Network Working Group S. Scott
Internet-Draft Cornell Tech
Intended status: Informational N. Sullivan
Expires: April 25, 2019 Cloudflare
C. Wood
Apple Inc.
October 22, 2018
Hashing to Elliptic Curves
draft-irtf-cfrg-hash-to-curve-02
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 . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1. Terminology . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1. Encoding . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2. Serialization . . . . . . . . . . . . . . . . . . . . 5
2.1.3. Random Oracle . . . . . . . . . . . . . . . . . . . . 5
3. Algorithm Recommendations . . . . . . . . . . . . . . . . . . 6
4. Utility Functions . . . . . . . . . . . . . . . . . . . . . . 6
5. Deterministic Encodings . . . . . . . . . . . . . . . . . . . 7
5.1. Interface . . . . . . . . . . . . . . . . . . . . . . . . 7
5.2. Encoding Variants . . . . . . . . . . . . . . . . . . . . 7
5.2.1. Icart Method . . . . . . . . . . . . . . . . . . . . 7
5.2.2. Shallue-Woestijne-Ulas Method . . . . . . . . . . . . 9
5.2.3. Simplified SWU Method . . . . . . . . . . . . . . . . 11
5.2.4. Elligator2 Method . . . . . . . . . . . . . . . . . . 12
5.3. Cost Comparison . . . . . . . . . . . . . . . . . . . . . 14
6. Random Oracles . . . . . . . . . . . . . . . . . . . . . . . 14
6.1. Interface . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2. General Construction (FFSTV13) . . . . . . . . . . . . . 14
7. Curve Transformations . . . . . . . . . . . . . . . . . . . . 15
8. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 15
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 17
10. Security Considerations . . . . . . . . . . . . . . . . . . . 17
11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 17
12. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 17
13. Normative References . . . . . . . . . . . . . . . . . . . . 17
Appendix A. Related Work . . . . . . . . . . . . . . . . . . . . 19
A.1. Probabilistic Encoding . . . . . . . . . . . . . . . . . 20
A.2. Naive Encoding . . . . . . . . . . . . . . . . . . . . . 20
A.3. Deterministic Encoding . . . . . . . . . . . . . . . . . 21
A.4. Supersingular Curves . . . . . . . . . . . . . . . . . . 21
A.5. Twisted Variants . . . . . . . . . . . . . . . . . . . . 21
Appendix B. Try-and-Increment Method . . . . . . . . . . . . . . 22
Appendix C. Sample Code . . . . . . . . . . . . . . . . . . . . 22
C.1. Icart Method . . . . . . . . . . . . . . . . . . . . . . 22
C.2. Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . . 23
C.3. Simplified SWU Method . . . . . . . . . . . . . . . . . . 25
C.4. Elligator2 Method . . . . . . . . . . . . . . . . . . . . 25
C.5. HashToBase . . . . . . . . . . . . . . . . . . . . . . . 26
C.5.1. Considerations . . . . . . . . . . . . . . . . . . . 27
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 28
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1. Introduction
Many cryptographic protocols require a procedure which maps arbitrary
input, e.g., passwords, to points on an elliptic curve (EC).
Prominent examples include Simple Password Exponential Key Exchange
[Jablon96], Password Authenticated Key Exchange [BMP00], Identity-
Based Encryption [BF01] and Boneh-Lynn-Shacham signatures [BLS01].
Unfortunately for implementors, the precise mapping which 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 address this lapse by providing a thorough set
of recommendations across a range of implementations, and curve
types. We provide implementation and performance details for each
mechanism, along with references to the security rationale behind
each recommendation and guidance for applications not yet covered.
Each algorithm conforms to a common interface, i.e., it maps a
bitstring {0, 1}^* to a point on an elliptic curve E. For each
variant, we describe the requirements for E to make it work. Sample
code for each variant is presented in the appendix. Unless otherwise
stated, all elliptic curve points are assumed to be represented as
affine coordinates, i.e., (x, y) points on a curve.
1.1. Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
2. Background
Here we give a brief definition of elliptic curves, with an emphasis
on defining important parameters and their relation to encoding.
Let F be the finite field GF(p^k). We say that F is a field of
characteristic p. For most applications, F is a prime field, in
which case k=1 and we will simply write GF(p).
Elliptic curves can be represented by equations of different standard
forms, including, but not limited to: Weierstrass, Montgomery, and
Edwards. Each of these variants correspond to a different category
of curve equation. For example, the short Weierstrass equation is
"y^2 = x^3 + Ax + B". Certain encoding functions may have
requirements on the curve form, the characteristic of the field, and
the parameters, such as A and B in the previous example.
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An elliptic curve E is specified by its equation, and a finite field
F. The curve E forms a group, whose elements correspond to those who
satisfy the curve equation, with values taken from the field F. As a
group, E has order n, which is the number of points on the curve.
For security reasons, it is a strong requirement that all
cryptographic operations take place in a prime order group. However,
not all elliptic curves generate groups of prime order. In those
cases, it is allowed to work with elliptic curves of order n = qh,
where q is a large prime, and h is a short number known as the
cofactor. Thus, we may wish an encoding that returns points on the
subgroup of order q. Multiplying a point P on E by the cofactor h
guarantees that hP is a point in the subgroup of order q.
