . . . . . . . . . . 39
8.6. Suites for curve25519 and edwards25519 . . . . . . . . . 40
8.7. Suites for curve448 and edwards448 . . . . . . . . . . . 41
8.8. Suites for secp256k1 . . . . . . . . . . . . . . . . . . 42
8.9. Suites for BLS12-381 . . . . . . . . . . . . . . . . . . 44
8.9.1. BLS12-381 G1 . . . . . . . . . . . . . . . . . . . . 44
8.9.2. BLS12-381 G2 . . . . . . . . . . . . . . . . . . . . 45
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 46
10. Security Considerations . . . . . . . . . . . . . . . . . . . 46
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 47
12. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 47
13. References . . . . . . . . . . . . . . . . . . . . . . . . . 47
13.1. Normative References . . . . . . . . . . . . . . . . . . 47
13.2. Informative References . . . . . . . . . . . . . . . . . 48
Appendix A. Related Work . . . . . . . . . . . . . . . . . . . . 54
Appendix B. Rational maps . . . . . . . . . . . . . . . . . . . 56
B.1. Twisted Edwards to Weierstrass and Montgomery curves . . 56
B.2. Montgomery to Weierstrass curves . . . . . . . . . . . . 57
Appendix C. Isogeny maps for Suites . . . . . . . . . . . . . . 58
C.1. 3-isogeny map for secp256k1 . . . . . . . . . . . . . . . 58
C.2. 11-isogeny map for BLS12-381 G1 . . . . . . . . . . . . . 59
C.3. 3-isogeny map for BLS12-381 G2 . . . . . . . . . . . . . 63
Appendix D. Sample Code . . . . . . . . . . . . . . . . . . . . 65
D.1. Interface and projective coordinate systems . . . . . . . 65
D.2. Simplified SWU for p = 3 (mod 4) . . . . . . . . . . . . 66
D.3. curve25519 (Elligator 2) . . . . . . . . . . . . . . . . 68
D.4. edwards25519 (Elligator 2) . . . . . . . . . . . . . . . 69
D.5. curve448 (Elligator 2) . . . . . . . . . . . . . . . . . 69
D.6. edwards448 (Elligator 2) . . . . . . . . . . . . . . . . 70
Appendix E. Scripts for parameter generation . . . . . . . . . . 72
E.1. Finding Z for the Shallue and van de Woestijne map . . . 72
E.2. Finding Z for Simplified SWU . . . . . . . . . . . . . . 72
E.3. Finding Z for Elligator 2 . . . . . . . . . . . . . . . . 73
Appendix F. sqrt functions . . . . . . . . . . . . . . . . . . . 73
F.1. p = 3 (mod 4) . . . . . . . . . . . . . . . . . . . . . . 74
F.2. p = 5 (mod 8) . . . . . . . . . . . . . . . . . . . . . . 74
F.3. p = 9 (mod 16) . . . . . . . . . . . . . . . . . . . . . 74
F.4. Constant-time Tonelli-Shanks algorithm . . . . . . . . . 75
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 77
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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], Identity-Based Encryption [BF01] and Boneh-Lynn-
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 arbitrary-length
bit string and produces as output a point on an elliptic curve. We
provide implementation details for each algorithm, describe the
security rationale behind each recommendation, and give guidance for
elliptic curves that are not explicitly covered.
This document does not cover rejection sampling methods, sometimes
known as "try-and-increment" or "hunt-and-peck," because the goal is
to describe algorithms that can plausibly be made constant time. Use
of these rejection methods is NOT RECOMMENDED, because they have been
a perennial cause of side-channel vulnerabilities.
1.1. How to use this document
This document is intended for use by both implementors and protocol
designers.
For implementors, the necessary and sufficient level of specification
is a hash-to-curve suite, which fixes all of the parameters listed in
Section 8, plus a domain separation tag (Section 3.1). Starting from
working operations on the target elliptic curve and its base field, a
hash-to-curve suite requires implementing the specified encoding
function (Section 3), its constituent subroutines (Section 5,
Section 6, Section 7), and a few utility functions (Section 4).
Correspondingly, designers specifying a protocol that requires
hashing to an elliptic curve should either choose an existing hash-
to-curve suite or specify a new one (see Section 8.1). In addition,
designers should choose a domain separation tag following the
guidelines in Section 3.1.
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1.2. Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
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 writes
elements of field extensions in a primitive element or polynomial
basis, i.e., as a vector of m elements of GF(p) written in ascending
order by degree. The entries of this vector are indexed in ascending
order starting from 1, i.e., x = (x_1, x_2, ..., x_m). For example,
if q = p^2 and the primitive element basis is (1, i), then x = (a, b)
corresponds to the element a + b * i, where x_1 = a and x_2 = b.
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 hash-to-curve algorithms restrict the form of the curve
equation, the characteristic of the field, and/or the parameters of
the curve. For each algorithm presented, this document lists the
relevant restrictions.
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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.
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
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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
construction is distinguishable from uniformly random, i.e., it does
not behave like a random oracle.
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Brier et al. [BCIMRT10] describe two generic constructions whose
outputs are indifferentiable from a random oracle when the
constructions are instantiated with appropriate hash functions
modeled as random oracles. 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. The
reverse operation, deserialization, converts a bit string to an
elliptic curve point. For example, [SEC1] and [p1363a] give standard
methods for serialization and deserialization.
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.
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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
arbitrary-length 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 high-level 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 arbitrary-length 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. This function is suitable for
applications requiring a random oracle returning points 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.
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hash_to_curve(alpha)
Input: alpha, an arbitrary-length 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
All uses of the encoding functions defined in this document MUST
include domain separation (Section 2.2.5) to avoid interfering with
other uses of similar functionality.
Protocols that instantiate multiple, independent hash functions based
on either hash_to_curve or encode_to_curve MUST enforce domain
separation between those hash functions. This requirement applies
both in the case of multiple hashes to the same curve and in the case
of multiple hashes to different curves. (This is because the
hash_to_base primitive (Section 5) requires domain separation to
guarantee independent outputs.)
Domain separation is enforced with a domain separation tag (DST),
which is an octet string. Care is required when selecting and using
a domain separation tag. The following requirements apply:
1. Tags MUST be supplied as the DST parameter to hash_to_base, as
described in Section 5.
2. Tags MUST begin with a fixed protocol identification string.
This identification string should be unique to the protocol.
3. Tags SHOULD include a protocol version number.
4. For protocols that define multiple ciphersuites, each
ciphersuite's tag MUST be different. For this purpose, it is
RECOMMENDED to include a ciphersuite identifier in each tag.
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5. For protocols that use multiple encodings, either to the same
curve or to different curves, each encoding MUST use a different
tag. For this purpose, it is RECOMMENDED to include the
encoding's Suite ID (Section 8) in the domain separation tag.
For independent encodings based on the same suite, each tag
should also include a distinct identifier, e.g., "ENC1" and
"ENC2".
As an example, consider a fictional protocol named Quux that defines
several different ciphersuites. A reasonable choice of tag is "QUUX-
V-CS", where and are two-digit numbers indicating
the version and ciphersuite, 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.
Reasonable choices of tags for the E1 and E2 oracles are "BAZ-V-
CS-E1" and "BAZ-V-CS-E2", respectively, where and
are as described 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 low-level 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 OR-ing 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
is_square(x) := { True, if x^((q - 1) / 2) is 0 or 1 in F;
{ False, otherwise.
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o sqrt(x): The sqrt operation is a multi-valued 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 6.4, any higher-level 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 several
algorithms in Appendix F. Regardless of the method chosen, the
sqrt function should be implemented in a way that resists timing
side channels, i.e., in constant time.
o sgn0(x): This function returns either +1 or -1 indicating the
"sign" of x, where sgn0(x) == -1 just when x is "negative". In
other words, this function always considers 0 to be positive.
