Network Working Group W. Ladd
Internet-Draft Grad Student UC Berkley
Intended status: Informational R. Salz
Expires: January 30, 2015 Akamai
S. Turner
IECA, Inc.
July 29, 2014
The Curve25519 Function
draft-turner-thecurve25519function-00
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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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 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). This document
specifies how to use Curve25519 for key exchange; it does not specify
how to use Curve25519 for use with digital signatures. This document
defines the algorithm, expected "wire format," and provides some
implementation guidance to avoid known side-channel exposures.
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.
mod: An abbreviation for modulo.
_#: Subscript notation, where # is a number or letter
=: Denotes equal to.
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^: Denotes exponentiation.
+, -, *, /: Denotes addition, subtraction, multiplication, and
division.
Note that all operations are performed mod 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 s[0] ... s[31], such that
s[0]+256_s[1]+256^2_s[2]+...+256^31*s[31] is congruent to x modulo p,
and s[31] is minimal. Implementations MUST only produce points in
this form, and MUST mask the high bit of byte 31 to zero on receiving
a point. The high bit is, following convention, 0x80.
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 elements
of 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 Curve25519
Let s be a 255 bits long integer, where s=sum s_i2^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 done over GF(p), i.e., they are performed modulo p. The
parameter a24 is a24 = (486662 - 2)/4 = 121665.
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Let x_1 = 1
x_2 = 1
z_2 = 0
x_3 = x
z_3 = 1
For t = 254 to 0:
Do constant time conditional swap of:
(x_2, z_2) and (x_3, z_3) if s_t is set
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)
Do constant time conditional swap of:
(x_2, z_2) and (x_3, z_3) if s_t is set
Return x_2*(z_2^(p-1))
In implementing this procedure, due to the existence of side-channels
in commodity hardware, it is vital that the pattern of memory
accesses and jumps not depend on the bits of s. It is also essential
that the arithmetic used not leak information about words.
To compute the conditional swap in constant time (independent of s_t)
use dummy = s_t*(x_2-x_3) x_2 = x_2 - dummy x_3 = x_3 + dummy where
s_t is 1 or 0, or dummy = s_t & (x_2 XOR x_3) x_2 = x_2 XOR x_3 x_3 =
x_3 XOR x_2 where s_t is regarded as the all-1 word of 255 bits. The
latter version is more efficient on most architectures.
5. Use of the Curve25519 function
The Curve25519 function can be used in an ECDH protocol as follows:
Alice takes 32 random bytes in s[0] to s[32]. She masks the lower
three bits of s[0] and the top bit of s[31] to zero and sets the
second top most bit of s[31] to 1. This means that s is of the form
2^254+8*{0,1, ...., 2^(251)-1} as a little-endian integer.
Alice then transmits K_A = Curve25519(s, 9) to Bob, where 9 is the
number 9. As a sequence of 32 bytes, t, the representation of 9 is
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t[0]=9, and the remaining bytes are all zero. The natural wire-
format representation of the value is in little-endian byte order.
Bob picks a random g, and computes K_B = Curve25519(g, 9) similarly,
and transmits it to Alice.
Alice computes Curve25519(s, Curve25519(g, 9)); Bob computes
Curve25519(g, Curve25519(s, 9)) using their secret values and the
received input.
Both of them now share K=Curve25519(s, Curve25519(g,
9))=Curve25519(g, Curve25519(s, 9)) as a shared secret. Alice and
Bob use a key-derivation function, such as hashing K, to compute a
shared secret.
6. Test Vectors
The following test vectors are taken from [NaCl]:
Alice's public key:
0x8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a
Alice's secret key
0x77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a
Bob's public key:
0xde9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f
Bob's secret key:
0x5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb
Shared secret:
0x4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742
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.
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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,
<http://www.iacr.org/cryptodb/archive/2006/
PKC/3351/3351.pdf>.
[Mont] Montgomery, P., "Speeding the Pollard and elliptic curve
methods of factorization", 1983,
<http://www.ams.org/journals/mcom/1987-48-177/
S0025-5718-1987-0866113-7/S0025-5718-1987-0866113-7.pdf>.
[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,
<http://cr.yp.to/highspeed/naclcrypto-20090310.pdf>.
Authors' Addresses
Watson Ladd
Grad Student UC Berkley
Email: watsonbladd@gmail.com
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Rich Salz
Akamai
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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