Summary of quantities:
+--------+-------------------+--------------------------------------+
| Symbol | Meaning | Relevance |
+--------+-------------------+--------------------------------------+
| p | Order of finite | Curve points need to be represented |
| | field, F = GF(p) | in terms of p. For prime power |
| | | extension fields, we write F = |
| | | GF(p^k). |
| | | |
| n | Number of curve | For map to E, needs to produce n |
| | points, #E(F) = n | elements. |
| | | |
| q | Order of the | If n is not prime, may need mapping |
| | largest prime | to q. |
| | subgroup of E, n | |
| | = qh | |
| | | |
| h | Cofactor | For mapping to subgroup, need to |
| | | multiply by cofactor. |
+--------+-------------------+--------------------------------------+
2.1. Terminology
In the following, we categorize the terminology for mapping
bitstrings to points on elliptic curves.
2.1.1. Encoding
In practice, the input of a given cryptographic algorithm will be a
bitstring of arbitrary length, denoted {0, 1}^*. Hence, a concern for
virtually all protocols involving elliptic curves is how to convert
this input into a curve point. The general term "encoding" refers to
the process of producing an elliptic curve point given as input a
bitstring. In some protocols, the original message may also be
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recovered through a decoding procedure. An encoding may be
deterministic or probabilistic, although the latter is problematic in
potentially leaking plaintext information as a side-channel.
Suppose as the input to the encoding function we wish to use a fixed-
length bitstring of length L. Comparing sizes of the sets, 2^L and
n, an encoding function cannot be both deterministic and bijective.
We can instead use an injective encoding from {0, 1}^L to E, with "L
< log2(n)- 1", which is a bijection over a subset of points in E.
This ensures that encoded plaintext messages can be recovered.
2.1.2. Serialization
A related issue is the conversion of an elliptic curve point to a
bitstring. We refer to this process as "serialization", since it is
typically used for compactly storing and transporting points, or for
producing canonicalized outputs. Since a deserialization algorithm
can often be used as a type of encoding algorithm, we also briefly
document properties of these functions.
A straightforward serialization algorithm maps a point (x, y) on E to
a bitstring of length 2*log(p), given that x, y are both elements in
GF(p). However, since there are only n points in E (with n
approximately equal to p), it is possible to serialize to a bitstring
of length log(n). For example, one common method is to store the
x-coordinate and a single bit to determine whether the point is (x,
y) or (x, -y), thus requiring log(p)+1 bits. This method reduces
storage, but adds computation, since the deserialization process must
recover the y coordinate.
2.1.3. Random Oracle
It is often the case that the output of the encoding function
Section 2.1.1 should be distributed uniformly at random on the
elliptic curve. That is, there is no discernible relation existing
between outputs that can be computed based on the inputs. In
practice, this requirement stems from needing a random oracle which
outputs elliptic curve points: one way to construct this is by first
taking a regular random oracle, operating entirely on bitstrings, and
applying a suitable encoding function to the output.
This motivates the term "hashing to the curve", since cryptographic
hash functions are typically modeled as random oracles. However,
this still leaves open the question of what constitutes a suitable
encoding method, which is a primary concern of this document.
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A random oracle onto an elliptic curve can also be instantiated using
direct constructions, however these tend to rely on many group
operations and are less efficient than hash and encode methods.
3. Algorithm Recommendations
The following table lists algorithms recommended by use-case:
+----------------+-----------------+--------------------------------+
| Application | Requirement | Additional Details |
+----------------+-----------------+--------------------------------+
| SPEKE | Naive | H(x)*G |
| [Jablon96] | | |
| | | |
| PAKE [BMP00] | Random Oracle | - |
| | | |
| BLS [BLS01] | Random Oracle | - |
| | | |
| IBE [BF01] | Random Oracle | Supersingular, pairing- |
| | | friendly curve |
| | | |
| PRF | Injective | F(k, m) = k*H(m) |
| | encoding | |
+----------------+-----------------+--------------------------------+
To find the suitable algorithm, lookup the requirement from above,
with the chosen curve in the below:
+------------+--------------------------+---------------+
| Curve | Inj. Encoding | Random Oracle |
+------------+--------------------------+---------------+
| P-256 | Simple SWU Section 5.2.3 | FFSTV(SWU) |
| | | |
| P-384 | Icart Section 5.2.1 | FFSTV(Icart) |
| | | |
| Curve25519 | Elligator2 Section 5.2.4 | ... |
| | | |
| Curve448 | Elligator2 Section 5.2.4 | ... |
+------------+--------------------------+---------------+
4. Utility Functions
Algorithms in this document make use of utility functions described
below.
o HashToBase(x, i). This method is parametrized by p and H, where p
is the prime order of the base field Fp, and H is a cryptographic
hash function which outputs at least floor(log2(p)) + 2 bits. The
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function first hashes x, converts the result to an integer, and
reduces modulo p to give an element of Fp.
We provide a more detailed algorithm in Appendix C.5. The value
of i is used to separate inputs when used multiple times in one
algorithm (see Section 6.2 for example). When i is omitted, we
set it to 0.
o CMOV(a, b, c): If c = 1, return a, else return b.