This function may be implemented in multiple ways; Section 4.1
defines two variants. Throughout the document, sgn0 is used
generically to mean either of these variants. Each suite in
Section 8 specifies the sgn0 variant to be used.
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 non-negative integer as described in
[RFC8017].
o a || b: denotes the concatenation of bit strings a and b.
4.1. sgn0 variants
This section defines two ways of determining the "sign" of an element
of F. The variant that should be used is a matter of convention.
Other sgn0 variants are possible, but the two given below cover
commonly used notions of sign.
It is RECOMMENDED to select the variant that matches the point
decompression method of the target curve. In particular, since point
decompression requires computing a square root and then choosing the
sign of the resulting point, all decompression methods specify,
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implicitly or explicitly, a method for determining the sign of an
element of F. It is convenient for hash-to-curve and decompression
to agree on a notion of sign, since this may permit simpler
implementations.
See Section 2.1 for a discussion of representing elements of field
extensions as vectors; this representation is used in both of the
sgn0 variants below.
Note that any valid sgn0 function for field extensions must iterate
over the entire vector representation of the input element. To see
why, imagine a function sgn0* that ignores the final entry in its
input vector, and consider a field element x = (0, x_2). Since sgn0*
ignores x_2, sgn0*(x) == sgn0*(-x), which is incorrect when x_2 != 0.
The same argument applies to all entries of any x, establishing the
claim.
4.1.1. Big endian variant
The following sgn0 variant is defined such that sgn0_be(x) = -1 just
when the big-endian encoding of x is lexically greater than the
encoding of -x.
This variant SHOULD be used when points on the target elliptic curve
are serialized using the SORT compression method given in IEEE
1363a-2004 [p1363a], Section 5.5.6.1.2, and other similar methods.
sgn0_be(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
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
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4.1.2. Little endian variant
The following sgn0 variant is defined such that sgn0_le(x) = -1 just
when x != 0 and the parity of the least significant nonzero entry of
the vector representation of x is 1.
This variant SHOULD be used when points on the target elliptic curve
are serialized using any of the following methods:
o the LSB compression method given in IEEE 1363a-2004 [p1363a],
Section 5.5.6.1.1,
o the method given in [SEC1] Section 2.3.3, or
o the method given in ANSI X9.62-1998 [x9.62], Section 4.2.1.
This variant is also compatible with the compression method specified
for the Ed25519 and Ed448 elliptic curves [RFC8032].
sgn0_le(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
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 (1, 2, ..., m):
3. sign_i = CMOV(1, -1, x_i mod 2 == 1)
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
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.
Implementors MUST NOT use rejection sampling to generate a uniformly
random element of F. The reason is that these procedures are
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difficult to implement in constant time, and later well-meaning
"optimizations" may silently render an implementation non-constant-
time.
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
level in bits. (This is needed for a birthday bound of
approximately 2^(-k).)
2. H is modeled as a random oracle, so care should be taken when
instantiating it. Hash functions in the SHA-2 [FIPS180-4] and
SHA-3 [FIPS202] families are typical and RECOMMENDED choices.
For example, for 128-bit 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 is modeled as a random oracle. 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 256-bit integer) modulo p
is slightly more likely to be in [0, 37] 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 k-bit security. To obtain such an integer, HKDF
[RFC5869] is used to expand the input msg to a L-byte 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 255-bit prime
and k = 128-bit security, L = ceil((255 + 128) / 8) = 48 bytes.
Finally, hash_to_base appends one zero byte to msg in the invocation
of HKDF-Extract. This ensures that the use of HKDF in hash_to_base
is indifferentiable from a random oracle (see [LBB19], Lemma 8 and
[DRST12], Theorems 4.3 and 4.4). (In particular, this approach works
because it ensures that the final byte of each HMAC invocation in
HKDF-Extract and HKDF-Expand is distinct.)
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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 REQUIRED method for applying domain separation.
Section 5.3 details the hash_to_base procedure.
5.2. Performance considerations
The hash_to_base function uses HKDF-Extract to combine the input msg
and domain separation tag DST into a short digest, which is then
passed to HKDF-Expand [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.
A related issue is that the random oracle construction described in
Section 3 requires evaluating two independent hash functions H0 and
H1 on msg. One 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 non-black-box 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 HKDF-Expand. This means that two
invocations of hash_to_base on the same msg with different ctr values
both start with identical invocations of HKDF-Extract. This is an
improvement because it allows sharing one evaluation of HKDF-Extract
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.
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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.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
- L = ceil((ceil(log2(p)) + k) / 8), where k is the security
parameter of the cryptosystem (e.g., k = 128).
- HKDF-Extract and HKDF-Expand 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. msg_prime = HKDF-Extract(DST, msg || I2OSP(0, 1))
2. info_pfx = "H2C" || I2OSP(ctr, 1) // "H2C" is a 3-byte ASCII string
3. for i in (1, ..., m):
4. info = info_pfx || I2OSP(i, 1)
5. t = HKDF-Expand(msg_prime, info, L)
6. e_i = OS2IP(t) mod p
7. u = (e_1, ..., e_m)
8. return u
5.4. Alternative hash_to_base functions
The hash_to_base function is suitable for use with a wide range of
hash functions, including SHA-2 [FIPS180-4], SHA-3 [FIPS202], BLAKE2
[RFC7693], and others. In some cases, however, implementors may wish
to replace the HKDF-based function defined in this section with one
built on a different pseudorandom function. This section briefly
describes the REQUIRED way of doing so.
The security considerations of Section 5.1 continue to apply. In
particular, an alternative hash_to_base function:
o MUST give collision resistance commensurate with the security
level of the target elliptic curve.
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o MUST be built on a pseudorandom function that is designed for use
in applications requiring cryptographic randomness.
o MUST NOT use rejection sampling.
o MUST output an element of F whose statistical distance from
uniform is commensurate with the security level of the target
elliptic curve. It is RECOMMENDED to follow the guidelines for
controlling bias in Section 5.1.
o MUST give independent output values for distinct (msg, ctr)
inputs.
o MUST support domain separation via a supplied domain separation
tag (DST). Care is required when implementing domain separation:
this document assumes that instantiating hash_to_base with
distinct DSTs yields independent hash functions.
The efficiency considerations of Section 5.2 should also be followed.
In particular, it SHOULD be possible to hash one msg with multiple
ctr values without requiring multiple passes over msg.
Finally, the Suite ID value MUST be modified to indicate that an
alternative hash_to_base function is being used. Section 8.2 gives
details.
6. Deterministic Mappings
The mappings in this section are suitable for constructing either
nonuniform or random oracle encodings using the constructions of
Section 3. Certain mappings restrict the form of the curve or its
parameters. For each mapping presented, this document lists the
relevant restrictions.
Note that mappings in this section are not interchangeable: different
mappings will almost certainly output different points when evaluated
on the same input.
6.1. Choosing a mapping function
This section gives brief guidelines on choosing a mapping function
for a given elliptic curve. Note that the suites given in Section 8
are recommended mappings for the respective curves.
If the target elliptic curve is a supersingular curve supported by
either the Boneh-Franklin method (Section 6.9.1) or the Elligator 2
method for A == 0 (Section 6.9.2), that mapping is the recommended
one.
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Otherwise, if the target elliptic curve is a Montgomery curve
(Section 6.7), the Elligator 2 method (Section 6.7.1) is recommended.
Similarly, if the target elliptic curve is a twisted Edwards curve
(Section 6.8), the twisted Edwards Elligator 2 method (Section 6.8.2)
is recommended.
The remaining cases are Weierstrass curves. For curves supported by
the Simplified SWU method (Section 6.6.2), that mapping is the
recommended one. Otherwise, the Simplified SWU method for AB == 0
(Section 6.6.3) is recommended if the goal is best performance, while
the Shallue-van de Woestijne method (Section 6.6.1) is recommended if
the goal is simplicity of implementation. (The reason for this
distinction is that the Simplified SWU method for AB == 0 requires
implementing an isogeny map in addition to the mapping function,
while the Shallue-van de Woestijne method does not.)
