Network Working Group W. Ladd
Internet-Draft Grad Student UC Berkley
Intended status: Informational R. Salz
Expires: February 13, 2015 Akamai Technologies
S. Turner
IECA, Inc.
August 12, 2014
The Curve25519 Function
draft-turner-thecurve25519function-01
Abstract
This document specifies the Curve25519 function, an ECDH (Elliptic-
Curve Diffie-Hellman) key-agreement scheme for use in cryptographic
applications. It was designed with performance and security in mind.
This document is based on information in the public domain.
Status of This Memo
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the Trust Legal Provisions and are provided without warranty as
described in the Simplified BSD License.
1. Introduction
This document specifies the Curve25519 function, an ECDH (Elliptic-
curve Diffie-Hellman) key-agreement scheme for use in cryptographic
applications. It was designed with performance and security in mind.
This document is based on information in the public domain.
This document provides a stable reference for the Curve25519 function
[Curve25519] to which other specifications may refer when defining
their use of Curve25519. It specifies how to use Curve25519 for key
exchange. This document defines the algorithm, the "wire format"
(how to serialize and parse bytes sent over a network, for example),
and provides some implementation guidance to avoid known side-channel
timing exposures.
This document does not specify the use of Curve25519 in any other
specific protocol, such as TLS (Transport Layer Security) or IPsec
(Internet Protocol Security). It does not specify how to use
Curve25519 for digital signatures.
Readers are assumed to be familiar with the concepts of elliptic
curves, modular arithmetic, group operations, and finite fields
[RFC6090] as well as rings [Curve25519].
1.1. Terminology
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. Notation and Definitions
The following notation and definitions are used in this document
(notation is to the left of the ":"):
A: A value used in the elliptic-curve equation E.
E: An elliptic-curve equation.
p: A prime.
GF(p): The field with p elements.
_#: Subscript notation, where # is a number or letter.
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Q
=: Assignment.
^: Exponentiation.
+, -, *, /: Addition, subtraction, multiplication, and division,
respectively.
Note that all operations are performed modulo p.
3. The Curve25519 Function
Let p = 2^255 - 19. Let E be the elliptic curve with the equation
y^2 = x^3 + 486662 * x^2 + x over GF(p).
Each element x of GF(p) has a unique little-endian representation as
32 bytes x[0] ... x[31], such that x[0] + 256 * x[1] + 256^2 * x[2] +
... + 256^31 * x[31] is congruent to x modulo p, and x[31] is
minimal. Implementations MUST only produce points in this form. On
receiving a point, implementations MUST mask the leftmost bit of byte
31 to zero. This is done to preserve compatibility with point
formats which reserve the sign bit for use in other protocols and
increase resistance to implementation fingerprinting.
Implementations MUST reject numbers in the range [2^255-19, 2^255-1],
inclusive.
Let X denote the projection map from a point (x,y) on E, to x,
extended so that X of the point at infinity is zero. X is surjective
onto GF(p) if the y coordinate takes on values in GF(p) and in a
quadratic extension of GF(p).
Then Curve25519(s, X(Q)) = X(sQ) is a function defined for all
integers s and elements X(Q) of GF(p). Proper implementations use a
restricted set of integers for s and only x-coordinates of points Q
defined over GF(p). The remainder of this document describes how to
compute this function quickly and securely, and use it in a Diffie-
Hellman scheme.
4. Implementing the Curve25519 Function
Let s be a 255 bits long integer, where
s = sum s_i * 2^i with s_i in {0, 1}.
Computing Curve25519(s, x) is done by the following procedure, taken
from [Curve25519] based on formulas from [Mont]. 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 = x
x_2 = 0
z_2 = 1
x_3 = x
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:
dummy = mask(s_t) AND (x_2 XOR x_3)
x_2 = x_2 XOR dummy
x_3 = x_3 XOR dummy
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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.
5. Use of the Curve25519 function
The Curve25519 function can be used in an ECDH protocol 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 = Curve25519(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 = Curve25519(g, 9) and transmits it to
Alice.
Alice computes Curve25519(f, Curve25519(g, 9)); Bob computes
Curve25519(g, Curve25519(f, 9)) using their generated values and the
received input.
Both of them now share K = Curve25519(f, Curve25519(g, 9)) =
Curve25519(g, Curve25519(f, 9)) as a shared secret. Alice and Bob
can then use a key-derivation function, such as hashing K, to compute
a key.
6. Test Vectors
The following test vectors are taken from [NaCl]. All numbers are
shown as little-endian hexadecimal byte strings:
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, Curve25519(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:
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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, Curve25519(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
7. Security Considerations
Curve25519 meets all standard assumptions on DH and DLP difficulty.
In addition, Curve25519 is twist secure: the co-factor of the curve
is 8, that of the twist is 4. Protocols that require contributory
behavior must ban outputs K_A = 0, K_B = 0 or K = 0.
Curve25519 is designed to enable very high performance software
implementations, thus reducing the cost of highly secure cryptography
to a point where it can be used more widely.
8. IANA Considerations
None.
9. Acknowledgements
We would like to thank Tanja Lange (Technische Universiteit
Eindhoven) for her review and comments.
10. References
10.1. Normative References
[Curve25519]
Bernstein, D., "Curve25519 - new Diffie-Hellman speed
records", April 2006,
.
[Mont] Montgomery, P., "Speeding the Pollard and elliptic curve
methods of factorization", 1983,
.
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[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC6090] McGrew, D., Igoe, K., and M. Salter, "Fundamental Elliptic
Curve Cryptography Algorithms", RFC 6090, February 2011.
10.2. Informative References
[NaCl] Bernstein, D., "Cryptography in NaCl", 2013,
.
Authors' Addresses
Watson Ladd
Grad Student UC Berkley
Email: watsonbladd@gmail.com
Rich Salz
Akamai Technologies
8 Cambridge Center
Cambridge, MA 02142
USA
Phone: +1-617-714-6169
Email: rsalz@akamai.com
Sean Turner
IECA, Inc.
Suite 106
Fairfax, VA 22031
USA
Phone: +1-703-628-3180
Email: turners@ieca.com
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