Network Working Group N. Sullivan
Internet-Draft Cloudflare
Intended status: Informational C. Wood
Expires: September 6, 2018 Apple Inc.
March 05, 2018
Hashing to Elliptic Curves
draft-sullivan-hash-to-curve-00
Abstract
This document specifies a number of algorithms that may be used to
hash arbitrary strings to Elliptic Curves.
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
2. Algorithm Recommendations . . . . . . . . . . . . . . . . . . 3
3. Generic Interface . . . . . . . . . . . . . . . . . . . . . . 4
3.1. Utility Functions . . . . . . . . . . . . . . . . . . . . 4
4. Hashing Variants . . . . . . . . . . . . . . . . . . . . . . 5
4.1. Icart Method . . . . . . . . . . . . . . . . . . . . . . 5
4.2. Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . . 6
4.3. Simplified SWU Method . . . . . . . . . . . . . . . . . . 6
4.4. Elligator2 Method . . . . . . . . . . . . . . . . . . . . 8
5. Curve Transformations . . . . . . . . . . . . . . . . . . . . 10
6. Cost Comparison . . . . . . . . . . . . . . . . . . . . . . . 10
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11
8. Security Considerations . . . . . . . . . . . . . . . . . . . 11
9. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 11
10. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 11
11. Normative References . . . . . . . . . . . . . . . . . . . . 11
Appendix A. Try-and-Increment Method . . . . . . . . . . . . . . 13
Appendix B. Sample Code . . . . . . . . . . . . . . . . . . . . 13
B.1. Icart Method . . . . . . . . . . . . . . . . . . . . . . 13
B.2. Shallue-Woestijne-Ulas Method . . . . . . . . . . . . . . 14
B.3. Simplified SWU Method . . . . . . . . . . . . . . . . . . 15
B.4. Elligator2 Method . . . . . . . . . . . . . . . . . . . . 16
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 17
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], and Boneh-
Lynn-Shacham signatures [BLS01].
Let E be an elliptic curve over base field GF(p). In practice,
efficient (polynomial-time) functions that hash arbitrary input to E
can be constructed by composing a cryptographically secure hash
function F1 : {0,1}^* ->GF(p) and an injection F2 : GF(p) -> E, i.e.,
Hash(m) = F2(F1(m)). Probabilistic constructions of Hash, e.g., the
MapToGroup function described by Boneh et al. [BLS01]. Their
algorithm fails with probability 2^I, where I is a tunable parameter
that one can control. Another variant, dubbed the "Try and
Increment" approach, was described by Boneh et al. [BLS01]. This
function works by hashing input m using a standard hash function,
e.g., SHA256, and then checking to see if the resulting point E(m,
f(m)), for curve function f, belongs on E. This algorithm is
expected to find a valid curve point after approximately two
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attempts, i.e., when ctr=1, on average. (See Appendix Appendix A for
a more detailed description of this algorithm.) Since the running
time of the 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.
Shallue and Woestijne [SWU] first introduced a deterministic
algorithm that maps elements in F_{q} to an EC 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].
This document specifies several algorithms for deterministically
hashing onto a curve with varying properties: Icart, SWU, Simplified
SWU, and Elligator2. Each algorithm conforms to a common interface,
i.e., it maps an element from a base field F to a curve E. For each
variant, we describe the requirements for F and 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. Algorithm Recommendations
The following table lists recommended algorithms to use for specific
curves.
+------------+------------------------+
| Curve | Algorithm |
+------------+------------------------+
| P-256 | SWU Section 4.3 |
| | |
| P-384 | Icart Section 4.1 |
| | |
| Curve25519 | Elligator2 Section 4.4 |
| | |
| Curve448 | Elligator2 Section 4.4 |
+------------+------------------------+
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The SWU variant from Section Section 4.2 applies to any curve. As
such, this algorithm SHOULD be used if no other better alternative is
known. More efficient variants and their curve requirements are
shown in the table below. These MAY be used if the target curve
meets the listed criteria.
+--------------------+----------------------------------------------+
| Algorithm | Requirement |
+--------------------+----------------------------------------------+
| Icart Section 4.1 | p = 2 mod 3 |
| | |
| SWU Section 4.2 | None |
| | |
| Simplified SWU | p = 3 mod 4 |
| Section 4.3 | |
| | |
| Elligator2 Section | p is large and there is a point of order two |
| 4.4 | and j-invariant != 1728 |
+--------------------+----------------------------------------------+
3. Generic Interface
The generic interface for hashing to elliptic curves is as follows:
hash_to_curve(alpha)
where alpha is a message to hash onto a curve.