The Shallue-van de Woestijne method (Section 6.6.1) works with any
curve, and may be used in cases where a generic mapping is required.
Note, however, that this mapping is almost always more
computationally expensive than the curve-specific recommendations
above.
6.2. 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.
6.3. 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.
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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.
6.4. 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.
To resolve this ambiguity, the mappings in this section specify the
sign of the y-coordinate 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 square-root
implementations.
6.5. 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 constant-time 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.
6.6. 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.
6.6.1. Shallue-van de Woestijne Method
Shallue and van de Woestijne [SW06] describe a mapping that applies
to essentially any elliptic curve. (Note, however, that this mapping
is more expensive to evaluate than the other mappings in this
document.)
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The parameterization given below is for Weierstrass curves; its
derivation is detailed in [W19]. This parameterization also works
for Montgomery (Section 6.7) and twisted Edwards (Section 6.8) curves
via the rational maps given in Appendix B: first evaluate the
Shallue-van de Woestijne mapping to an equivalent Weierstrass curve,
then map that point to the target Montgomery or twisted Edwards curve
using the corresponding rational map.
Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B over F =
GF(p^m) where p > 5 and odd.
Constants:
o A and B, the parameter of the Weierstrass curve.
o Z, an element of F meeting the below criteria. Appendix E.1 gives
a Sage [SAGE] script that outputs the RECOMMENDED Z.
1. g(Z) != 0 in F.
2. -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.
3. -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.
4. At least one of g(Z) and g(-Z / 2) is square in F.
Sign of y: Inputs u and -u give the same x-coordinate. Thus, we set
sgn0(y) == sgn0(u).
Exceptions: The exceptional cases for u occur when (1 + u^2 * g(Z)) *
(1 - u^2 * g(Z)) == 0. The restrictions on Z given above ensure that
implementations that use inv0 to invert this product are exception
free.
Operations:
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1. t1 = u^2 * g(Z)
2. t2 = 1 + t1
3. t1 = 1 - t1
4. t3 = inv0(t1 * t2)
5. t4 = u * t1 * t3 * sqrt(-g(Z) * (3 * Z^2 + 4 * A))
6. x1 = -Z / 2 - t4
7. x2 = -Z / 2 + t4
8. t5 = 2 * t2^2 * t3 * sqrt(-g(Z) / (3 * Z^2 + 4 * A))
9. x3 = Z + t5^2
10. If is_square(g(x1)), set x = x1 and y = sqrt(g(x1))
11. Else If is_square(g(x2)), set x = x2 and y = sqrt(g(x2))
12. Else set x = x3 and y = sqrt(g(x3))
13. If sgn0(u) != sgn0(y), set y = -y
14. return (x, y)
6.6.1.1. Implementation
The following procedure implements the Shallue and van de Woestijne
method in a straight-line fashion.
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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 = -Z / 2
3. c3 = sqrt(-g(Z) * (3 * Z^2 + 4 * A)) // sgn0(c3) MUST equal 1
4. c4 = -4 * g(Z) / (3 * Z^2 + 4 * A)
Steps:
1. t1 = u^2
2. t1 = t1 * c1
3. t2 = 1 + t1
4. t1 = 1 - t1
5. t3 = t1 * t2
6. t3 = inv0(t3)
7. t4 = u * t1
8. t4 = t4 * t3
9. t4 = t4 * c3
10. x1 = c2 - t4
11. gx1 = x1^2
12. gx1 = gx1 + A
13. gx1 = gx1 * x1
14. gx1 = gx1 + B
15. e1 = is_square(gx1)
16. x2 = c2 + t4
17. gx2 = x2^2
18. gx2 = gx2 + A
19. gx2 = gx2 * x2
20. gx2 = gx2 + B
21. e2 = is_square(gx2) AND NOT e1 // Avoid short-circuit logic ops
22. x3 = t2^2
23. x3 = x3 * t3
24. x3 = x3^2
25. x3 = x3 * c4
26. x3 = x3 + Z
27. x = CMOV(x3, x1, e1) // x = x1 if gx1 is square, else x = x3
28. x = CMOV(x, x2, e2) // x = x2 if gx2 is square and gx1 is not
29. gx = x^2
30. gx = gx + A
31. gx = gx * x
32. gx = gx + B
33. y = sqrt(gx)
34. e3 = sgn0(u) == sgn0(y)
35. y = CMOV(-y, y, e3) // Select correct sign of y
36. return (x, y)
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6.6.2. Simplified Shallue-van de Woestijne-Ulas Method
The function map_to_curve_simple_swu(u) implements a simplification
of the Shallue-van de Woestijne-Ulas mapping [U07] described by Brier
et al. [BCIMRT10], which they call the "simplified SWU" map. Wahby
and Boneh [WB19] generalize this mapping to curves over fields of odd
characteristic p > 3.
Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B over F =
GF(p^m) where p > 5 and odd, A != 0, and B != 0.
Constants:
o A and B, the parameters of the Weierstrass curve.
o Z, an element of F meeting the below criteria. Appendix E.2 gives
a Sage [SAGE] script that outputs the RECOMMENDED Z. The criteria
are:
1. Z is non-square in F,
2. Z != -1 in F,
3. the polynomial g(x) - Z is irreducible over F, and
4. g(B / (Z * A)) is square in F.
Sign of y: Inputs u and -u give the same x-coordinate. 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)
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6.6.2.1. Implementation
The following procedure implements the simplified SWU mapping in a
straight-line fashion. Appendix D gives an optimized straight-line
procedure for P-256 [FIPS186-4]. For more information on optimizing
this mapping, see [WB19] Section 4 or the example code found at
[hash2curve-repo].
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)
6.6.3. Simplified SWU for AB == 0
Wahby and Boneh [WB19] show how to adapt the simplified SWU mapping
to Weierstrass curves having A == 0 or B == 0, which the mapping of
Section 6.6.2 does not support. (The case A == B == 0 is excluded
because y^2 = x^3 is not an elliptic curve.)
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This method applies to curves like secp256k1 [SEC2] and to pairing-
friendly curves in the Barreto-Lynn-Scott [BLS03], Barreto-Naehrig
[BN05], and other families.
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]; for details, see [WB19],
Appendix A.) 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 6.6.2 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_simple_swu. No further sign adjustments are necessary.
Exceptions: map_to_curve_simple_swu handles its exceptional cases.
Exceptional cases of iso_map MUST 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)
See [hash2curve-repo] or [WB19], Section 4.3 for details on
implementing the isogeny map.
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6.7. Mappings for Montgomery curves
The mapping defined in Section 6.7.1 implements Elligator 2 [BHKL13]
for curves defined by the Weierstrass equation y^2 = x^3 + A * x^2 +
B * x.
Such a Weierstrass curve is related to the Montgomery curve B' * t^2
= s^3 + A' * s^2 + s by the following change of variables:
o A = A' / B'
o B = 1 / B'^2
o x = s / B'
o y = t / 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 (s, t) on the original Montgomery curve by computing
o s = B' * x
o t = 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].
6.7.1. 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 non-zero and non-square in F.
Constants:
o A and B, the parameters of the elliptic curve.
o Z, a non-square element of F. Appendix E.3 gives a Sage [SAGE]
script that outputs the RECOMMENDED Z.
Sign of y: Inputs u and -u give the same x-coordinate. 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).
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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 set x = x2 and y = sqrt(gx2)
8. If sgn0(u) != sgn0(y), set y = -y
9. return (x, y)
6.7.1.1. Implementation
The following procedure implements Elligator 2 in a straight-line
fashion. Appendix D gives optimized straight-line 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. e1 = t1 == -1 // exceptional case: Z * u^2 == -1
4. t1 = CMOV(t1, 0, e1) // if t1 == -1, set t1 = 0
5. x1 = t1 + 1
6. x1 = inv0(x1)
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)
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6.8. Mappings for Twisted Edwards curves
Twisted Edwards curves (a class of curves that includes Edwards
curves) are given by the equation a * v^2 + w^2 = 1 + d * v^2 * w^2,
with a != 0, d != 0, and a != d [BBJLP08].
