Elliptic Curves for Security
draftirtfcfrgcurves00
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This is an older version of an InternetDraft that was ultimately published as RFC 7748.



Author  Adam Langley  
Last updated  20150128  
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draftirtfcfrgcurves00
CFRG A. Langley InternetDraft Google Intended status: Informational January 28, 2015 Expires: August 1, 2015 Elliptic Curves for Security draftirtfcfrgcurves00 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 ~128bit security level. Status of This Memo This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79. InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current Internet Drafts is at http://datatracker.ietf.org/drafts/current/. InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress." This InternetDraft will expire on August 1, 2015. Copyright Notice Copyright (c) 2015 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (http://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Langley Expires August 1, 2015 [Page 1] InternetDraft cfrgcurve January 2015 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. Wireformat of field elements . . . . . . . . . . . . . . . . 8 9. Elliptic Curve DiffieHellman . . . . . . . . . . . . . . . . 9 9.1. DiffieHellman 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 "draftblackrpgecc01" (by Benjamin Black, Joppe W. Bos, Craig Costello, Patrick Longa and Michael Naehrig) and "draftturnerthecurve25519function01" (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 sidechannel 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 constanttime, exceptionfree 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 Langley Expires August 1, 2015 [Page 2] InternetDraft cfrgcurve January 2015 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. Langley Expires August 1, 2015 [Page 3] InternetDraft cfrgcurve January 2015 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 nontrivial 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 nontrivial 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 xcoordinate of the elliptic curve point P. Y(P) The ycoordinate 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 nonsquare 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. Langley Expires August 1, 2015 [Page 4] InternetDraft cfrgcurve January 2015 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 nonsquare 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. Langley Expires August 1, 2015 [Page 5] InternetDraft cfrgcurve January 2015 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. Langley Expires August 1, 2015 [Page 6] InternetDraft cfrgcurve January 2015 Input: a prime p and curve parameters nonsquare 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 ~128bit security level, the prime 2^25519 is recommended for performance over a widerange of architectures. This prime is congruent to 1 mod 4 and the above procedure results in the following twisted Edwards curve, called "intermediate25519": p 2^25519 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 2^25519 Langley Expires August 1, 2015 [Page 7] InternetDraft cfrgcurve January 2015 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)/(1y), sqrt(1)*sqrt(486664)*u/x) (x, y) = (sqrt(1)*sqrt(486664)*u/v, (u1)/(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. Wireformat of field elements When transmitting field elements in the DiffieHellman protocol below, they MUST be encoded as an array of bytes, x, in littleendian 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 (8log2(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: draftturnerthecurve25519function also says "Implementations MUST reject numbers in the range [2^25519, 2^2551], inclusive." but I'm not aware of any implementations that do so.) Langley Expires August 1, 2015 [Page 8] InternetDraft cfrgcurve January 2015 9. Elliptic Curve DiffieHellman This section describes how to perform DiffieHellman using curves generated by the above procedure. For safety reasons, DiffieHellman 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 ucoordinates 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. Langley Expires August 1, 2015 [Page 9] InternetDraft cfrgcurve January 2015 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 sidechannels 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: Langley Expires August 1, 2015 [Page 10] InternetDraft cfrgcurve January 2015 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 all1 or all0 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. DiffieHellman 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 littleendian 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 keyderivation 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 littleendian hexadecimal byte strings: Langley Expires August 1, 2015 [Page 11] InternetDraft cfrgcurve January 2015 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, <http://www.ecc brainpool.org/download/Domainparameters.pdf>. [NIST] National Institute of Standards, "Recommended Elliptic Curves for Federal Government Use", July 1999, <http://csrc.nist.gov/groups/ST/toolkit/documents/dss/ NISTReCur.pdf>. Langley Expires August 1, 2015 [Page 12] InternetDraft cfrgcurve January 2015 [S] Semaev, I., "Evaluation of discrete logarithms on some elliptic curves", 1998. [SC] Bernstein, D. and T. Lange, "SafeCurves: choosing safe curves for ellipticcurve cryptography", June 2014, <http://safecurves.cr.yp.to/>. [SEC1] Certicom Research, "SEC 1: Elliptic Curve Cryptography", September 2000, <http://www.secg.org/collateral/sec1_final.pdf>. [Smart] Smart, N., "The discrete logarithm problem on elliptic curves of trace one", 1999. [curve25519] Bernstein, D., "Curve25519  new DiffieHellman speed records", 2006, <http://www.iacr.org/cryptodb/archive/2006/ PKC/3351/3351.pdf>. [ed25519] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. Yang, "Highspeed highsecurity signatures", 2011, <http://ed25519.cr.yp.to/ed2551920110926.pdf>. [montgomery] Montgomery, P., "Speeding the Pollard and elliptic curve methods of factorization", 1983, <http://www.ams.org/journals/mcom/198748177/ S00255718198708661137/S00255718198708661137.pdf>. [nacl] Bernstein, D., "Cryptography in NaCl", 2009, <http://cr.yp.to/highspeed/naclcrypto20090310.pdf>. Author's Address Adam Langley Google 345 Spear St San Francisco, CA 94105 US Email: agl@google.com Langley Expires August 1, 2015 [Page 13]