CFRG A. Langley
Internet-Draft Google
Intended status: Informational January 6, 2015
Expires: July 10, 2015
Elliptic Curves for Security
draft-agl-cfrgcurve-00
Abstract
This memo describes an algorithm for deterministically generating
parameters for elliptic curves over prime fields offering high
practical security in cryptographic applications, including Transport
Layer Security (TLS) and X.509 certificates. It also specifies a
specific curve at the ~128-bit security level.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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This Internet-Draft will expire on July 10, 2015.
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document authors. All rights reserved.
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Table of Contents
1. Note on authorship . . . . . . . . . . . . . . . . . . . . . 2
2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
3. Requirements Language . . . . . . . . . . . . . . . . . . . . 3
4. Security Requirements . . . . . . . . . . . . . . . . . . . . 3
5. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3
6. Parameter Generation . . . . . . . . . . . . . . . . . . . . 4
6.1. Edwards Curves . . . . . . . . . . . . . . . . . . . . . 4
6.2. Twisted Edwards Curves . . . . . . . . . . . . . . . . . 5
6.3. Generators . . . . . . . . . . . . . . . . . . . . . . . 6
7. Recommended Curves . . . . . . . . . . . . . . . . . . . . . 7
8. Wire-format of field elements . . . . . . . . . . . . . . . . 8
9. Elliptic Curve Diffie-Hellman . . . . . . . . . . . . . . . . 9
9.1. Diffie-Hellman protocol . . . . . . . . . . . . . . . . . 11
10. Test vectors . . . . . . . . . . . . . . . . . . . . . . . . 11
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 12
11.1. Normative References . . . . . . . . . . . . . . . . . . 12
11.2. Informative References . . . . . . . . . . . . . . . . . 12
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 13
1. Note on authorship
This document is a merging of "draft-black-rpgecc-01" (by Benjamin
Black, Joppe W. Bos, Craig Costello, Patrick Longa and Michael
Naehrig) and "draft-turner-thecurve25519function-01" (by Watson Ladd,
Rich Salz and Sean Turner). They are the actual authors of the words
and figures, but authorship also implies support and so are not
listed as authors until they have confirmed that they support this
document. None the less, they deserve any credit for the contents.
2. Introduction
Since the initial standardization of elliptic curve cryptography
(ECC) in [SEC1] there has been significant progress related to both
efficiency and security of curves and implementations. Notable
examples are algorithms protected against certain side-channel
attacks, different 'special' prime shapes which allow faster modular
arithmetic, and a larger set of curve models from which to choose.
There is also concern in the community regarding the generation and
potential weaknesses of the curves defined in [NIST].
This memo describes a deterministic algorithm for generation of
elliptic curves for cryptography. The constraints in the generation
process produce curves that support constant-time, exception-free
scalar multiplications that are resistant to a wide range of side-
channel attacks including timing and cache attacks, thereby offering
high practical security in cryptographic applications. The
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deterministic algorithm operates without any hidden parameters,
reliance on randomness or any other processes offering opportunities
for manipulation of the resulting curves. The selection between
curve models is determined by choosing the curve form that supports
the fastest (currently known) complete formulas for each modularity
option of the underlying field prime. Specifically, the Edwards
curve x^2 + y^2 = 1 + dx^2y^2 is used with primes p with p = 3 mod 4,
and the twisted Edwards curve -x^2 + y^2 = 1 + dx^2y^2 is used for
primes p with p = 1 mod 4.
3. Requirements Language
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 RFC 2119 [RFC2119].
4. Security Requirements
For each curve at a specific security level:
1. The domain parameters SHALL be generated in a simple,
deterministic manner, without any secret or random inputs. The
derivation of the curve parameters is defined in Section 6.
2. The trace of Frobenius MUST NOT be in {0, 1} in order to rule out
the attacks described in [Smart], [AS], and [S], as in [EBP].
3. MOV Degree: the embedding degree k MUST be greater than (r - 1) /
100, as in [EBP].
4. CM Discriminant: discriminant D MUST be greater than 2^100, as in
[SC].
5. Notation
Throughout this document, the following notation is used:
p Denotes the prime number defining the underlying field.
GF(p) The finite field with p elements.
d An element in the finite field GF(p), not equal to -1 or zero.
Ed An Edwards curve: an elliptic curve over GF(p) with equation x^2 +
y^2 = 1 + dx^2y^2.
tEd A twisted Edwards curve where a=-1: an elliptic curve over GF(p)
with equation -x^2 + y^2 = 1 + dx^2y^2.