These curves are closely related to Montgomery curves (Section 6.7):
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.
6.8.1. 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 4-isogeny 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
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* v^2 + w^2 = 1 + d * v^2 * w^2, first compute A and B, the
parameters of the equivalent Weierstrass curve given by y^2 = x^3 + A
* x^2 + B * x, as follows:
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 6.7.1. The rational map from
the point (x, y) on this Weierstrass curve to the point (v, w) on the
twisted Edwards curve is given by
o B' = 1 / sqrt(B) = 4 / (a - d)
o v = x / y
o w = (B' * x - 1) / (B' * x + 1)
For completeness, we give the inverse map in Appendix B.1. Note that
the inverse map is not used when hashing to a twisted Edwards curve.
Rational maps may be undefined on certain inputs, e.g., when the
denominator of one of the rational functions is zero. In the map
described above, the exceptional cases are y == 0 or B' * x == -1.
Implementations MUST detect exceptional cases and return the value
(v, w) = (0, 1), which is a valid point on all twisted Edwards curves
given by the equation above.
The following straight-line implementation of the above rational map
handles the exceptional cases. Implementations of other rational
maps (e.g., the ones give in [RFC7748]) are analogous.
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rational_map(x, y)
Input: (x, y), a point on the curve y^2 = x^3 + A * x^2 + B * x.
Output: (v, w), a point on an equivalent twisted Edwards curve.
1. t1 = x * B'
2. t2 = t1 + 1
3. t3 = y * t2
4. t3 = inv0(t3)
5. v = t2 * t3
6. v = v * x
7. w = t1 - 1
8. w = w * y
9. w = w * t3
10. e = w == 0
11. w = CMOV(w, 1, e)
12. return (v, w)
6.8.2. Elligator 2 Method
Preconditions: A twisted Edwards curve E and an equivalent curve M
meeting the requirements in Section 6.8.1.
Helper functions:
o map_to_curve_elligator2 is the mapping of Section 6.7.1 to the
curve M.
o rational_map is a function that takes a point (x, y) on M and
returns a point (v, w) on E, as defined in Section 6.8.1.
Sign of y (and w): 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 6.7.1. The exceptions for the rational map are as given
in Section 6.8.1. No other exceptions are possible.
The following procedure implements the Elligator 2 mapping for a
twisted Edwards curve. (Note that the output point is denoted (v, w)
because it is a point on the target twisted Edwards curve.)
map_to_curve_elligator2_edwards(u)
Input: u, an element of F.
Output: (v, w), a point on E.
1. (x, y) = map_to_curve_elligator2(u) // (x, y) is on M
2. (v, w) = rational_map(x, y) // (v, w) is on E
3. return (v, w)
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6.9. Mappings for Supersingular curves
6.9.1. Boneh-Franklin Method
The function map_to_curve_bf(u) implements the Boneh-Franklin 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:
1. w = (2 * q - 1) / 3 // Integer arithmetic
2. x = (u^2 - B)^w
3. y = u
4. return (x, y)
6.9.1.1. Implementation
The following procedure implements the Boneh-Franklin's algorithm in
a straight-line 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)
6.9.2. 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).
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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 x-coordinate. Thus, we set
sgn0(y) == sgn0(u).
Exceptions: none.
Operations:
1. x1 = u
2. gx1 = x1^3 + B * x1
3. x2 = -x1
4. gx2 = -gx1
5. If is_square(gx1), set x = x1 and y = sqrt(gx1)
6. Else set x = x2 and y = sqrt(gx2)
7. If sgn0(u) != sgn0(y), set y = -y.
8. return (x, y)
6.9.2.1. Implementation
The following procedure implements the Elligator 2 mapping for A == 0
in a straight-line 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)
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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 P-256, P-384, and P-521 [FIPS186-4].
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 pairing-friendly curves having subgroup G2 over an
extension field, Scott et al. [SBCDK09] describe a method for
fast cofactor clearing that exploits an efficiently-computable
endomorphism. Fuentes-Castaneda et al. [FKR11] propose an
alternative method that is sometimes more efficient. Budroni and
Pintore [BP18] give concrete instantiations of these methods for
Barreto-Lynn-Scott pairing-friendly 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
not generally give the same result as the fast method, and SHOULD NOT
be used.
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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. The ID
also indicates whether a suite is a random oracle or nonuniform
encoding (Section 2.2.3, Section 3). Section 8.2 discusses the
naming conventions for suite IDs.
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 sgn0, one of the variants specified in Section 4.1.
o H, the hash function used by hash_to_base (Section 5.1).
o L, the length of HKDF-Expand output in hash_to_base (Section 5.1).
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
MUST be specified when applicable.
All applications MUST choose a domain separation tag (DST) for use
with hash_to_base (Section 5), in accordance with the guidelines of
Section 3.1. In addition, applications whose security requires a
random oracle MUST use a suite specifying hash_to_curve (Section 3);
see Section 8.2.
The below table lists the curves for which suites are defined and the
subsection that gives the corresponding parameters.
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+---------------------------+-------------+
| E | Section |
+---------------------------+-------------+
| NIST P-256 | Section 8.3 |
| | |
| NIST P-384 | Section 8.4 |
| | |
| NIST P-521 | Section 8.5 |
| | |
| curve25519 / edwards25519 | Section 8.6 |
| | |
| curve448 / edwards448 | Section 8.7 |
| | |
| secp256k1 | Section 8.8 |
| | |
| BLS12-381 | Section 8.9 |
+---------------------------+-------------+
8.1. Defining a new hash-to-curve suite
The RECOMMENDED way to define a new hash-to-curve suite is:
1. E, F, p, and m are determined by the elliptic curve and the
field.
2. Choose a sgn0 variant following the guidelines in Section 4.1.
3. Choose a hash function H meeting the requirements in Section 5.1,
and compute L as described in that section.
4. Choose a mapping following the guidelines in Section 6.1, and
select any required parameters for that mapping.
5. Choose h_eff to be either the cofactor of E or, if a fast
cofactor clearing method is to be used, a value appropriate to
that method as discussed in Section 7.
6. Construct a Suite ID following the guidelines in Section 8.2.
When hashing to an elliptic curve not listed in this section,
corresponding hash-to-curve suites SHOULD be specified as described
in this section.
8.2. Suite ID naming conventions
Suite IDs MUST be constructed as follows:
CURVE_ID || "-" || HASH_ID || "-" || MAP_ID || "-" || ENC_VAR || "-"
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The fields CURVE_ID, HASH_ID, MAP_ID, and ENC_VAR are ASCII-encoded
strings of at most 64 characters each. Fields can contain only ASCII
characters between 0x21 and 0x7E (inclusive) other than hyphen and
underscore (i.e., 0x2d, and 0x5f). As indicated above, each field
(including the last) is followed by a hyphen ("-", ASCII 0x2d); this
helps to ensure that Suite IDs are prefix free.
Fields MUST be chosen as follows:
o CURVE_ID: a human-readable representation of the target elliptic
curve.
o HASH_ID: a human-readable representation of the hash function used
in hash_to_base (Section 5).
If a suite uses an alternative hash_to_base function
(Section 5.4), a short descriptive name MUST be chosen for that
function using only the allowed characters listed above. That
name MUST be appended to the HASH_ID field, separated by a colon.
For example, a hash_to_base function based on KMAC128 [SP.800-185]
might use the short name "h2b/kmac128", and a reasonable value for
the HASH_ID field would be "SHA3:h2b/kmac128".
o MAP_ID: a human-readable representation of the map_to_curve
function (Section 6).
o ENC_VAR: a string indicating the encoding type and other
information. The first two characters of this string indicate
whether the suite represents a hash_to_curve or an encode_to_curve
operation (Section 3), as follows:
* If ENC_VAR begins with "RO", the suite uses hash_to_curve.
