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diff --git a/src/crypto/crypto_ops_builder/ietf.txt b/src/crypto/crypto_ops_builder/ietf.txt index 0736f71ec..609f5e75a 100644 --- a/src/crypto/crypto_ops_builder/ietf.txt +++ b/src/crypto/crypto_ops_builder/ietf.txt @@ -1,1402 +1,4 @@ - +https://tools.ietf.org/id/draft-josefsson-eddsa-ed25519-02.txt -[Docs] [txt|pdf] [Tracker] [Email] [Diff1] [Diff2] [Nits] - -Versions: 00 01 02 - -Network Working Group S. Josefsson -Internet-Draft SJD AB -Intended status: Informational N. Moeller -Expires: August 26, 2015 - February 22, 2015 - - - EdDSA and Ed25519 - - draft-josefsson-eddsa-ed25519-02 - - -Abstract - - The elliptic curve signature scheme EdDSA and one instance of it - called Ed25519 is described. An example implementation and test - vectors are provided. - -Status of This Memo - - This Internet-Draft is submitted in full conformance with the - provisions of BCP 78 and BCP 79. - - Internet-Drafts are working documents of the Internet Engineering - Task Force (IETF). Note that other groups may also distribute - working documents as Internet-Drafts. The list of current Internet- - Drafts is at http://datatracker.ietf.org/drafts/current/. - - Internet-Drafts 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 Internet-Drafts as reference - material or to cite them other than as "work in progress." - - This Internet-Draft will expire on August 26, 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/license-info) 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. - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 1] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -Table of Contents - - 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2 - 2. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 3 - 3. Background . . . . . . . . . . . . . . . . . . . . . . . . . 3 - 4. EdDSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 - 4.1. Encoding . . . . . . . . . . . . . . . . . . . . . . . . 4 - 4.2. Keys . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 4.3. Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 4.4. Verify . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 5. Ed25519 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 - 5.1. Modular arithmetic . . . . . . . . . . . . . . . . . . . 6 - 5.2. Encoding . . . . . . . . . . . . . . . . . . . . . . . . 6 - 5.3. Decoding . . . . . . . . . . . . . . . . . . . . . . . . 6 - 5.4. Point addition . . . . . . . . . . . . . . . . . . . . . 7 - 5.5. Key Generation . . . . . . . . . . . . . . . . . . . . . 8 - 5.6. Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 8 - 5.7. Verify . . . . . . . . . . . . . . . . . . . . . . . . . 9 - 5.8. Python illustration . . . . . . . . . . . . . . . . . . . 9 - 6. Test Vectors for Ed25519 . . . . . . . . . . . . . . . . . . 14 - 7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 17 - 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 18 - 9. Security Considerations . . . . . . . . . . . . . . . . . . . 18 - 9.1. Side-channel leaks . . . . . . . . . . . . . . . . . . . 18 - 10. References . . . . . . . . . . . . . . . . . . . . . . . . . 18 - 10.1. Normative References . . . . . . . . . . . . . . . . . . 18 - 10.2. Informative References . . . . . . . . . . . . . . . . . 18 - Appendix A. Ed25519 Python Library . . . . . . . . . . . . . . . 19 - Appendix B. Library driver . . . . . . . . . . . . . . . . . . . 23 - Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 24 - -1. Introduction - - - The Edwards-curve Digital Signature Algorithm (EdDSA) is a variant of - Schnorr's signature system with Twisted Edwards curves. EdDSA needs - to be instantiated with certain parameters and this document describe - Ed25519 - an instantiation of EdDSA in a curve over GF(2^255-19). To - facilitate adoption in the Internet community of Ed25519, this - document describe the signature scheme in an implementation-oriented - way, and we provide sample code and test vectors. - - The advantages with EdDSA and Ed25519 include: - - 1. High-performance on a variety of platforms. - - 2. Does not require the use of a unique random number for each - signature. - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 2] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - 3. More resilient to side-channel attacks. - - 4. Small public keys (32 bytes) and signatures (64 bytes). - - 5. The formulas are "strongly unified", i.e., they are valid for all - points on the curve, with no exceptions. This obviates the need - for EdDSA to perform expensive point validation on untrusted - public