Common software implementations of constant-time selects assume c
= 1 or c = 0. CMOV may be implemented by computing the desired
selector (0 or 1) by ORing all bits of c together. The end result
will be either 0 if all bits of c are zero, or 1 if at least one
bit of c is 1.
o CTEQ(a, b): Returns a == b. Inputs a and b must be the same
length (as bytestrings) and the comparison must be implemented in
constant time.
o Legendre(x, p): x^((p-1)/2). The Legendre symbol computes whether
the value x is a "quadratic residue" modulo p, and takes values 1,
-1, 0, for when x is a residue, non-residue, or zero,
respectively. Due to Euler's criterion, this can be computed in
constant time, with respect to a fixed p, using the equation
x^((p-1)/2). For clarity, we will generally prefer using the
formula directly, and annotate the usage with this definition.
5. Deterministic Encodings
5.1. Interface
The generic interface for deterministic encoding functions to
elliptic curves is as follows:
map2curve(alpha)
where alpha is a message to encode on a curve.
5.2. Encoding Variants
5.2.1. Icart Method
The following map2curve_icart(alpha) implements the Icart method from
[Icart09]. This algorithm works for any curve over F_{p^n}, where
p^n = 2 mod 3 (or p = 2 mod 3 and for odd n), including:
o P384
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o Curve1174
o Curve448
Unsupported curves include: P224, P256, P521, and Curve25519 since,
for each, p = 1 mod 3.
Mathematically, given input alpha, and A and B from E, the Icart
method works as follows:
u = HashToBase(alpha)
v = ((3A - u^4) / 6u)
x = (v^2 - B - (u^6 / 27))^(1/3) + (u^2 / 3)
y = ux + v
The following procedure implements this algorithm in a straight-line
fashion. It requires knowledge of A and B, the constants from the
curve Weierstrass form. It outputs a point with affine coordinates.
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map2curve_icart(alpha)
Input:
alpha - value to be hashed, an octet string
Output:
(x, y) - a point in E
Steps:
1. u = HashToBase(alpha) // {0,1}^* -> Fp
2. u2 = u^2 (mod p) // u^2
3. t2 = u2^2 (mod p) // u^4
4. v1 = 3 * A (mod p) // 3A in Fp
5. v1 = v1 - t2 (mod p) // 3A - u^4
6. t1 = 6 * u (mod p) // 6u
7. t3 = t1 ^ (-1) (mod p) // modular inverse
8. v = v1 * t3 (mod p) // (3A - u^4)/(6u)
9. x = v^2 (mod p) // v^2
10. x = x - B (mod p) // v^2 - B
11. t1 = 27 ^ (-1) (mod p) // 1/27
12. t1 = t1 * u2 (mod p) // u^4 / 27
13. t1 = t1 * t2 (mod p) // u^6 / 27
14. x = x - t1 (mod p) // v^2 - B - u^6/27
15. t1 = (2 * p) - 1 // 2p - 1 in ZZ
16. t1 = t1 / 3 // (2p - 1)/3 in ZZ
17. x = x^t1 (mod p) // (v^2 - B - u^6/27) ^ (1/3)
18. t2 = u2 / 3 (mod p) // u^2 / 3
19. x = x + t2 (mod p) // (v^2 - B - u^6/27) ^ (1/3) + (u^2 / 3)
20. y = u * x (mod p) // ux
21. y = y + v (mod p) // ux + v
22. Output (x, y)
5.2.2. Shallue-Woestijne-Ulas Method
The Shallue-Woestijne-Ulas (SWU) method, originated in part by
Shallue and Woestijne [SW06] and later simplified and extended by
Ulas [SWU07], deterministically encodes an arbitrary string to a
point on a curve. This algorithm works for any curve over F_{p^n}.
Given curve equation g(x) = x^3 + Ax + B, this algorithm works as
follows:
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1. t = HashToBase(alpha, 0)
2. u = HashToBase(alpha, 1)
3. X1 = u
4. X2 = (-B / A)(1 + 1 / (t^4 * g(u)^2 + t^2 * g(u)))
5. X3 = t^3 * g(u)^2 * g(X2)
6. If g(X1) is square, output (X1, sqrt(g(X1)))
7. If g(X2) is square, output (X2, sqrt(g(X2)))
8. Output (X3(t, u), sqrt(g(X3)))
The algorithm relies on the following equality:
t^3 * g(u)^2 * g(X2(t, u)) = g(X1(t, u)) * g(X2(t, u)) * g(X3(t, u))
The algorithm computes three candidate points, constructed such that
at least one of them lies on the curve.
The following procedure implements this algorithm. It outputs a
point with affine coordinates. It requires knowledge of A and B, the
constants from the curve Weierstrass form.
map2curve_swu(alpha)
Input:
alpha - value to be hashed, an octet string
Output:
(x, y) - a point in E
Steps:
1. t = HashToBase(alpha, 0) // {0,1}^* -> Fp
2. u = HashToBase(alpha, 1) // {0,1}^* -> Fp
3. t2 = t^2
4. t4 = t2^2
5. gu = u^3
6. gu = gu + (A * u)
7. gu = gu + B // gu = g(u)
8. x1 = u // x1 = X1(t, u) = u
9. x2 = B * -1
10. x2 = x2 / A
11. gx1 = x1^3
12. gx1 = gx1 + (A * x1)
13. gx1 = gx1 + B // gx1 = g(X1(t, u))
14. d1 = gu^2
15. d1 = d1 * t4
16. d2 = t2 * gu
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17. d3 = d1 + d2
18. d3 = d3^(-1)
19. n1 = 1 + d3
20. x2 = x2 * n1 // x2 = X2(t, u)
21. gx2 = x2^3
22. gx2 = gx2 + (A * x2)
23. gx2 = gx2 + B // gx2 = g(X2(t, u))
24. x3 = t2 * gu
25. x3 = x3 * x2 // x3 = X3(t, u)
26. gx3 = x3^3
27. gx3 = gx3 + (A * x3)
28. gx3 = gx3 + B // gx3 = g(X3(t, u))
29. l1 = gx1^((p - 1) / 2)
30. l2 = gx2^((p - 1) / 2)
31. s1 = gx1^(1/2)
32. s2 = gx2^(1/2)
33. s3 = gx3^(1/2)
34. if l1 == 1:
35. Output (x1, s1)
36. if l2 == 1:
37. Output (x2, s2)
38. Output (x3, s3)
5.2.3. Simplified SWU Method
The following map2curve_simple_swu(alpha) implements the simplified
Shallue-Woestijne-Ulas algorithm from [SimpleSWU]. This algorithm
works for any curve over F_{p^n}, where p = 3 mod 4, including:
o P256
o ...