* If ENC_VAR begins with "NU", the suite uses encode_to_curve.
* ENC_VAR MUST NOT begin with any other string.
ENC_VAR MAY also be used to encode other information used to
identify variants, for example, a version number. The RECOMMENDED
way to do so is to add one or more subfields separated by colons.
For example, "RO:V02" is an appropriate ENC_VAR value for the
second version of a random-oracle suite, while
"RO:V02:FOO01:BAR17" might be used to indicate a variant of that
suite.
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8.3. Suites for NIST P-256
This section defines ciphersuites for the NIST P-256 elliptic curve
[FIPS186-4].
The suites P256-SHA256-SSWU-RO- and P256-SHA256-SSWU-NU- share the
following parameters, in addition to the common parameters below.
o f: Simplified SWU method, Section 6.6.2
o Z: -10
The suites P256-SHA256-SVDW-RO- and P256-SHA256-SVDW-NU- share the
following parameters, in addition to the common parameters below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: -3
The common parameters for the above suites are:
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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-256
o L: 48
o h_eff: 1
An optimized example implementation of the Simplified SWU mapping to
P-256 is given in Appendix D.2.
8.4. Suites for NIST P-384
This section defines ciphersuites for the NIST P-384 elliptic curve
[FIPS186-4].
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The suites P384-SHA512-SSWU-RO- and P384-SHA512-SSWU-NU- share the
following parameters, in addition to the common parameters below.
o f: Simplified SWU method, Section 6.6.2
o Z: -12
The suites P384-SHA512-SVDW-RO- and P384-SHA512-SVDW-NU- share the
following parameters, in addition to the common parameters below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: -1
The common parameters for the above suites are:
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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-512
o L: 72
o h_eff: 1
An optimized example implementation of the Simplified SWU mapping to
P-384 is given in Appendix D.2.
8.5. Suites for NIST P-521
This section defines ciphersuites for the NIST P-521 elliptic curve
[FIPS186-4].
The suites P521-SHA512-SSWU-RO- and P521-SHA512-SSWU-NU- share the
following parameters, in addition to the common parameters below.
o f: Simplified SWU method, Section 6.6.2
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o Z: -4
The suites P521-SHA512-SVDW-RO- and P521-SHA512-SVDW-NU- share the
following parameters, in addition to the common parameters below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: 1
The common parameters for the above suites are:
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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-512
o L: 96
o h_eff: 1
An optimized example implementation of the Simplified SWU mapping to
P-521 is given in Appendix D.2.
8.6. Suites for curve25519 and edwards25519
This section defines ciphersuites for curve25519 and edwards25519
[RFC7748].
The suites curve25519-SHA256-ELL2-RO- and curve25519-SHA256-ELL2-NU-
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
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o f: Elligator 2 method, Section 6.7.1
The suites edwards25519-SHA256-EDELL2-RO- and
edwards25519-SHA256-EDELL2-NU- 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 6.8.2
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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-256
o L: 48
o Z: 2
o h_eff: 8
Optimized example implementations of the above mappings are given in
Appendix D.3 and Appendix D.4.
8.7. Suites for curve448 and edwards448
This section defines ciphersuites for curve448 and edwards448
[RFC7748].
The suites curve448-SHA512-ELL2-RO- and curve448-SHA512-ELL2-NU-
share the following parameters, in addition to the common parameters
below.
o E: B * y^2 = x^3 + A * x^2 + x, where
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* A = 156326
* B = 1
o f: Elligator 2 method, Section 6.7.1
The suites edwards448-SHA512-EDELL2-RO- and
edwards448-SHA512-EDELL2-NU- 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 6.8.2
o M: curve448, defined in [RFC7748], Section 4.2
o rational_map: the 4-isogeny 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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-512
o L: 84
o Z: -1
o h_eff: 4
Optimized example implementations of the above mappings are given in
Appendix D.5 and Appendix D.6.
8.8. Suites for secp256k1
This section defines ciphersuites for the secp256k1 elliptic curve
[SEC2].
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The suites secp256k1-SHA256-SSWU-RO- and secp256k1-SHA256-SSWU-NU-
share the following parameters, in addition to the common parameters
below.
o f: Simplified SWU for AB == 0, Section 6.6.3
o Z: -11
o E': y'^2 = x'^3 + A' * x' + B', where
* A': 0x3f8731abdd661adca08a5558f0f5d272e953d363cb6f0e5d405447c01
a444533
* B': 1771
o iso_map: the 3-isogeny map from E' to E given in Appendix C.1
The suites secp256k1-SHA256-SVDW-RO- and secp256k1-SHA256-SVDW-NU-
share the following parameters, in addition to the common parameters
below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: 1
The common parameters for all of the above suites are:
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 sgn0: sgn0_le (Section 4.1.2)
o H: SHA-256
o L: 48
o h_eff: 1
An optimized example implementation of the Simplified SWU mapping to
the curve E' isogenous to secp256k1 is given in Appendix D.2.
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8.9. Suites for BLS12-381
This section defines ciphersuites for groups G1 and G2 of the
BLS12-381 elliptic curve [draft-yonezawa-pfc-01].
8.9.1. BLS12-381 G1
The suites BLS12381G1-SHA256-SSWU-RO- and BLS12381G1-SHA256-SSWU-NU-
share the following parameters, in addition to the common parameters
below.
o f: Simplified SWU for AB == 0, Section 6.6.3
o Z: 11
o E': y'^2 = x'^3 + A' * x' + B', where
* A' = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aef
d881ac98936f8da0e0f97f5cf428082d584c1d
* B' = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14f
cef35ef55a23215a316ceaa5d1cc48e98e172be0
o iso_map: the 11-isogeny map from E' to E given in Appendix C.2
The suites BLS12381G1-SHA256-SVDW-RO- and BLS12381G1-SHA256-SVDW-NU-
share the following parameters, in addition to the common parameters
below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: -3
The common parameters for the above suites are:
o E: y^2 = x^3 + 4
o p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f
6241eabfffeb153ffffb9feffffffffaaab
o m: 1
o sgn0: sgn0_be (Section 4.1.1)
o H: SHA-256
o L: 64
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o h_eff: 0xd201000000010001
Note that this h_eff value is chosen for compatibility with the fast
cofactor clearing method described by Scott ([WB19] Section 5).
An optimized example implementation of the Simplified SWU mapping to
the curve E' isogenous to BLS12-381 G1 is given in Appendix D.2.
8.9.2. BLS12-381 G2
Group G2 of BLS12-381 is defined over a field F = GF(p^m) defined as:
o p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f
6241eabfffeb153ffffb9feffffffffaaab
o m: 2
o (1, I) is the basis for F, where I^2 + 1 == 0 in F
The suites BLS12381G2-SHA256-SSWU-RO- and BLS12381G2-SHA256-SSWU-NU-
share the following parameters, in addition to the common parameters
below.
o f: Simplified SWU for AB == 0, Section 6.6.3
o Z: -(2 + 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.3
The suites BLS12381G2-SHA256-SVDW-RO- and BLS12381G2-SHA256-SVDW-NU-
share the following parameters, in addition to the common parameters
below.
o f: Shallue-van de Woestijne method, Section 6.6.1
o Z: I
The common parameters for the above suites are:
o E: y^2 = x^3 + 4 * (1 + I)
o p, m, F: defined above
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o sgn0: sgn0_be (Section 4.1.1)
o H: SHA-256
o L: 64
o h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff0315
08ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689
f6a359894c0adebbf6b4e8020005aaa95551
Note that this h_eff value is chosen for compatibility with the fast
cofactor clearing method described by Budroni and Pintore ([BP18],
Section 4.1).
9. IANA Considerations
This document has no IANA actions.
10. Security Considerations
When constant-time 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.