values. - - 6. Collision resilience, meaning that hash-function collisions do - not break this system. - - For further background, see the original EdDSA paper [EDDSA]. - -2. Notation - - - The following notation is used throughout the document: - - GF(p) finite field with p elements - - x^y x multiplied by itself y times - - B generator of the group or subgroup of interest - - n B B added to itself n times. - - h_i the i'th bit of h - - a || b (bit-)string a concatenated with (bit-)string b - -3. Background - - - EdDSA is defined using an elliptic curve over GF(p) of the form - - -x^2 + y^2 = 1 + d x^2 y^2 - - In general, p could be a prime power, but it is usually chosen as a - prime number. It is required that p = 1 modulo 4 (which implies that - -1 is a square modulo p) and that d is a non-square modulo p. For - Ed25519, the curve used is equivalent to Curve25519 [CURVE25519], - under a change of coordinates, which means that the difficulty of the - discrete logarithm problem is the same as for Curve25519. - - Points on this curve form a group under addition, (x3, y3) = (x1, y1) - + (x2, y2), with the formulas - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 3] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - x1 y2 + x2 y1 y1 y2 + x1 x2 - x3 = -------------------, y3 = ------------------- - 1 + d x1 x2 y1 y2 1 - d x1 x2 y1 y2 - - The neutral element in the group is (0, 1). - - Unlike manyy other curves used for cryptographic applications, these - formulas are "strongly unified": they are valid for all points on the - curve, with no exceptions. In particular, the denominators are non- - zero for all input points. - - There are more efficient formulas, which are still strongly unified, - which use homogeneous coordinates to avoid the expensive modulo p - inversions. See [Faster-ECC] and [Edwards-revisited]. - -4. EdDSA - - - EdDSA is a digital signature system with several parameters. The - generic EdDSA digital signature system is normally not implemented - directly, but instead a particular instance of EdDSA (like Ed25519) - is implemented. A precise explanation of the generic EdDSA is thus - not particulary useful for implementers, but for background and - completeness, a succint description of the generic EdDSA algorithm is - given here. - - EdDSA has seven parameters: - - 1. an integer b >= 10. - - 2. a cryptographic hash function H producing 2b-bit outputs. - - 3. a prime power p congruent to 1 modulo 4. - - 4. a (b-1)-bit encoding of elements of the finite field GF(p). - - 5. a non-square element d of GF(p) - - 6. an element B != (0,1) of the set E = { (x,y) is a member of GF(p) - x GF(p) such that -x^2 + y^2 = 1 + dx^2y^2 }. - - 7. a prime q, of size b-3 bits, such that qB = (0, 1), i.e., q is - the order of B or a multiple thereof. - -4.1. Encoding - - - An element (x,y) of E is encoded as a b-bit string called ENC(x,y) - which is the (b-1)-bit encoding of y concatenated with one bit that - is 1 if x is negative and 0 if x is not negative. Negative elements - - - -Josefsson & Moeller Expires August 26, 2015 [Page 4] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - of GF(q) are those x which the (b-1)-bit encoding of x is - lexicographically larger than the (b-1)-bit encoding of -x. - -4.2. Keys - - - An EdDSA secret key is a b-bit string k. Let the hash H(k) = (h_0, - h_1, ..., h_(2b-1)) determine an integer a which is 2^(b-2) plus the - sum of m = 2^i * h_i for all i equal or larger than 3 and equal to or - less than b-3 such that m is a member of the set { 2^(b-2), 2^(b-2) + - 8, ..., 2^(b-1) - 8 }. The EdDSA public key is ENC(A) = ENC(aB). - The bits h_b, ..., h_(2b-1) is used below during signing. - -4.3. Sign - - - The signature of a message M under a secret key k is the 2b-bit - string ENC(R) || ENC'(S), where ENC'(S) is defined as the b-bit - little-endian encoding of S. R and S are derived as follows. First - define r = H(h_b, ... h_(2b-1)), M) interpreting 2b-bit strings in - little-endian form as integers in {0, 1, ..., 2^(2b)-1}. Let R=rB - and S=(r+H(ENC(R) || ENC(A) || M)a) mod l. - -4.4. Verify - - - To verify a signature ENC(R) || ENC'(S) on a message M under a public - key ENC(A), proceed as follows. Parse the inputs so that A and R is - an element of E, and S is a member of the set {0, 1, ..., l-1 }. - Compute H' = H(ENC(R) || ENC(A) || M) and check the group equation - 8SB = 8R + 8H'A in E. Verification is rejected if parsing fails or - the group equation does not hold. - -5. Ed25519 - - - Theoretically, Ed25519 is EdDSA instantiated with b=256, H being - SHA-512 [RFC4634], p