3.1. Utility Functions
Algorithms in this document make use of utility functions described
below.
o HashToBase(x): H(x)[0:log2(p) + 1], i.e., hash-truncate-reduce,
where H is a cryptographic hash function, such as SHA256, and p is
the prime order of base field Fp.
o CMOV(a, b, c): If c = 1, return a, else return b.
Note: We assume that HashToBase maps its input to the base field
uniformly. In practice, there may be inherent biases in p, e.g., p =
2^k - 1 will have non-negligible bias in higher bits.
((TODO: expand on this problem))
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4. Hashing Variants
4.1. Icart Method
The following hash_to_curve_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
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)
x = (v^2 - b - (u^6 / 27))^(1/3) + (u^2 / 3)
y = ux + v
where v = ((3A - u^4) / 6u).
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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hash_to_curve_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
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 (mod p) // 2p - 1
16. t1 = t1 / 3 (mod p) // (2p - 1)/3
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)
4.2. Shallue-Woestijne-Ulas Method
((TODO: write this section))
4.3. Simplified SWU Method
The following hash_to_curve_simple_swu(alpha) implements the
simplfied Shallue-Woestijne-Ulas algorithm from [SimpleSWU]. This
algorithm works for any curve over F_{p^n}, where p = 3 mod 4,
including:
o P256
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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. z = t^3 * g(alpha)
5. Output (-g * alpha) * (g * beta)
The following procedure implements this algorithm. It outputs a
point with affine coordinates.
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hash_to_curve_simple_swu(alpha)
Input:
alpha - value to be hashed, 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 = (y1 ^ 2 == h2)
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)
4.4. Elligator2 Method
The following hash_to_curve_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 f(x) = y^2 = x(x^2 + Ax + B), i.e., a Montgomery form with
the point of order 2 at (0,0), 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. If f(-A/(1+ur^2)) is square, then output f(-A/(1+ur^2))^(1/2)
3. Else, output f(-Aur^2/(1+ur^2))^(1/2)
Another way to express this algorithm is as follows:
1. r = HashToBase(alpha)
2. d = -A / (1 + ur^2)
3. e = f(d)^((p-1)/2)
4. u = ed - (1 - e)A/u
Here, e is the Legendre symbol of y = (d^3 + Ad^2 + d), which will be
1 if y is a quadratic residue (square) mod p, and -1 otherwise.
(Note that raising y to ((p -1) / 2) is a common way to compute the
Legendre symbol.)
The following procedure implements this algorithm.
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hash_to_curve_elligator2(alpha)
Input:
alpha - value to be hashed, an octet string
u - fixed non-square value in Fp.
f() - Curve function
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 symbol
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. u = v - v2 (mod p)
19. Output (u, f(u))
Elligator2 can be simplified with projective coordinates.
((TODO: write this variant))
5. Curve Transformations
((TODO: write this section))
6. Cost Comparison
The following table summarizes the cost of each hash_to_curve
variant. We express this cost in terms of additions (A),
multiplications (M), squares (SQ), and square roots (SR).
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((TODO: finish this section))
+--------------------------+-------------------+
| Algorithm | Cost (Operations) |
+--------------------------+-------------------+
| hash_to_curve_icart | TODO |
| | |
| hash_to_curve_swu | TODO |
| | |
| hash_to_curve_simple_swu | TODO |
| | |
| hash_to_curve_elligator2 | TODO |
+--------------------------+-------------------+
7. IANA Considerations
This document has no IANA actions.
8. Security Considerations
Each hash 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. Some
variants variants are not 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.
9. 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.
10. Contributors
o Sharon Goldberg
Boston University
goldbe@cs.bu.edu
11. Normative References
[BLS01] "Short signatures from the Weil pairing", n.d.,
.
[BMP00] "Provably secure password-authenticated key exchange using
diffie-hellman", n.d..
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[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.,
.
[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, .
[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..
[SWU] "Rational points on certain hyperelliptic curves over
finite fields", n.d., .
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Appendix A. 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 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
3. h = "INVALID"
4. While h is "INVALID" or h is EC point at infinity:
A. CTR = I2OSP(ctr, 4)
B. ctr = ctr + 1
C. attempted_hash = Hash(m || CTR)
D. h = RS2ECP(attempted_hash)
E. 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 is a
function that converts of a random 2n-octet string to an EC point as
specified in Section 5.1.3 of [RFC8032].