Given curve equation g(x) = x^3 + Ax + B, this algorithm works as
follows:
1. t = HashToBase(alpha)
2. alpha = (-B / A) * (1 + (1 / (t^4 + t^2)))
3. beta = -t^2 * alpha
4. If g(alpha) is square, output (alpha, sqrt(g(alpha)))
5. Output (beta, sqrt(g(beta)))
The following procedure implements this algorithm. It outputs a
point with affine coordinates. It requires knowledge of A and B, the
constants from the curve Weierstrass form.
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map2curve_simple_swu(alpha)
Input:
alpha - value to be encoded, an octet string
Output:
(x, y) - a point in E
Steps:
1. t = HashToBase(alpha)
2. alpha = t^2 (mod p)
3. alpha = alpha * -1 (mod p)
4. right = alpha^2 + alpha (mod p)
5. right = right^(-1) (mod p)
6. right = right + 1 (mod p)
7. left = B * -1 (mod p)
8. left = left / A (mod p)
9. x2 = left * right (mod p)
10. x3 = alpha * x2 (mod p)
11. h2 = x2 ^ 3 (mod p)
12. i2 = x2 * A (mod p)
13. i2 = i2 + B (mod p)
14. h2 = h2 + i2 (mod p)
15. h3 = x3 ^ 3 (mod p)
16. i3 = x3 * A (mod p)
17. i3 = i3 + B (mod p)
18. h3 = h3 + i3 (mod p)
19. y1 = h2 ^ ((p + 1) / 4) (mod p)
20. y2 = h3 ^ ((p + 1) / 4) (mod p)
21. e = CTEQ(y1 ^ 2, h2) // Constant-time equality
22. x = CMOV(x2, x3, e) // If e = 1, choose x2, else choose x3
23. y = CMOV(y1, y2, e) // If e = 1, choose y1, else choose y2
24. Output (x, y)
5.2.4. Elligator2 Method
The following map2curve_elligator2(alpha) implements the Elligator2
method from [Elligator2]. This algorithm works for any curve with a
point of order 2 and j-invariant != 1728. Given curve equation y^2 =
x(x^2 + Ax + B), i.e., a Montgomery form with (0,0), a point of order
2, this algorithm works as shown below. (Note that any curve with a
point of order 2 is isomorphic to this representation.)
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1. r = HashToBase(alpha)
2. Let u be a non-square value in Fp
3. v = -A/(1+ur^ 2)
4. e = Legendre(v^3+Av^2+Bv)
5.1. If r != 0, then
5.2. x = ev - (1 - e)A/2
5.3. y = -e*sqrt(x^3+Ax^2+x)
5.4. Else, x=0 and y=0
6. Output (x,y)
Here, e is the Legendre symbol defined as in Section 4.
The following procedure implements this algorithm.
map2curve_elligator2(alpha)
Input:
alpha - value to be encoded, an octet string
u - fixed non-square value in Fp.
Output:
(x, y) - a point in E
Steps:
1. r = HashToBase(alpha)
2. r = r^2 (mod p)
3. nu = r * u (mod p)
4. r = nu
5. r = r + 1 (mod p)
6. r = r^(-1) (mod p)
7. v = A * r (mod p)
8. v = v * -1 (mod p) // -A / (1 + ur^2)
9. v2 = v^2 (mod p)
10. v3 = v * v2 (mod p)
11. e = v3 + v (mod p)
12. v2 = v2 * A (mod p)
13. e = v2 + e (mod p)
14. e = e^((p - 1) / 2) // = Legendre(e)
15. nv = v * -1 (mod p)
16. v = CMOV(v, nv, e) // If e = 1, choose v, else choose nv
17. v2 = CMOV(0, A, e) // If e = 1, choose 0, else choose A
18. x = v - v2 (mod p)
19. y = -e*sqrt(x^3+Ax^2+Bx)
19. Output (x, y)
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Elligator2 can be simplified with projective coordinates.
((TODO: write this variant))
5.3. Cost Comparison
The following table summarizes the cost of each map2curve variant.
We express this cost in terms of additions (A), multiplications (M),
squares (SQ), and square roots (SR).