When the hash_to_curve function (Section 3) is instantiated with
hash_to_base (Section 5), the resulting function is indifferentiable
from a random oracle. In most cases such a function can be safely
used in protocols whose security analysis assumes a random oracle
that outputs points on an elliptic curve. As Ristenpart et al.
discuss in [RSS11], however, not all security proofs that rely on
random oracles continue to hold when those oracles are replaced by
indifferentiable functionalities. This limitation should be
considered when analyzing the security of protocols relying on the
hash_to_curve function.
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When hashing passwords using any function described in this document,
an adversary who learns the output of the hash function (or
potentially any intermediate value, e.g., the output of hash_to_base)
may be able to carry out a dictionary attack. To mitigate such
attacks, it is recommended to first execute a more costly key
derivation function (e.g., PBKDF2 [RFC2898] or scrypt [RFC7914]) on
the password, then hash the output of that function to the target
elliptic curve. For collision resistance, the hash underlying the
key derivation function should be chosen according to the guidelines
listed in Section 5.1.
11. Acknowledgements
The authors would like to thank Adam Langley for his detailed writeup
of Elligator 2 with Curve25519 [L13]; Christopher Patton and Benjamin
Lipp for educational discussions; and Sean Devlin, Justin Drake, Dan
Harkins, Thomas Icart, Leonid Reyzin, Michael Scott, and Mathy
Vanhoef for helpful feedback.
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
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,
.
[RFC2898] Kaliski, B., "PKCS #5: Password-Based Cryptography
Specification Version 2.0", RFC 2898,
DOI 10.17487/RFC2898, September 2000,
.
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010,
.
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[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, .
[RFC7914] Percival, C. and S. Josefsson, "The scrypt Password-Based
Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
August 2016, .
[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,
.
[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
13.2. Informative References
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Zapalowicz, "Binary Elligator squared", In Selected Areas
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and M. Tibouchi, "Efficient Indifferentiable Hashing into
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the Weil pairing", In Advances in Cryptology - CRYPTO
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"Elligator: elliptic-curve points indistinguishable from
uniform random strings", In Proceedings of the 2013 ACM
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November 2013, .
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Winnerlein, "BLAKE2X", December 2016,
.
[BLMP19] Bernstein, D., Lange, T., Martindale, C., and L. Panny,
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evaluation of isogenies", In Advances in Cryptology -
EUROCRYPT 2019, DOI 10.1007/978-3-030-17656-3, 2019,
.
[BLS01] Boneh, D., Lynn, B., and H. Shacham, "Short signatures
from the Weil pairing", In Journal of Cryptology, vol 17,
pages 297-319, DOI 10.1007/s00145-004-0314-9, July 2004,
.
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[BMP00] Boyko, V., MacKenzie, P., and S. Patel, "Provably secure
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Curves of Prime Order", In Selected Areas in Cryptography
2005, pages 319-331, DOI 10.1007/11693383_22, 2006,
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[BP18] Budroni, A. and F. Pintore, "Hashing to G2 on BLS pairing-
friendly curves", In ACM Communications in Computer
Algebra, pages 63-66, DOI 10.1145/3313880.3313884,
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Theory", publisher Springer-Verlag, ISBN 9783642081422,
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[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
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.
[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 266-281, DOI 10.1016/j.jsc.2011.11.003, 2012,
.
[draft-yonezawa-pfc-01]
Yonezawa, S., Chikara, S., Kobayashi, T., and T. Saito,
"Pairing-friendly Curves", March 2019,
.
[DRST12] Dodis, Y., Ristenpart, T., Steinberger, J., and S.
Tessaro, "To hash or not to hash again?
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[F11] Farashahi, R., "Hashing into Hessian curves",
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elliptic and hyperelliptic curves", In Math. Comp. vol 82,
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[FIPS180-4]
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.
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[FIPS186-4]
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2013, .
[FIPS202] National Institute of Standards and Technology (NIST),
"SHA-3 Standard: Permutation-Based Hash and Extendable-
Output Functions", August 2015,
.
[FJT13] Fouque, P., Joux, A., and M. Tibouchi, "Injective
encodings to elliptic curves", In ACISP 2013,
pages 203-218, DOI 10.1007/978-3-642-39059-3_14, 2013,
.
[FKR11] Fuentes-Castaneda, L., Knapp, E., and F. Rodriguez-
Henriquez, "Fast Hashing to G2 on Pairing-Friendly
Curves", In Selected Areas in Cryptography, pages 412-430,
DOI 10.1007/978-3-642-28496-0_25, 2011,
.
[FSV09] Farashahi, R., Shparlinski, I., and J. Voloch, "On hashing
into elliptic curves", In Journal of Mathematical
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[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 81-91, DOI 10.1007/978-3-642-14712-8_5, 2010,
.
[FT12] Fouque, P. and M. Tibouchi, "Indifferentiable Hashing to
Barreto-Naehrig Curves", In Progress in Cryptology -
LATINCRYPT 2012, pages 1-7,
DOI 10.1007/978-3-642-33481-8_1, 2012,
.
[hash2curve-repo]
"Hashing to Elliptic Curves - GitHub repository", 2019,
.
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[Icart09] Icart, T., "How to Hash into Elliptic Curves", In Advances
in Cryptology - CRYPTO 2009, pages 303-316,
DOI 10.1007/978-3-642-03356-8_18, 2009,
.
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exchange", In SIGCOMM Computer Communication Review, vol
26 issue 5, pages 5-26, DOI 10.1145/242896.242897, 1996,
.
[jubjub-fq]
"zkcrypto/jubjub - fq.rs", 2019,
.
[KLR10] Kammerer, J., Lercier, R., and G. Renault, "Encoding
points on hyperelliptic curves over finite fields in
deterministic polynomial time", In PAIRING 2010,
pages 278-297, DOI 10.1007/978-3-642-17455-1_18, 2010,
.
[L13] Langley, A., "Implementing Elligator for Curve25519",
2013, .
[LBB19] Lipp, B., Blanchet, B., and K. Bhargavan, "A Mechanised
Proof of the WireGuard Virtual Private Network Protocol",
In INRIA Research Report No. 9269, April 2019,
.
[p1363a] IEEE Computer Society, "IEEE Standard Specifications for
Public-Key Cryptography---Amendment 1: Additional
Techniques", March 2004,
.
[RFC7693] Saarinen, M-J., Ed. and J-P. Aumasson, "The BLAKE2
Cryptographic Hash and Message Authentication Code (MAC)",
RFC 7693, DOI 10.17487/RFC7693, November 2015,
.
[RSS11] Ristenpart, T., Shacham, H., and T. Shrimpton, "Careful
with Composition: Limitations of the Indifferentiability
Framework", In Advances in Cryptology - EUROCRYPT 2011,
pages 487-506, DOI 10.1007/978-3-642-20465-4_27, May 2011,
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[S05] Skalba, M., "Points on elliptic curves over finite
fields", In Acta Arithmetica, vol 117 no 3, pages 293-301,
DOI 10.4064/aa117-3-7, 2005,
.
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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
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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
x-coordinate 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
input string, which means that it is not safe to use in protocols
sensitive to timing side channels.
Schinzel and Skalba [SS04] introduce a 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. Fouque and
Tibouchi [FT12] give a parameterization of this mapping for Barreto-
Naehrig pairing-friendly curves [BN05].
Ulas [U07] describes a simpler version of the Shallue-van de
Woestijne map, and Brier et al. [BCIMRT10] give a further
simplification, which the authors call the "simplified SWU" map.
That simplified map applies only to fields of characteristic p = 3
(mod 4); Wahby and Boneh [WB19] generalize to fields of any
characteristic, and give further optimizations.
Boneh and Franklin give a deterministic algorithm mapping to certain
supersingular curves over fields of characteristic p = 2 (mod 3)
[BF01]. 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
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having a point of order 2. This includes Curve25519 and Curve448,
both of which are CFRG-recommended 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 r, 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.
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
This section gives several useful rational maps.