Appendix B. Sample Code
B.1. Icart Method
The following Sage program implements hash_to_curve_icart(alpha) for
P-384.
p = 394020061963944792122790401001436138050797392704654466679482934042 \
45721771496870329047266088258938001861606973112319
F = GF(p)
A = p - 3
B = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875a \
c656398d8a2ed19d2a85c8edd3ec2aef
q = 394020061963944792122790401001436138050797392704654466679469052796 \
27659399113263569398956308152294913554433653942643
E = EllipticCurve([F(A), F(B)])
g = E(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a \
385502f25dbf55296c3a545e3872760ab7, \
0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c0 \
0a60b1ce1d7e819d7a431d7c90ea0e5f)
E.set_order(q)
def icart(u):
u = F(u)
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v = (3*A - u^4)//(6*u)
x = (v^2 - B - u^6/27)^((2*p-1)//3) + u^2/3
y = u*x + v
return E(x, y)
def icart_straight(u):
u = F(u)
u2 = u ^ 2
t2 = u2 ^ 2
assert t2 == u^4
v1 = 3 * A
v1 = v1 - t2
t1 = 6 * u
t3 = t1 ^ (-1)
v = v1 * t3
assert v == (3 * A - u^4) // (6 * u)
x = v ^ 2
x = x - B
assert x == (v^2 - B)
t1 = F(27) ^ (-1)
t1 = t1 * u2
t1 = t1 * t2
assert t1 == ((u^6) / 27)
x = x - t1
t1 = (2 * p) - 1
t1 = t1 / 3
assert t1 == ((2*p) - 1) / 3
x = x ^ t1
t2 = u2 / 3
x = x + t2
y = u * x
y = y + v
return E(x, y)
B.2. Shallue-Woestijne-Ulas Method
((TODO: write this section))
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B.3. Simplified SWU Method
The following Sage program implements hash_to_curve_swu(alpha) for
P-256.
p = 115792089210356248762697446949407573530086143415290314195533631308 \
867097853951
F = GF(p)
A = F(p - 3)
B = F(ZZ("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2 \
604b", 16))
E = EllipticCurve([A, B])
def simple_swu(alpha):
t = F(alpha)
alpha = -(t^2)
frac = (1 / (alpha^2 + alpha))
x2 = (-B / A) * (1 + frac)
x3 = alpha * x2
h2 = x2^3 + A * x2 + B
h3 = x3^3 + A * x3 + B
if is_square(h2):
return E(x2, h2^((p + 1) // 4))
else:
return E(x3, h3^((p + 1) // 4))
def simple_swu_straight(alpha):
t = F(alpha)
alpha = t^2
alpha = alpha * -1
right = alpha^2 + alpha
right = right^(-1)
right = right + 1
left = B * -1
left = left / A
x2 = left * right
x3 = alpha * x2
h2 = x2 ^ 3
i2 = x2 * A
i2 = i2 + B
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h2 = h2 + i2
h3 = x3 ^ 3
i3 = x3 * A
i3 = i3 + B
h3 = h3 + i3
y1 = h2^((p + 1) // 4)
y2 = h3^((p + 1) // 4)
# Is it square?
e = y1^2 == h2
x = x2
if e != 1:
x = x3
y = y1
if e != 1:
y = y2
return E(x, y)
B.4. Elligator2 Method
The following Sage program implements hash_to_curve_elligator2(alpha)
for Curve25519.
p = 2**255 - 19
F = GF(p)
A = 486662
B = 1
E = EllipticCurve(F, [0, A, 0, 1, 0])
def curve25519(x):
return x^3 + (A * x^2) + x
def elligator2(alpha):
r = F(alpha)
# u is a fixed nonsquare value, eg -1 if p==3 mod 4.
u = F(2) # F(2)
assert(not u.is_square())
# If f(-A/(1+ur^2)) is square, return its square root.
# Else, return the square root of f(-Aur^2/(1+ur^2)).
x = -A / (1 + (u * r^2))
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y = curve25519(x)
if y.is_square(): # is this point square?
y = y.square_root()
else:
x = (-A * u * r^2) / (1 + (u * r^2))
y = curve25519(x).square_root()
return (x, curve25519(x))
def elligator2_straight(alpha):
r = F(alpha)
r = r^2
r = r * 2
r = r + 1
r = r^(-1)
v = A * r
v = v * -1 # d
v2 = v^2
v3 = v * v2
e = v3 + v
v2 = v2 * A
e = v2 + e
# Legendre symbol
e = e^((p - 1) / 2)
nv = v * -1
if e != 1:
v = nv
v2 = 0
if e != 1:
v2 = A
u = v - v2
return (u, curve25519(u))
Authors' Addresses
Sullivan & Wood Expires September 6, 2018 [Page 17]
Internet-Draft hash-to-curve March 2018
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
Sullivan & Wood Expires September 6, 2018 [Page 18]