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oddDivisor The largest odd divisor of the number of GF(p)-rational
points on a (twisted) Edwards curve.
oddDivisor' The largest odd divisor of the number of GF(p)-rational
points on the non-trivial quadratic twist of a (twisted) Edwards
curve.
cofactor The cofactor of the subgroup of order oddDivisor in the
group of GF(p)-rational points of a (twisted) Edwards curve.
cofactor' The cofactor of the subgroup of order oddDivisor in the
group of GF(p)-rational points on the non-trivial quadratic twist
of a (twisted) Edwards curve.
trace The trace of Frobenius of Ed or tEd such that #Ed(GF(p)) = p +
1 - trace or #tEd(GF(p)) = p + 1 - trace, respectively.
P A generator point defined over GF(p) of prime order oddDivisor on
Ed or tEd.
X(P) The x-coordinate of the elliptic curve point P.
Y(P) The y-coordinate of the elliptic curve point P.
6. Parameter Generation
This section describes the generation of the curve parameter, namely
d, of the elliptic curve. The input to this process is p, the prime
that defines the underlying field. The size of p determines the
amount of work needed to compute a discrete logarithm in the elliptic
curve group and choosing a precise p depends on many implementation
concerns. The performance of the curve will be dominated by
operations in GF(p) and thus carefully choosing a value that allows
for easy reductions on the intended architecture is critical for
performance. This document does not attempt to articulate all these
considerations.
6.1. Edwards Curves
For p = 3 mod 4, the elliptic curve Ed in Edwards form is determined
by the non-square element d from GF(p) (not equal to -1 or zero) with
smallest absolute value such that #Ed(GF(p)) = cofactor * oddDivisor,
#Ed'(GF(p)) = cofactor' * oddDivisor', cofactor = cofactor' = 4, and
both subgroup orders oddDivisor and oddDivisor' are prime. In
addition, care must be taken to ensure the MOV degree and CM
discriminant requirements from Section 4 are met.
These cofactors are chosen because they are minimal.
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Input: a prime p, with p = 3 mod 4
Output: the parameter d defining the curve Ed
1. Set d = 0
2. repeat
repeat
if (d > 0) then
d = -d
else
d = -d + 1
end if
until d is not a square in GF(p)
Compute oddDivisor, oddDivisor', cofactor and cofactor' where #Ed(GF(p)) =
cofactor * oddDivisor, #Ed'(GF(p)) = cofactor' * oddDivisor', cofactor and
cofactor' are powers of 2 and oddDivisor, oddDivisor' are odd.
until ((cofactor = cofactor' = 4), oddDivisor is prime and oddDivisor' is prime)
3. Output d
GenerateCurveEdwards
6.2. Twisted Edwards Curves
For a prime p = 1 mod 4, the elliptic curve tEd in twisted Edwards
form is determined by the non-square element d from GF(p) (not equal
to -1 or zero) with smallest absolute value such that #tEd(GF(p)) =
cofactor * oddDivisor, #tEd'(GF(p)) = cofactor' * oddDivisor',
cofactor = 8, cofactor' = 4 and both subgroup orders oddDivisor and
oddDivisor' are prime. In addition, care must be taken to ensure the
MOV degree and CM discriminant requirements from Section 4 are met.
These cofactors are chosen so that they are minimal such that the
cofactor of the main curve is greater than the cofactor of the twist.
It's not possible in this case for the cofactors to be equal, but it
is possible for the twist cofactor to be larger. The latter is
considered dangerous because algorithms that depend on the cofactor
of the curve may be vulnerable if a point on the twist is accepted.
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Input: a prime p, with p = 1 mod 4
Output: the parameter d defining the curve tEd
1. Set d = 0
2. repeat
repeat
if (d > 0) then
d = -d
else
d = -d + 1
end if
until d is not a square in GF(p)
Compute oddDivisor, oddDivisor', cofactor, cofactor' where #tEd(GF(p)) =
cofactor * oddDivisor, #tEd'(GF(p)) = cofactor' * oddDivisor', cofactor
and cofactor' are powers of 2 and oddDivisor, oddDivisor' are odd.
until (cofactor = 8 and cofactor' = 4 and rd is prime and rd' is prime)
3. Output d
GenerateCurveTEdwards
6.3. Generators
Any point with the correct order will serve as a generator for the
group. The following algorithm computes a possible generator by
taking the smallest positive value x in GF(p) (when represented as an
integer) such that (x, y) is on the curve and such that (X(P),Y(P)) =
8 * (x, y) has large prime order oddDivisor.