B.1. Twisted Edwards to Weierstrass and Montgomery curves
The inverse of the rational map specified in Section 6.8.1, i.e., the
map from the point (v, w) on the twisted Edwards curve a * v^2 + w^2
= 1 + d * v^2 * w^2 to the point (x, y) on the Weierstrass curve y^2
= x^3 + A * x^2 + B * x is given by:
o A = (a + d) / 2
o B = (a - d)^2 / 16
o B' = 1 / sqrt(B) = 4 / (a - d)
o x = (1 + w) / (B' * (1 - w))
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o y = (1 + w) / (B' * v * (1 - w))
This map is undefined when w == 1 or v == 0. In this case, return
the point (x, y) = (0, 0).
It may also be useful to map to a Montgomery curve of the form B' *
t^2 = s^3 + A' * s^2 + s. 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 s = (1 + w) / (1 - w)
o t = (1 + w) / (v * (1 - w))
whose inverse is given by:
o v = s / t
o w = (s - 1) / (s + 1)
Composing the mapping immediately above with the mapping from
Montgomery to Weierstrass curves in Appendix B.2 yields a mapping
from twisted Edwards curves to Weierstrass curves of the form
required by the mappings in Section 6.6. This mapping can be used to
apply the Shallue-van de Woestijne method (Section 6.6.1) to twisted
Edwards curves.
B.2. Montgomery to Weierstrass curves
The rational map from the point (s, t) on the Montgomery curve B' *
t^2 = s^3 + A' * s^2 + s to the point (x, y) on the equivalent
Weierstrass curve y^2 = x^3 + C * x + D is given by:
o C = (3 - A'^2) / (3 * B'^2)
o D = (2 * A'^3 - 9 * A') / (27 * B'^3)
o x = (3 * s + A') / (3 * B')
o y = t / B'
The inverse map, from the point (x, y) to the point (s, t), is given
by
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o s = (3 * B' * x - A') / 3
o t = y * B'
This mapping can be used to apply the Shallue-van de Woestijne method
(Section 6.6.1) to Montgomery curves.
Appendix C. Isogeny maps for Suites
This section specifies the isogeny maps for the secp256k1 and
BLS12-381 suites listed in Section 8.
These maps are given in terms of affine coordinates. Wahby and Boneh
([WB19], Section 4.3) show how to evaluate these maps in a projective
coordinate system (Appendix D.1), which avoids modular inversions.
Refer to the draft repository [hash2curve-repo] for a Sage [SAGE]
script that constructs these isogenies.
C.1. 3-isogeny map for secp256k1
This section specifies the isogeny map for the secp256k1 suite listed
in Section 8.8.
The 3-isogeny map from (x', y') on E' to (x, y) on 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_(1,1) * x' +
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_(3,1) * x' +
k_(3,0)
* y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0)
The constants used to compute x_num are as follows:
o k_(1,0) =
0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa8c7
o k_(1,1) =
0x7d3d4c80bc321d5b9f315cea7fd44c5d595d2fc0bf63b92dfff1044f17c6581
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o k_(1,2) =
0x534c328d23f234e6e2a413deca25caece4506144037c40314ecbd0b53d9dd262
o k_(1,3) =
0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa88c
The constants used to compute x_den are as follows:
o k_(2,0) =
0xd35771193d94918a9ca34ccbb7b640dd86cd409542f8487d9fe6b745781eb49b
o k_(2,1) =
0xedadc6f64383dc1df7c4b2d51b54225406d36b641f5e41bbc52a56612a8c6d14
The constants used to compute y_num are as follows:
o k_(3,0) =
0x4bda12f684bda12f684bda12f684bda12f684bda12f684bda12f684b8e38e23c
o k_(3,1) =
0xc75e0c32d5cb7c0fa9d0a54b12a0a6d5647ab046d686da6fdffc90fc201d71a3
o k_(3,2) =
0x29a6194691f91a73715209ef6512e576722830a201be2018a765e85a9ecee931
o k_(3,3) =
0x2f684bda12f684bda12f684bda12f684bda12f684bda12f684bda12f38e38d84
The constants used to compute y_den are as follows:
o k_(4,0) =
0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffff93b
o k_(4,1) =
0x7a06534bb8bdb49fd5e9e6632722c2989467c1bfc8e8d978dfb425d2685c2573
o k_(4,2) =
0x6484aa716545ca2cf3a70c3fa8fe337e0a3d21162f0d6299a7bf8192bfd2a76f
C.2. 11-isogeny map for BLS12-381 G1
The 11-isogeny map from (x', y') on E' to (x, y) on 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_(1,9) * x'^9 +
... + k_(1,0)
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* x_den = x'^10 + k_(2,9) * x'^9 + k_(2,8) * x'^8 + ... + k_(2,0)
o y = y' * y_num / y_den, where
* y_num = k_(3,15) * x'^15 + k_(3,14) * x'^14 + k_(3,13) * x'^13
+ ... + k_(3,0)
* y_den = x'^15 + k_(4,14) * x'^14 + k_(4,13) * x'^13 + ... +
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
o k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c
928c5d1de4fa295f296b74e956d71986a8497e317
o k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314
baf4bb1b7fa3190b2edc0327797f241067be390c9e
o k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af
36003b14866f69b771f8c285decca67df3f1605fb7b
o k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf
1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229
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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
o k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f9063
43eb67ad34d6c56711962fa8bfe097e75a2e41c696
o k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b
91400da7d26d521628b00523b8dfe240c72de1f6
o k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3e
c03251cf9de405aba9ec61deca6355c77b0e5f4cb
o k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd
221351adc2ee7f8dc099040a841b6daecf2e8fedb
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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:
o k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec323
2b5be72e7a07f3688ef60c206d01479253b03663c1
o k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a103
56f453e01f78a4260763529e3532f6102c2e49a03d
o k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5d
d279cd2eca6757cd636f96f891e2538b53dbf67f2
o k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da4
89e726af41727364f2c28297ada8d26d98445f5416
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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
C.3. 3-isogeny map for BLS12-381 G2
The 3-isogeny map from (x', y') on E' to (x, y) on 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_(1,1) * x' +
k_(1,0)
* x_den = x'^2 + k_(2,1) * x' + k_(2,0)
o y = y' * y_num / y_den, where
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* y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' +
k_(3,0)
* y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + 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
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
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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 [hash2curve-repo].
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
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);
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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. Simplified SWU for p = 3 (mod 4)
The following is a straight-line implementation of the Simplified SWU
mapping that applies to any curve over GF(p) for p = 3 (mod 4). This
includes the ciphersuites for NIST curves P-256, P-384, and P-521
[FIPS186-4] given in Section 8. It also includes the curves
isogenous to secp256k1 (Section 8.8) and BLS12-381 G1
(Section 8.9.1).
The implementations for these curves differ only in the constants and
the base field. The constant definitions below are given in terms of
the parameters for the Simplified SWU mapping; for parameter values
for the curves listed above, see Section 8.3 (P-256), Section 8.4
(P-384), Section 8.5 (P-521), Section 8.8 (E' isogenous to
secp256k1), and Section 8.9.1 (E' isogenous to BLS12-381 G1).
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map_to_curve_simple_swu_3mod4(u)
Input: u, an element of F.
Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
point on the target curve.
Constants: defined per curve; see above.
1. c1 = B / 3
2. c2 = (p - 3) / 4 // Integer arithmetic
3. c3 = sqrt(-Z^3)
Steps:
1. t1 = u^2
2. t3 = Z * t1
3. t2 = t3^2
4. xd = t2 + t3
5. x1n = xd + 1
6. x1n = x1n * B
7. xd = -A * xd
8. e1 = xd == 0
9. xd = CMOV(xd, Z * A, e1) // If xd == 0, set xd = Z * A
10. t2 = xd^2
11. gxd = t2 * xd // gxd == xd^3
12. t2 = A * 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 = -10 * u^2 * x1n / xd
24. y2 = y1 * c3 // y2 = y1 * sqrt(-Z^3)
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)
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D.3. curve25519 (Elligator 2)
The following is a straight-line implementation of Elligator 2 for
curve25519 [RFC7748] as specified in Section 8.6.