((TODO: finish this section))
+----------------------+-------------------+
| Algorithm | Cost (Operations) |
+----------------------+-------------------+
| map2curve_icart | TODO |
| | |
| map2curve_swu | TODO |
| | |
| map2curve_simple_swu | TODO |
| | |
| map2curve_elligator2 | TODO |
+----------------------+-------------------+
6. Random Oracles
6.1. Interface
The generic interface for deterministic encoding functions to
elliptic curves is as follows:
hash2curve(alpha)
where alpha is a message to encode on a curve.
6.2. General Construction (FFSTV13)
When applications need a Random Oracle (RO), they can be constructed
from deterministic encoding functions. In particular, let F :
{0,1}^* -> E be a deterministic encoding function onto curve E, and
let H0 and H1 be two hash functions modeled as random oracles that
map input messages to the base field of E, i.e., Z_q. Farashahi et
al. [FFSTV13] showed that the following mapping is indistinguishable
from a RO:
hash2curve(alpha) = F(H0(alpha)) + F(H1(alpha))
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This construction works for the Icart, SWU, and Simplfied SWU
encodings.
Here, H0 and H1 are constructed as follows:
H0(alpha) = HashToBase(alpha, 2)
H1(alpha) = HashToBase(alpha, 3)
7. Curve Transformations
Every elliptic curve can be converted to an equivalent curve in short
Weierstrass form ([BL07] Theorem 2.1), making SWU a generic algorithm
that can be used for all curves. Curves in either Edwards or Twisted
Edwards form can be transformed into equivalent curves in Montgomery
form [BL17] for use with Elligator2. [RFC7748] describes how to
convert between points on Curve25519 and Ed25519, and between
Curve448 and its Edwards equivalent, Goldilocks.
8. Ciphersuites
To provide concrete recommendations for algorithms we define a hash-
to-curve "ciphersuite" as a four-tuple containing:
o Destination Group (e.g. P256 or Curve25519)
o HashToBase algorithm
o HashToCurve algorithm (e.g. SSWU, Icart)
o (Optional) Transformation (e.g. FFSTV, cofactor clearing)
A ciphersuite defines an algorithm that takes an arbitrary octet
string and returns an element of the Destination Group defined in the
ciphersuite by applying HashToCurve and Transformation (if defined).
This document describes the following set of ciphersuites: * H2C-
P256-SHA256-SSWU- * H2C-P384-SHA512-Icart- * H2C-
Curve25519-SHA512-Elligator2-Clear * H2C-
Curve448-SHA512-Elligator2-Clear * H2C-
Curve25519-SHA512-Elligator2-FFSTV * H2C-
Curve448-SHA512-Elligator2-FFSTV
H2C-P256-SHA256-SWU- is defined as follows:
o The destination group is the set of points on the NIST P-256
elliptic curve, with curve parameters as specified in [DSS]
(Section D.1.2.3) and [RFC5114] (Section 2.6).
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o HashToBase is defined as {#hashtobase} with the hash function
defined as SHA-256 as specified in [RFC6234], and p set to the
prime field used in P-256 (2^256 - 2^224 + 2^192 + 2^96 - 1).
o HashToCurve is defined to be {#sswu} with A and B taken from the
definition of P-256 (A=-3, B=4105836372515214212932612978004726840
9114441015993725554835256314039467401291).
H2C-P384-SHA512-Icart- is defined as follows:
o The destination group is the set of points on the NIST P-384
elliptic curve, with curve parameters as specified in [DSS]
(Section D.1.2.4) and [RFC5114] (Section 2.7).
o HashToBase is defined as {#hashtobase} with the hash function
defined as SHA-512 as specified in [RFC6234], and p set to the
prime field used in P-384 (2^384 - 2^128 - 2^96 + 2^32 - 1).
o HashToCurve is defined to be {#icart} with A and B taken from the
definition of P-384 (A=-3, B=2758019355995970587784901184038904809
305690585636156852142870730198868924130986086513626076488374510776
5439761230575).
H2C-Curve25519-SHA512-Elligator2-Clear is defined as follows:
o The destination group is the points on Curve25519, with curve
parameters as specified in [RFC7748] (Section 4.1).
o HashToBase is defined as {#hashtobase} with the hash function
defined as SHA-512 as specified in [RFC6234], and p set to the
prime field used in Curve25519 (2^255 - 19).
o HashToCurve is defined to be {#elligator2} with the curve function
defined to be the Montgomery form of Curve25519 (y^2 = x^3 +
486662x^2 + x) and u = 2.
o The final output is multiplied by the cofactor of Curve25519, 8.
H2C-Curve448-SHA512-Elligator2-Clear is defined as follows:
o The destination group is the points on Curve448, with curve
parameters as specified in [RFC7748] (Section 4.1).
o HashToBase is defined as {#hashtobase} with the hash function
defined as SHA-512 as specified in [RFC6234], and p set to the
prime field used in Curve448 (2^448 - 2^224 - 1).
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o HashToCurve is defined to be {#elligator2} with the curve function
defined to be the Montgomery form of Curve448 (y^2 = x^3 +
156326x^2 + x) and u = -1.
o The final output is multiplied by the cofactor of Curve448, 4.
H2C-Curve25519-SHA512-Elligator2-FFSTV is defined as in H2C-
Curve25519-SHA-512-Elligator2-Clear except HashToCurve is defined to
be {#ffstv} where F is {#elligator2}.
H2C-Curve448-SHA512-Elligator2-FFSTV is defined as in H2C-Curve448-
SHA-512-Elligator2-Clear except HashToCurve is defined to be {#ffstv}
where F is {#elligator2}.