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Input: a prime p and curve parameters non-square d and
a = -1 for twisted Edwards (p = 1 mod 4) or
a = 1 for Edwards (p = 3 mod 4)
Output: a generator point P = (X(P), Y(P)) of order oddDivisor
1. Set x = 0 and found_gen = false
2. while (not found_gen) do
x = x + 1
while ((1 - a * x^2) * (1 - d * x^2) is not a quadratic
residue mod p) do
x = x + 1
end while
Compute an integer s, 0 < s < p, such that
s^2 * (1 - d * x^2) = 1 - a * x^2 mod p
Set y = min(s, p - s)
(X(P), Y(P)) = 8 * (x, y)
if ((X(P), Y(P)) has order oddDivisor on Ed or tEd, respectively) then
found_gen = true
end if
end while
3. Output (X(P),Y(P))
GenerateGen
7. Recommended Curves
For the ~128-bit security level, the prime 2^255-19 is recommended
for performance over a wide-range of architectures. This prime is
congruent to 1 mod 4 and the above procedure results in the following
twisted Edwards curve, called "intermediate25519":
p 2^255-19
d 121665
order 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
cofactor 8
In order to be compatible with widespread existing practice, the
recommended curve is an isogeny of this curve. An isogeny is a
"renaming" of the points on the curve and thus cannot affect the
security of the curve:
p 2^255-19
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d 370957059346694393431380835087545651895421138798432190163887855330
85940283555
order 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed
cofactor 8
X(P) 151122213495354007725011514095885315114540126930418572060461132
83949847762202
Y(P) 463168356949264781694283940034751631413079938662562256157830336
03165251855960
The d value in the this curve is much larger than the generated curve
and this might slow down some implementations. If this is a problem
then implementations are free to calculate on the original curve,
with small d as the isogeny map can be merged into the affine
transform without any performance impact.
The latter curve is isomorphic to a Montgomery curve defined by v^2 =
u^3 + 486662u^2 + u where the maps are:
(u, v) = ((1+y)/(1-y), sqrt(-1)*sqrt(486664)*u/x)
(x, y) = (sqrt(-1)*sqrt(486664)*u/v, (u-1)/(u+1)
The base point maps onto the Montgomery curve such that u = 9, v = 14
781619447589544791020593568409986887264606134616475288964881837755586
237401.
The Montgomery curve defined here is equal to the one defined in
[curve25519] and the isomorphic twisted Edwards curve is equal to the
one defined in [ed25519].
8. Wire-format of field elements
When transmitting field elements in the Diffie-Hellman protocol
below, they MUST be encoded as an array of bytes, x, in little-endian
order such that x[0] + 256 * x[1] + 256^2 * x[2] + ... + 256^n * x[n]
is congruent to the value modulo p and x[n] is minimal. On receiving
such an array, implementations MUST mask the (8-log2(p)%8)%8 most-
significant bits in the final byte. This is done to preserve
compatibility with point formats which reserve the sign bit for use
in other protocols and to increase resistance to implementation
fingerprinting.
(NOTE: draft-turner-thecurve25519function also says "Implementations
MUST reject numbers in the range [2^255-19, 2^255-1], inclusive." but
I'm not aware of any implementations that do so.)
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9. Elliptic Curve Diffie-Hellman
This section describes how to perform Diffie-Hellman using curves
generated by the above procedure. For safety reasons, Diffie-Hellman
is performed on the Montgomery isomorphism of the curve and the
public values transmitted are u coordinates.
Let U denote the projection map from a point (u,v) on E, to u,
extended so that U of the point at infinity is zero. U is surjective
onto GF(p) if the v coordinate takes on values in GF(p) and in a
quadratic extension of GF(p).
Then DH(s, U(Q)) = U(sQ) is a function defined for all integers s and
elements U(Q) of GF(p). Proper implementations use a restricted set
of integers for s and only u-coordinates of points Q defined over
GF(p). The remainder of this section describes how to compute this
function quickly and securely, and use it in a Diffie- Hellman
scheme.
Let s be a 255 bits long integer, where s = sum s_i * 2^i with s_i in
{0, 1}.
Computing DH(s, u) is done by the following procedure, taken from
[curve25519] based on formulas from [montgomery]. All calculations
are performed in GF(p), i.e., they are performed modulo p. The
parameter a24 is a24 = (486662 - 2) / 4 = 121665.
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x_1 = u
x_2 = 0
z_2 = 1
x_3 = u
z_3 = 1
For t = 254 down to 0:
// Conditional swap; see text below.
(x_2, x_3) = cswap (s_t, x_2, x_3)
(z_2, z_3) = cswap (s_t, z_2, z_3)
A = x_2 + z_2
AA = A^2
B = x_2 - z_2
BB = B^2
E = AA - BB
C = x_3 + z_3
D = x_3 - z_3
DA = D * A
CB = C * B
x_3 = (DA + CB)^2
z_3 = x_1 * (DA - CB)^2
x_2 = AA * BB
z_2 = E * (AA + a24 * E)
// Conditional swap; see text below.
