| CFRG | A. Langley |
| Internet-Draft | |
| Intended status: Informational | R. Salz |
| Expires: September 25, 2015 | Akamai Technologies |
| S. Turner | |
| IECA, Inc. | |
| March 24, 2015 |
Elliptic Curves for Security
draft-irtf-cfrg-curves-02
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 and a specific curve at the ~224-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 A, of the Montgomery curve y^2 = x^3 + Ax^2 + x. 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.
The value (A-2)/4 is used in several of the elliptic curve point arithmetic formulas. For simplicity and performance reasons, it is beneficial to make this constant small, i.e. to choose A so that (A-2) is a small integer which is divisible by four.
For primes congruent to 1 mod 4, the minimal cofactors of the curve and its twist are either {4, 8} or {8, 4}. We choose a curve with the latter cofactors so that any algorithms that take the cofactor into account don't have to worry about checking for points on the twist, because the twist cofactor is larger.
To generate the Montgomery curve we find the minimal, positive A value such (A-2) is divisible by four and where the cofactors are as desired. The find1Mod4 function in the following Sage script returns this value given p:
def findCurve(prime, curveCofactor, twistCofactor):
F = GF(prime)
for A in xrange(1, 1e9):
if (A-2) % 4 != 0:
continue
try:
E = EllipticCurve(F, [0, A, 0, 1, 0])
except:
continue
order = E.order()
twistOrder = 2*(prime+1)-order
if (order % curveCofactor == 0 and is_prime(order // curveCofactor) and
twistOrder % twistCofactor == 0 and is_prime(twistOrder // twistCofactor)):
return A
def find1Mod4(prime):
assert((prime % 4) == 1)
return findCurve(prime, 8, 4)
Generating a curve where p = 1 mod 4
For a prime congruent to 3 mod 4, both the curve and twist cofactors can be 4 and this is minimal. Thus we choose the curve with these cofactors and minimal, positive A such that (A-2) is divisible by four. The find3Mod4 function in the following Sage script returns this value given p:
def find3Mod4(prime):
assert((prime % 4) == 3)
return findCurve(prime, 4, 4)
Generating a curve where p = 3 mod 4
The base point for a curve is the point with minimal, positive u value that is in the correct subgroup. The findBasepoint function in the following Sage script returns this value given p and A:
def findBasepoint(prime, A):
F = GF(prime)
E = EllipticCurve(F, [0, A, 0, 1, 0])
for uInt in range(1, 1e3):
u = F(uInt)
v2 = u^3 + A*u^2 + u
if not v2.is_square():
continue
v = v2.sqrt()
point = E(u, v)
order = point.order()
if order > 8 and order.is_prime():
return point
Generating the base point
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 Montgomery curve, called curve25519:
The base point is u = 9, v = 14781619447589544791020593568409986887264606134616475288964881837755586237401.
This curve is isomorphic to the twisted Edwards curve -x^2 + y^2 = 1 - d*x^2*y^2 where:
The isomorphism 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 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].
For the ~224-bit security level, the prime 2^448-2^224-1 is recommended for performance on a wide-range of architectures. This prime is congruent to 3 mod 4 and the above procedure results in the following Montgomery curve, called curve448:
The base point is u = 5, v = 3552939267855681752641275020637833348089763993877142718318808984351 \ 69088786967410002932673765864550910142774147268105838985595290606362.
This curve is isomorphic to the Edwards curve x^2 + y^2 = 1 + d*x^2*y^2 where:
The isomorphism maps are:
(u, v) = ((y-1)/(y+1), sqrt(156324)*u/x) (x, y) = (sqrt(156324)*u/v, (1+u)/(1-u)
That curve is also 4-isogenous to the following Edward's curve x^2 + y^2 = 1 + d*x^2*y^2, called Ed448-Goldilocks:
The 4-isogeny maps between the Montgomery curve and the Edward's curve are:
(u, v) = (x^2/y^2, (2 - x^2 - y^2)*x/y^3) (x, y) = (4*v*(u^2 - 1)/(u^4 - 2*u^2 + 4*v^2 + 1), (u^5 - 2*u^3 - 4*u*v^2 + u)/(u^5 - 2*u^2*v^2 - 2*u^3 - 2*v^2 + u))
The curve25519 and curve448 functions performs scalar multiplication on the Montgomery form of the above curves. (This is used when implementing Diffie-Hellman.) The functions take a scalar and a u-coordinate as inputs and produce a u-coordinate as output. Although the functions work 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) or GF(2^448-2^224-1) 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 of curve25519 (but not curve448) 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. Here the "bits" parameter should be set to 255 for curve25519 and 448 for curve448:
def decodeLittleEndian(b,bits):
return sum([b[i] << 8*i for i in range((bits+7)/8)])
def decodeUCoordinate(u,bits):
u_list = [ord(b) for b in u]
if bits % 8:
u_list[-1] &= (1<<(bits%8))-1
return decodeLittleEndian(u_list,bits)
def encodeUCoordinate(u,bits):
u = u % p
return ''.join([chr((u >> 8*i) & 0xff) for i in range((bits+7)/8)])
(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. For curve25519, 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}. Likewise, for curve448, set the two least significant bits of the first byte to 0, and the most significant bit of the last byte to 1. This means that the resulting integer is of the form 2^447 + 4 * {0, 1, ..., 2^(445) - 1}.