9. IANA Considerations
This document has no IANA actions.
10. Security Considerations
Each encoding function variant accepts arbitrary input and maps it to
a pseudorandom point on the curve. Points are close to
indistinguishable from randomly chosen elements on the curve. Not
all encoding functions are full-domain hashes. Elligator2, for
example, only maps strings to "about half of all curve points,"
whereas Icart's method only covers about 5/8 of the points.
11. Acknowledgements
The authors would like to thank Adam Langley for this detailed
writeup up Elligator2 with Curve25519 [ElligatorAGL]. We also thank
Sean Devlin and Thomas Icart for feedback on earlier versions of this
document.
12. Contributors
o Sharon Goldberg
Boston University
goldbe@cs.bu.edu
13. Normative References
[BF01] "Identity-based encryption from the Weil pairing", n.d..
[BL07] "Faster addition and doubling on elliptic curves", n.d.,
.
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[BL17] "Montgomery curves and the Montgomery ladder", n.d.,
.
[BLS01] "Short signatures from the Weil pairing", n.d.,
.
[BMP00] "Provably secure password-authenticated key exchange using
diffie-hellman", n.d..
[DSS] National Institute of Standards and Technology, U.S.
Department of Commerce, "Digital Signature Standard,
version 4", NIST FIPS PUB 186-4, 2013.
[ECOPRF] "EC-OPRF - Oblivious Pseudorandom Functions using Elliptic
Curves", n.d..
[Elligator2]
"Elligator -- Elliptic-curve points indistinguishable from
uniform random strings", n.d., .
[ElligatorAGL]
"Implementing Elligator for Curve25519", n.d.,
.
[FFSTV13] "Indifferentiable deterministic hashing to elliptic and
hyperelliptic curves", n.d..
[FIPS-186-4]
"Digital Signature Standard (DSS), FIPS PUB 186-4, July
2013", n.d.,
.
[hacspec] "hacspec", n.d., .
[Icart09] "How to Hash into Elliptic Curves", n.d.,
.
[Jablon96]
"Strong password-only authenticated key exchange", n.d..
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997, .
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[RFC5114] Lepinski, M. and S. Kent, "Additional Diffie-Hellman
Groups for Use with IETF Standards", RFC 5114,
DOI 10.17487/RFC5114, January 2008, .
[RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
Key Derivation Function (HKDF)", RFC 5869,
DOI 10.17487/RFC5869, May 2010, .
[RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and SHA-based HMAC and HKDF)", RFC 6234,
DOI 10.17487/RFC6234, May 2011, .
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
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, .
[SECG1] "SEC 1 -- Elliptic Curve Cryptography", n.d.,
.
[SimpleSWU]
"Efficient Indifferentiable Hashing into Ordinary Elliptic
Curves", n.d..
[SW06] "Construction of rational points on elliptic curves over
finite fields", n.d..
[SWU07] "Rational points on certain hyperelliptic curves over
finite fields", n.d., .
Appendix A. Related Work
In this chapter, we give a background to some common methods to
encode or hash to the curve, motivated by the similar exposition in
[Icart09]. Understanding of this material is not required in order
to choose a suitable encoding function - we defer this to Section 3 -
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the background covered here can work as a template for analyzing
encoding functions not found in this document, and as a guide for
further research into the topics covered.
A.1. Probabilistic Encoding
As mentioned in Section 2, as a rule of thumb, for every x in GF(p),
there is approximately a 1/2 chance that there exist a corresponding
y value such that (x, y) is on the curve E.
This motivates the construction of the MapToGroup method described by
Boneh et al. [BLS01]. For an input message m, a counter i, and a
standard hash function H : {0, 1}^* -> GF(p) x {0, 1}, one computes
(x, b) = H(i || m), where i || m denotes concatenation of the two
values. Next, test to see whether there exists a corresponding y
value such that (x, y) is on the curve, returning (x, y) if
successful, where b determines whether to take +/- y. If there does
not exist such a y, then increment i and repeat. A maximum counter
value is set to I, and since each iteration succeeds with probability
approximately 1/2, this process fails with probability 2^-I. (See
Appendix B for a more detailed description of this algorithm.)
Although MapToGroup describes a method to hash to the curve, it can
also be adapted to a simple encoding mechanism. For a bitstring of
length strictly less than log2(p), one can make use of the spare bits
in order to encode the counter value. Allocating more space for the
counter increases the expansion, but reduces the failure probability.
Since the running time of the MapToGroup algorithm depends on m, this
algorithm is NOT safe for cases sensitive to timing side channel
attacks. Deterministic algorithms are needed in such cases where
failures are undesirable.
A.2. Naive Encoding
A naive solution includes computing H(m)*G as map2curve(m), where H
is a standard hash function H : {0, 1}^* -> GF(p), and G is a
generator of the curve. Although efficient, this solution is
unsuitable for constructing a random oracle onto E, since the
discrete logarithm with respect to G is known. For example, given y1
= map2curve(m1) and y2 = map2curve(m2) for any m1 and m2, it must be
true that y2 = H(m2) / H(m1) * map2curve(m1). This relationship
would not hold (with overwhelming probability) for truly random
values y1 and y2. This causes catastrophic failure in many cases.
However, one exception is found in SPEKE [Jablon96], which constructs
a base for a Diffie-Hellman key exchange by hashing the password to a
curve point. Notably the use of a hash function is purely for
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encoding an arbitrary length string to a curve point, and does not
need to be a random oracle.