(x_2, x_3) = cswap (s_t, x_2, x_3)
(z_2, z_3) = cswap (s_t, z_2, z_3)
Return x_2 * (z_2^(p - 1))
In implementing this procedure, due to the existence of side-channels
in commodity hardware, it is important that the pattern of memory
accesses and jumps not depend on the values of any of the bits of s.
It is also important that the arithmetic used not leak information
about the integers modulo p (such as having b * c distinguishable
from c * c).
The cswap instruction SHOULD be implemented in constant time
(independent of s_t) as follows:
cswap(s_t, x_2, x_3)
dummy = s_t * (x_2 - x_3)
x_2 = x_2 - dummy
x_3 = x_3 + dummy
Return (x_2, x_3)
where s_t is 1 or 0. Alternatively, an implementation MAY use the
following:
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cswap(s_t, x_2, x_3)
dummy = mask(s_t) AND (x_2 XOR x_3)
x_2 = x_2 XOR dummy
x_3 = x_3 XOR dummy
Return (x_2, x_3)
where mask(s_t) is the all-1 or all-0 word of the same length as x_2
and x_3, computed, e.g., as mask(s_t) = 1 - s_t. The latter version
is often more efficient.
9.1. Diffie-Hellman protocol
The DH function can be used in an ECDH protocol with the recommended
curve as follows:
Alice generates 32 random bytes in f[0] to f[31]. She masks the
three rightmost bits of f[0] and the leftmost bit of f[31] to zero
and sets the second leftmost bit of f[31] to 1. This means that f is
of the form 2^254 + 8 * {0, 1, ..., 2^(251) - 1} as a little-endian
integer.
Alice then transmits K_A = DH(f, 9) to Bob, where 9 is the number 9.
Bob similarly generates 32 random bytes in g[0] to g[31], applies the
same masks, computes K_B = DH(g, 9) and transmits it to Alice.
Alice computes DH(f, DH(g, 9)); Bob computes DH(g, DH(f, 9)) using
their generated values and the received input.
Both of them now share K = DH(f, DH(g, 9)) = DH(g, DH(f, 9)) as a
shared secret. Alice and Bob can then use a key-derivation function,
such as hashing K, to compute a key.
10. Test vectors
The following test vectors are taken from [nacl]. All numbers are
shown as little-endian hexadecimal byte strings:
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Alice's private key, f:
77 07 6d 0a 73 18 a5 7d 3c 16 c1 72 51 b2 66 45
df 4c 2f 87 eb c0 99 2a b1 77 fb a5 1d b9 2c 2a
Alice's public key, DH(f, 9):
85 20 f0 09 89 30 a7 54 74 8b 7d dc b4 3e f7 5a
0d bf 3a 0d 26 38 1a f4 eb a4 a9 8e aa 9b 4e 6a
Bob's private key, g:
5d ab 08 7e 62 4a 8a 4b 79 e1 7f 8b 83 80 0e e6
6f 3b b1 29 26 18 b6 fd 1c 2f 8b 27 ff 88 e0 eb
Bob's public key, DH(g, 9):
de 9e db 7d 7b 7d c1 b4 d3 5b 61 c2 ec e4 35 37
3f 83 43 c8 5b 78 67 4d ad fc 7e 14 6f 88 2b 4f
Their shared secret, K:
4a 5d 9d 5b a4 ce 2d e1 72 8e 3b f4 80 35 0f 25
e0 7e 21 c9 47 d1 9e 33 76 f0 9b 3c 1e 16 17 42
11. References
11.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
11.2. Informative References
[AS] Satoh, T. and K. Araki, "Fermat quotients and the
polynomial time discrete log algorithm for anomalous
elliptic curves", 1998.
[EBP] ECC Brainpool, "ECC Brainpool Standard Curves and Curve
Generation", October 2005, .
[NIST] National Institute of Standards, "Recommended Elliptic
Curves for Federal Government Use", July 1999,
.
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[S] Semaev, I., "Evaluation of discrete logarithms on some
elliptic curves", 1998.
[SC] Bernstein, D. and T. Lange, "SafeCurves: choosing safe
curves for elliptic-curve cryptography", June 2014,
.
[SEC1] Certicom Research, "SEC 1: Elliptic Curve Cryptography",
September 2000,
.
[Smart] Smart, N., "The discrete logarithm problem on elliptic
curves of trace one", 1999.
[curve25519]
Bernstein, D., "Curve25519 -- new Diffie-Hellman speed
records", 2006,
.
[ed25519] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
Yang, "High-speed high-security signatures", 2011,
.
[montgomery]
Montgomery, P., "Speeding the Pollard and elliptic curve
methods of factorization", 1983,
.
[nacl] Bernstein, D., "Cryptography in NaCl", 2009,
.
Author's Address
Adam Langley
Google
345 Spear St
San Francisco, CA 94105
US
Email: agl@google.com
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