CFRG | A. Langley |
Internet-Draft | |
Intended status: Informational | R. Salz |
Expires: August 1, 2015 | Akamai Technologies |
S. Turner | |
IECA, Inc. | |
January 28, 2015 |
Elliptic Curves for Security
draft-irtf-cfrg-curves-01
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.
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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, various '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 generating cryptographic elliptic curves over a given prime field. 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 deterministic algorithm operates without any input parameters that would permit 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 when p = 1 mod 4.
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].
For each curve at a specific security level:
Throughout this document, the following notation is used:
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. This document does not attempt to articulate all these considerations.
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 3 are met.
These cofactors are chosen because they are minimal.
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
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 3 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. For 1 mod 4 primes, the cofactors are never equal. If the cofactor of the twist is larger than the cofactor of the curve, algorithms may be vulnerable to a small-subgroup attack if a point on the twist is incorrectly accepted.
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
For the ~128-bit security level, the prime 2^255-19 is recommended for performance on 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:
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:
The d value in 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 = 14781619447589544791020593568409986887264606134616475288964881837755586237401.
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].
The curve25519 function performs scalar multiplication on the Montgomery form of the above curve. (This is used when implementing Diffie-Hellman.) The function takes a scalar and a u-coordinate as inputs and produces a u-coordinate as output. Although the function works internally with integers, the inputs and outputs are 32-byte strings and this specification defines their encoding.
U-coordinates are elements of the underlying field GF(2^255-19) and are encoded as an array of bytes, u, in little-endian order such that u[0] + 256 * u[1] + 256^2 * u[2] + ... + 256^n * u[n] is congruent to the value modulo p and u[n] is minimal. When receiving such an array, implementations MUST mask the most-significant bit 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.
For example, the following functions implement this in Python, although the Python code is not intended to be performant nor side-channel free:
def decodeLittleEndian(b): return sum([b[i] << 8*i for i in range(32)]) def decodeUCoordinate(u): u_list = [ord(b) for b in u] u_list[31] &= 0x7f return decodeLittleEndian(u_list) def encodeUCoordinate(u): u = u % p return ''.join([chr((u >> 8*i) & 0xff) for i in range(32)])
(EDITORS 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.)
Scalars are assumed to be randomly generated bytes. In order to decode 32 bytes into an integer scalar, set the three least significant bits of the first byte and the most significant bit of the last to zero, set the second most significant bit of the last byte to 1 and, finally, decode as little-endian. This means that resulting integer is of the form 2^254 + 8 * {0, 1, ..., 2^(251) - 1}.
def decodeScalar(k): k_list = [ord(b) for b in k] k_list[0] &= 248 k_list[31] &= 127 k_list[31] |= 64 return decodeLittleEndian(k_list)
To implement the curve25519(k, u) function (where k is the scalar and u is the u-coordinate) first decode k and u and then perform the following procedure, taken from [curve25519] and based on formulas from [montgomery]. All calculations are performed in GF(p), i.e., they are performed modulo p. The constant a24 is (486662 - 2) / 4 = 121665.
x_1 = u x_2 = 1 z_2 = 0 x_3 = u z_3 = 1 swap = 0 For t = 254 down to 0: k_t = (k >> t) & 1 swap ^= k_t // Conditional swap; see text below. (x_2, x_3) = cswap(swap, x_2, x_3) (z_2, z_3) = cswap(swap, z_2, z_3) swap = k_t 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(swap, x_2, x_3) (z_2, z_3) = cswap(swap, z_2, z_3) Return x_2 * (z_2^(p - 2))
(TODO: Note the difference in the formula from Montgomery's original paper. See https://www.ietf.org/mail-archive/web/cfrg/current/msg05872.html.)
Finally, encode the resulting value as 32 bytes in little-endian order.
When 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 k. It is also important that the arithmetic used not leak information about the integers modulo p (such as having b*c be distinguishable from c*c).
The cswap instruction SHOULD be implemented in constant time (independent of swap) as follows:
cswap(swap, x_2, x_3): dummy = swap * (x_2 - x_3) x_2 = x_2 - dummy x_3 = x_3 + dummy Return (x_2, x_3)
where swap is 1 or 0. Alternatively, an implementation MAY use the following:
cswap(swap, x_2, x_3): dummy = mask(swap) 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(swap) is the all-1 or all-0 word of the same length as x_2 and x_3, computed, e.g., as mask(swap) = 1 - swap. The latter version is often more efficient.
Input scalar: a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4 Input scalar as a number (base 10): 31029842492115040904895560451863089656472772604678260265531221036453811406496 Input U-coordinate: e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c Input U-coordinate as a number: 34426434033919594451155107781188821651316167215306631574996226621102155684838 Output U-coordinate: c3da55379de9c6908e94ea4df28d084f32eccf03491c71f754b4075577a28552 Input scalar: 4b66e9d4d1b4673c5ad22691957d6af5c11b6421e0ea01d42ca4169e7918ba0d Input scalar as a number (base 10): 35156891815674817266734212754503633747128614016119564763269015315466259359304 Input U-coordinate: e5210f12786811d3f4b7959d0538ae2c31dbe7106fc03c3efc4cd549c715a493 Input U-coordinate as a number: 8883857351183929894090759386610649319417338800022198945255395922347792736741 Output U-coordinate: 95cbde9476e8907d7aade45cb4b873f88b595a68799fa152e6f8f7647aac7957
The curve25519 function can be used in an ECDH protocol as follows:
Alice generates 32 random bytes in f[0] to f[31] and transmits K_A = curve25519(f, 9) to Bob, where 9 is the u-coordinate of the base point and is encoded as a byte with value 9, followed by 31 zero bytes.
Bob similarly generates 32 random bytes in g[0] to g[31] and computes K_B = curve25519(g, 9) and transmits it to Alice.
Alice computes curve25519(f, K_B); Bob computes curve25519(g, K_A) using their generated values and the received input.
Both 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.
Note that this Diffie-Hellman protocol is not contributory, e.g. if the u-coordinate is zero then the output will always be zero. A contributory Diffie-Hellman function would ensure that the output was unpredictable no matter what the peer's input. This is not a problem for the vast majority of cases but, if a contributory function is specifically required, then curve25519 should not be used.
Alice's private key, f: 77076d0a7318a57d3c16c17251b26645df4c2f87ebc0992ab177fba51db92c2a Alice's public key, curve25519(f, 9): 8520f0098930a754748b7ddcb43ef75a0dbf3a0d26381af4eba4a98eaa9b4e6a Bob's private key, g: 5dab087e624a8a4b79e17f8b83800ee66f3bb1292618b6fd1c2f8b27ff88e0eb Bob's public key, curve25519(g, 9): de9edb7d7b7dc1b4d35b61c2ece435373f8343c85b78674dadfc7e146f882b4f Their shared secret, K: 4a5d9d5ba4ce2de1728e3bf480350f25e07e21c947d19e3376f09b3c1e161742
This document merges draft-black-rpgecc-01 and draft-turner-thecurve25519function-01. The following authors of those documents wrote much of the text and figures but are not listed as authors on this document: Benjamin Black, Joppe W. Bos, Craig Costello, Patrick Longa, Michael Naehrig and Watson Ladd.
The authors would also like to thank Tanja Lange and Rene Struik for their reviews.
[RFC2119] | Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, March 1997. |
[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. |
[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. |