A.3. Deterministic Encoding
Shallue, Woestijne, and Ulas [SW06] first introduced a deterministic
algorithm that maps elements in F_{q} to a curve in time O(log^4 q),
where q = p^n for some prime p, and time O(log^3 q) when q = 3 mod 4.
Icart introduced yet another deterministic algorithm which maps F_{q}
to any EC where q = 2 mod 3 in time O(log^3 q) [Icart09]. Elligator
(2) [Elligator2] is yet another deterministic algorithm for any odd-
characteristic EC that has a point of order 2. Elligator2 can be
applied to Curve25519 and Curve448, which are both CFRG-recommended
curves [RFC7748].
However, an important caveat to all of the above deterministic
encoding functions, is that none of them map injectively to the
entire curve, but rather some fraction of the points. This makes
them unable to use to directly construct a random oracle on the
curve.
Brier et al. [SimpleSWU] proposed a couple of solutions to this
problem, The first applies solely to Icart's method described above,
by computing F(H0(m)) + F(H1(m)) for two distinct hash functions H0,
H1. The second uses a generator G, and computes F(H0(m)) + H1(m)*G.
Later, Farashahi et al. [FFSTV13] showed the generality of the
F(H0(m)) + F(H1(m)) method, as well as the applicability to
hyperelliptic curves (not covered here).
A.4. Supersingular Curves
For supersingular curves, for every y in GF(p) (with p>3), there
exists a value x such that (x, y) is on the curve E. Hence we can
construct a bijection F : GF(p) -> E (ignoring the point at
infinity). This is the case for [BF01], but is not common.
A.5. Twisted Variants
We can also consider curves which have twisted variants, E^d. For
such curves, for any x in GF(p), there exists y in GF(p) such that
(x, y) is either a point on E or E^d. Hence one can construct a
bijection F : GF(p) x {0,1} -> E ∪ E^d, where the extra bit is
needed to choose the sign of the point. This can be particularly
useful for constructions which only need the x-coordinate of the
point. For example, x-only scalar multiplication can be computed on
Montgomery curves. In this case, there is no need for an encoding
function, since the output of F in GF(p) is sufficient to define a
point on one of E or E^d.
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Appendix B. Try-and-Increment Method
In cases where constant time execution is not required, the so-called
try-and-increment method may be appropriate. As discussion in
Section 1, this variant works by hashing input m using a standard
hash function ("Hash"), e.g., SHA256, and then checking to see if the
resulting point E(m, f(m)), for curve function f, belongs on E. This
is detailed below.
1. ctr = 0
2. h = "INVALID"
3. While h is "INVALID" or h is EC point at infinity:
4.1 CTR = I2OSP(ctr, 4)
4.2 ctr = ctr + 1
4.3 attempted_hash = Hash(m || CTR)
4.4 h = RS2ECP(attempted_hash)
4.5 If h is not "INVALID" and cofactor > 1, set h = h * cofactor
5. Output h
I2OSP is a function that converts a nonnegative integer to octet
string as defined in Section 4.1 of [RFC8017], and RS2ECP(h) =
OS2ECP(0x02 || h), where OS2ECP is specified in Section 2.3.4 of
[SECG1], which converts an input string into an EC point.
Appendix C. Sample Code
This section contains reference implementations for each map2curve
variant built using [hacspec].
C.1. Icart Method
The following hacspec program implements map2curve_icart(alpha) for
P-384.
from hacspec.speclib import *
prime = 2**384 - 2**128 - 2**96 + 2**32 - 1
felem_t = refine(nat, lambda x: x < prime)
affine_t = tuple2(felem_t, felem_t)
@typechecked
def to_felem(x: nat_t) -> felem_t:
return felem_t(nat(x % prime))
@typechecked
def fadd(x: felem_t, y: felem_t) -> felem_t:
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return to_felem(x + y)
@typechecked
def fsub(x: felem_t, y: felem_t) -> felem_t:
return to_felem(x - y)
@typechecked
def fmul(x: felem_t, y: felem_t) -> felem_t:
return to_felem(x * y)
@typechecked
def fsqr(x: felem_t) -> felem_t:
return to_felem(x * x)
@typechecked
def fexp(x: felem_t, n: nat_t) -> felem_t:
return to_felem(pow(x, n, prime))
@typechecked
def finv(x: felem_t) -> felem_t:
return to_felem(pow(x, prime-2, prime))
a384 = to_felem(prime - 3)
b384 = to_felem(27580193559959705877849011840389048093056905856361568521428707301988689241309860865136260764883745107765439761230575)
@typechecked
def map2p384(u:felem_t) -> affine_t:
v = fmul(fsub(fmul(to_felem(3), a384), fexp(u, 4)), finv(fmul(to_felem(6), u)))
u2 = fmul(fexp(u, 6), finv(to_felem(27)))
x = fsub(fsqr(v), b384)
x = fsub(x, u2)
x = fexp(x, (2 * prime - 1) // 3)
x = fadd(x, fmul(fsqr(u), finv(to_felem(3))))
y = fadd(fmul(u, x), v)
return (x, y)
C.2. Shallue-Woestijne-Ulas Method
The following hacspec program implements map2curve_swu(alpha) for
P-256.