def decodeScalar25519(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,255)
def decodeScalar448(k):
k_list = [ord(b) for b in k]
k_list[0] &= 252
k_list[55] |= 128
return decodeLittleEndian(k_list,448)
To implement the curve25519(k, u) and curve448(k, u) functions (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 for curve25519, and (156326 - 2) / 4 = 39081 for curve448.
x_1 = u
x_2 = 1
z_2 = 0
x_3 = u
z_3 = 1
swap = 0
For t = bits-1 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 or 56 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.
curve25519: 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 curve448: Input scalar: 3d262fddf9ec8e88495266fea19a34d28882acef045104d0d1aae121 700a779c984c24f8cdd78fbff44943eba368f54b29259a4f1c600ad3 Input scalar as a number (base 10): 5991891753738964027837560161452132561572308560850261299268914594686 \ 22403380588640249457727683869421921443004045221642549886377526240828 Input U-coordinate: 06fce640fa3487bfda5f6cf2d5263f8aad88334cbd07437f020f08f9 814dc031ddbdc38c19c6da2583fa5429db94ada18aa7a7fb4ef8a086 Input U-coordinate as a number: 3822399108141073301162299612348993770314163652405713251483465559224 \ 38025162094455820962429142971339584360034337310079791515452463053830 Output U-coordinate: ce3e4ff95a60dc6697da1db1d85e6afbdf79b50a2412d7546d5f239f e14fbaadeb445fc66a01b0779d98223961111e21766282f73dd96b6f Input scalar: 203d494428b8399352665ddca42f9de8fef600908e0d461cb021f8c5 38345dd77c3e4806e25f46d3315c44e0a5b4371282dd2c8d5be3095f Input scalar as a number (base 10): 6332543359069705927792594815348623723825251552520289610564040013321 \ 22152890562527156973881968934311400345568203929409663925541994577184 Input U-coordinate: 0fbcc2f993cd56d3305b0b7d9e55d4c1a8fb5dbb52f8e9a1e9b6201b 165d015894e56c4d3570bee52fe205e28a78b91cdfbde71ce8d157db Input U-coordinate as a number: 6227617977583254444629220684312341806495903900248112997616251537672 \ 28042600197997696167956134770744996690267634159427999832340166786063 Output U-coordinate: 884a02576239ff7a2f2f63b2db6a9ff37047ac13568e1e30fe63c4a7 ad1b3ee3a5700df34321d62077e63633c575c1c954514e99da7c179d
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. Both MUST check, without leaking extra information about the value of K, whether K is the all-zero value and abort if so (see below). Alice and Bob can then use a key-derivation function, such as hashing K, to compute a key.
The check for the all-zero value results from the fact that the curve25519 function produces that value if it operates on an input corresponding to a point with order dividing the co-factor, h, of the curve. This check is cheap and so MUST always be carried out. The check may be performed by ORing all the bytes together and checking whether the result is zero as this eliminates standard side-channels in software implementations.
curve25519: 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 curve448: Alice's private key, f: 9a8f4925d1519f5775cf46b04b5800d4ee9ee8bae8bc5565d498c28d d9c9baf574a9419744897391006382a6f127ab1d9ac2d8c0a598726b Alice's public key, curve448(f, 5): 9b08f7cc31b7e3e67d22d5aea121074a273bd2b83de09c63faa73d2c 22c5d9bbc836647241d953d40c5b12da88120d53177f80e532c41fa0 Bob's private key, g: 1c306a7ac2a0e2e0990b294470cba339e6453772b075811d8fad0d1d 6927c120bb5ee8972b0d3e21374c9c921b09d1b0366f10b65173992d Bob's public key, curve448(g, 5): 3eb7a829b0cd20f5bcfc0b599b6feccf6da4627107bdb0d4f345b430 27d8b972fc3e34fb4232a13ca706dcb57aec3dae07bdc1c67bf33609 Their shared secret, K: fe2d52f1ca113e5441538037dc4a9d4cb381035fb4a990ac50ac4333 63dc072301d1d4f2e82883b35103be96068c11e7c84b8fff74bb6ab0
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, Mike Hamburg, 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. |