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), t1 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
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31. e2 = t2 == gx2
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 straight-line implementation of Elligator 2 for
edwards25519 [RFC7748] as specified in Section 8.6. 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) // sgn0(c1) MUST equal 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. t1 = xd * yd
8. e = t1 == 0
9. xn = CMOV(xn, 0, e)
10. xd = CMOV(xd, 1, e)
11. yn = CMOV(yn, 1, e)
12. yd = CMOV(yd, 1, e)
13. return (xn, xd, yn, yd)
D.5. curve448 (Elligator 2)
The following is a straight-line implementation of Elligator 2 for
curve448 [RFC7748] as specified in Section 8.7.
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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. e1 = t1 == 1
3. t1 = CMOV(t1, 0, e1) // If Z * u^2 == -1, set t1 = 0
4. xd = 1 - t1
5. x1n = -156326
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. y2 = CMOV(y2, 0, e1)
21. t2 = y1^2
22. t2 = t2 * gxd
23. e2 = t2 == gx1
24. xn = CMOV(x2n, x1n, e2) // If e2, x = x1, else x = x2
25. y = CMOV(y2, y1, e2) // If e2, y = y1, else y = y2
26. e3 = sgn0(u) == sgn0(y) // Fix sign of y
27. y = CMOV(-y, y, e3)
28. return (xn, xd, y, 1)
D.6. edwards448 (Elligator 2)
The following is a straight-line implementation of Elligator 2 for
edwards448 [RFC7748] as specified in Section 8.7. The subroutine
map_to_curve_elligator2_curve448 is defined in Appendix D.5.
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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. t1 = xEd * yEd
33. e = t1 == 0
34. xEn = CMOV(xEn, 0, e)
35. xEd = CMOV(xEd, 1, e)
36. yEn = CMOV(yEn, 1, e)
37. yEd = CMOV(yEd, 1, e)
38. return (xEn, xEd, yEn, yEd)
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Appendix E. Scripts for parameter generation
This section gives Sage [SAGE] scripts used to generate parameters
for the mappings of Section 6.
E.1. Finding Z for the Shallue and van de Woestijne map
The below function outputs an appropriate Z for the Shallue and van
de Woestijne map (Section 6.6.1).
def find_z_svdw(F, A, B):
g = lambda x: F(x)^3 + F(A) * F(x) + F(B)
h = lambda Z: -(F(3) * Z^2 + F(4) * A) / (F(4) * g(Z))
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
if g(Z_cand) == F(0):
# Criterion 1: g(Z) != 0 in F.
continue
if h(Z_cand) == F(0):
# Criterion 2: -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.
continue
if not h(Z_cand).is_square():
# Criterion 3: -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.
continue
if g(Z_cand).is_square() or g(-Z_cand / F(2)).is_square():
# Criterion 4: At least one of g(Z) and g(-Z / 2) is square in F.
return Z_cand
ctr += 1
E.2. Finding Z for Simplified SWU
The below function outputs an appropriate Z for the Simplified SWU
map (Section 6.6.2).
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# Arguments:
# - F, a field object, e.g., F = GF(2^521 - 1)
# - A and B, the coefficients of the curve equation y^2 = x^3 + A * x + B
def find_z_sswu(F, A, B):
R. = F[] # Polynomial ring over F
g = xx^3 + F(A) * xx + F(B) # y^2 = g(x) = x^3 + A * x + B
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
if Z_cand.is_square():
# Criterion 1: Z is non-square in F.
continue
if Z_cand == F(-1):
# Criterion 2: Z != -1 in F.
continue
if not (g - Z_cand).is_irreducible():
# Criterion 3: g(x) - Z is irreducible over F.
continue
if g(B / (Z_cand * A)).is_square():
# Criterion 4: g(B / (Z * A)) is square in F.
return Z_cand
ctr += 1
E.3. Finding Z for Elligator 2
The below function outputs an appropriate Z for the Elligator 2 map
(Section 6.7.1).
# Argument:
# - F, a field object, e.g., F = GF(2^255 - 19)
def find_z_ell2(F):
ctr = F.gen()
while True:
for Z_cand in (F(ctr), F(-ctr)):
if Z_cand.is_square():
# Z must be a non-square in F.
continue
return Z_cand
ctr += 1
Appendix F. sqrt functions
This section defines special-purpose sqrt functions for the three
most common cases, p = 3 (mod 4), p = 5 (mod 8), and p = 9 (mod 16).
In addition, it gives a generic constant-time algorithm that works
for any prime modulus.
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F.1. p = 3 (mod 4)
sqrt_3mod4(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
Input: x, an element of F.
Output: s, an element of F such that (s^2) == x.
Constants:
1. c1 = (q + 1) / 4 // Integer arithmetic
Procedure:
1. return x^c1
F.2. p = 5 (mod 8)
sqrt_5mod8(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
Input: x, an element of F.
Output: s, an element of F such that (s^2) == x.
Constants:
1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
2. c2 = (q + 3) / 8 // Integer arithmetic
Procedure:
1. t1 = x^c2
2. e = (t1^2) == x
3. s = CMOV(t1 * c1, t1, e)
3. return s
F.3. p = 9 (mod 16)
Note that this case also applies to GF(p^2) when p = 3 (mod 8).
[AR13] and [S85] describe methods that work for other field
extensions.
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sqrt_9mod16(x)
Parameters:
- F, a finite field of characteristic p and order q = p^m.
- p, the characteristic of F (see immediately above).
- m, the extension degree of F, m >= 1 (see immediately above).
Input: x, an element of F.
Output: s, an element of F such that (s^2) == x.
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
F.4. Constant-time Tonelli-Shanks algorithm
This algorithm is a constant-time version of the classic Tonelli-
Shanks algorithm ([C93], Algorithm 1.5.1) due to Sean Bowe, Jack
Grigg, and Eirik Ogilvie-Wigley [jubjub-fq], adapted and optimized by
Michael Scott.
This algorithm applies to GF(p) for any p. Note, however, that the
special-purpose algorithms given in the prior sections are faster,
when they apply.
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sqrt_ts_ct(x)
Parameters:
- F, a finite field of order p
- p, the characteristic of F (see immediately above)
Input x, an element of F.
Output: r, an element of F such that (r^2) == 2.
Constants (see discussion below):
1. c1, the largest integer such that 2^c1 divides p - 1.
2. c2 = (p - 1) / (2^c1) // Integer arithmetic
3. c3 = (c2 - 1) / 2 // Integer arithmetic
4. c4, a non-square value in F
5. c5 = c4^c2 in F
Procedure:
1. r = x^c3
2. t = r * r * x
3. r = r * x
4. b = t
5. c = c5
6. for k in (m, m - 1, ..., 2):
7. for j in (1, 2, ..., k - 1):
8. b = b * b
9. r = CMOV(r, r * c, b != 1)
10. c = c * c
11. t = CMOV(t, t * c, b != 1)
12. b = t
13. return r
The constants used in this procedure can be computed as follows:
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precompute_ts(p)
Input: p, a prime
Output: the required constants c1, ..., c5
Procedure:
1. c1 = 0
2. c2 = p - 1
3. while c2 is even:
4. c2 = c2 / 2 // Integer arithmetic
5. c1 = c1 + 1
6. c3 = (c2 - 1) / 2 // Integer arithmetic
7. c4 = 1
8. while c4 is square mod p:
9. c4 = c4 + 1
10. c5 = c4^c2 mod p
11. return (c1, c2, c3, c4, c5)
Authors' Addresses
Armando Faz-Hernandez
Cloudflare
101 Townsend St
San Francisco
United States of America
Email: armfazh@cloudflare.com
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
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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
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