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from p256 import *
from hacspec.speclib import *
a256 = to_felem(prime - 3)
b256 = to_felem(41058363725152142129326129780047268409114441015993725554835256314039467401291)
@typechecked
def f_p256(x:felem_t) -> felem_t:
return fadd(fexp(x, 3), fadd(fmul(to_felem(a256), x), to_felem(b256)))
@typechecked
def x1(t:felem_t, u:felem_t) -> felem_t:
return u
@typechecked
def x2(t:felem_t, u:felem_t) -> felem_t:
coefficient = fmul(to_felem(-b256), finv(to_felem(a256)))
t2 = fsqr(t)
t4 = fsqr(t2)
gu = f_p256(u)
gu2 = fsqr(gu)
denom = fadd(fmul(t4, gu2), fmul(t2, gu))
return fmul(coefficient, fadd(to_felem(1), finv(denom)))
@typechecked
def x3(t:felem_t, u:felem_t) -> felem_t:
return fmul(fsqr(t), fmul(f_p256(u), x2(t, u)))
@typechecked
def map2p256(t:felem_t) -> felem_t:
u = fadd(t, to_felem(1))
x1v = x1(t, u)
x2v = x2(t, u)
x3v = x3(t, u)
exp = to_felem((prime - 1) // 2)
e1 = fexp(f_p256(x1v), exp)
e2 = fexp(f_p256(x2v), exp)
if e1 == 1:
return x1v
elif e2 == 1:
return x2v
else:
return x3v
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C.3. Simplified SWU Method
The following hacspec program implements map2curve_simple_swu(alpha)
for P-256.
from p256 import *
from hacspec.speclib import *
a256 = to_felem(prime - 3)
b256 = to_felem(41058363725152142129326129780047268409114441015993725554835256314039467401291)
def f_p256(x:felem_t) -> felem_t:
return fadd(fexp(x, 3), fadd(fmul(to_felem(a256), x), to_felem(b256)))
def map2p256(t:felem_t) -> affine_t:
alpha = to_felem(-(fsqr(t)))
frac = finv((fadd(fsqr(alpha), alpha)))
coefficient = fmul(to_felem(-b256), finv(to_felem(a256)))
x2 = fmul(coefficient, fadd(to_felem(1), frac))
x3 = fmul(alpha, x2)
h2 = fadd(fexp(x2, 3), fadd(fmul(a256, x2), b256))
h3 = fadd(fexp(x3, 3), fadd(fmul(a256, x3), b256))
exp = fmul(fadd(to_felem(prime), to_felem(-1)), finv(to_felem(2)))
e = fexp(h2, exp)
exp = to_felem((prime + 1) // 4)
if e == 1:
return (x2, fexp(f_p256(x2), exp))
else:
return (x3, fexp(f_p256(x3), exp))
C.4. Elligator2 Method
The following hacspec program implements map2curve_elligator2(alpha)
for Curve25519.
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from curve25519 import *
from hacspec.speclib import *
a25519 = to_felem(486662)
b25519 = to_felem(1)
u25519 = to_felem(2)
@typechecked
def f_25519(x:felem_t) -> felem_t:
return fadd(fmul(x, fsqr(x)), fadd(fmul(a25519, fsqr(x)), x))
@typechecked
def map2curve25519(r:felem_t) -> felem_t:
d = fsub(to_felem(p25519), fmul(a25519, finv(fadd(to_felem(1), fmul(u25519, fsqr(r))))))
power = nat((p25519 - 1) // 2)
e = fexp(f_25519(d), power)
x = 0
if e != 1:
x = fsub(to_felem(-d), to_felem(a25519))
else:
x = d
return x
C.5. HashToBase
The following procedure implements HashToBase.
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HashToBase(x, i)
Parameters:
H - cryptographic hash function to use
hbits - number of bits output by H
p - order of the base field Fp
label - context label for domain separation
Preconditions:
floor(log2(p)) + 1 >= hbits
Input:
x - value to be hashed, an octet string
i - hash call index, a non-negative integer
Output:
y - a value in the field Fp
Steps:
1. t1 = H("h2c" || label || I2OSP(i, 4) || x)
2. t2 = OS2IP(t1)
3. y = t2 (mod p)
4. Output y
where I2OSP, OS2IP [RFC8017] are used to convert an octet string to
and from a non-negative integer, and a || b denotes concatenation of
a and b.
C.5.1. Considerations
Performance: HashToBase requires hashing the entire input x. In some
algorithms/ciphersuite combinations, HashToBase is called multiple
times. For large inputs, implementers can therefore consider hashing
x before calling HashToBase. I.e. HashToBase(H'(x)).
Most algorithms assume that HashToBase maps its input to the base
field uniformly. In practice, there will be inherent biases. For
example, taking H as SHA256, over the finite field used by Curve25519
we have p = 2^255 - 19, and thus when reducing from 255 bits, the
values of 0 .. 19 will be twice as likely to occur. This is a
standard problem in generating uniformly distributed integers from a
bitstring. In this example, the resulting bias is negligible, but
for others this bias can be significant.
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To address this, our HashToBase algorithm greedily takes as many bits
as possible before reducing mod p, in order to smooth out this bias.
This is preferable to an iterated procedure, such as rejection
sampling, since this can be hard to reliably implement in constant
time.
Authors' Addresses
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
Christopher A. Wood
Apple Inc.
One Apple Park Way
Cupertino, California 95014
United States of America
Email: cawood@apple.com
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