is the prime 2^255-19, the 255-bit encoding of - GF(2^255-19) being the little-endian encoding of {0, 1, ..., - 2^255-20}, q is the prime 2^252 + 0x14def9dea2f79cd65812631a5cf5d3ed, - d = -121665/121666 which is a member of GF(p), and B is the unique - point (x, 4/5) in E for which x is "positive", which with the - encoding used simply means that the least significant bit of x is 0. - The curve p, prime q, d and B follows from [I-D.irtf-cfrg-curves]. - - Written out explicitly, B is the point (15112221349535400772501151409 - 588531511454012693041857206046113283949847762202, 4631683569492647816 - 9428394003475163141307993866256225615783033603165251855960). - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 5] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -5.1. Modular arithmetic - - - For advise on how to implement arithmetic modulo p = 2^255 - 1 - efficiently and securely, see Curve25519 [CURVE25519]. For inversion - modulo p, it is recommended to use the identity x^-1 = x^(p-2) (mod - p). - - For point decoding or "decompression", square roots modulo p are - needed. They can be computed using the Tonelli-Shanks algorithm, or - the special case for p = 5 (mod 8). To find a square root of a, - first compute the candidate root x = a^((p+3)/8) (mod p). Then there - are three cases: - - x^2 = a (mod p). Then x is a square root. - - x^2 = -a (mod p). Then 2^((p-1)/4) x is a square root. - - a is not a square modulo p. - -5.2. Encoding - - - All values are coded as octet strings, and integers are coded using - little endian convention. I.e., a 32-octet string h h[0],...h[31] - represents the integer h[0] + 2^8 h[1] + ... + 2^248 h[31]. - - A curve point (x,y), with coordiantes in the range 0 <= x,y < p, is - coded as follows. First encode the y-coordinate as a little-endian - string of 32 octets. The most significant bit of the final octet is - always zero. To form the encoding of the point, copy the least - significant bit of the x-coordinate to the most significant bit of - the final octet. - -5.3. Decoding - - - Decoding a point, given as a 32-octet string, is a little more - complicated. - - 1. First interpret the string as an integer in little-endian - representation. Bit 255 of this number is the least significant - bit of the x-coordinate, and denote this value x_0. The - y-coordinate is recovered simply by clearing this bit. If the - resulting value is >= p, decoding fails. - - 2. To recover the x coordinate, the curve equation implies x^2 = - (y^2 - 1) / (d y^2 + 1) (mod p). Since d is a non-square and -1 - is a square, the numerator, (d y^2 + 1), is always invertible - modulo p. Let u = y^2 - 1 and v = d y^2 + 1. To compute the - square root of (u/v), the first step is to compute the candidate - - - -Josefsson & Moeller Expires August 26, 2015 [Page 6] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - root x = (u/v)^((p+3)/8). This can be done using the following - trick, to use a single modular powering for both the inversion of - v and the square root: - - (p+3)/8 3 (p-5)/8 - x = (u/v) = u v (u v^7) (mod p) - - 3. Again, there are three cases: - - 1. If v x^2 = u (mod p), x is a square root. - - 2. If v x^2 = -u (mod p), set x <-- x 2^((p-1)/4), which is a - square root. - - 3. Otherwise, no square root exists modulo p, and decoding - fails. - - 4. Finally, use the x_0 bit to select the right square root. If x = - 0, and x_0 = 1, decoding fails. Otherwise, if x_0 != x mod 2, - set x <-- p - x. Return the decoded point (x,y). - -5.4. Point addition - - - For point addition, the following method is recommended. A point - (x,y) is represented in extended homogeneous coordinates (X, Y, Z, - T), with x = X/Z, y = Y/Z, x y = T/Z. - - The following formulas for adding two points, (x3,y3) = - (x1,y1)+(x2,y2) are described in [Edwards-revisited], section 3.1. - They are strongly unified, i.e., they work for any pair of valid - input points. - - A = (Y1-X1)*(Y2-X2) - B = (Y1+X1)*(Y2+X2) - C = T1*2*d*T2 - D = Z1*2*Z2 - E = B-A - F = D-C - G = D+C - H = B+A - X3 = E*F - Y3 = G*H - T3 = E*H - Z3 = F*G - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 7] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -5.5. Key Generation - - - The secret is 32 octets (256 bits, corresponding to b) of - cryptographically-secure random data. See [RFC4086] for a discussion - about randomness. - - The 32-byte public key is generated by the following steps. - - 1. Hash the 32-byte secret using SHA-512, storing the digest in a - 64-octet large buffer, denoted h. Only the lower 32 bytes are - used for generating the public key. - - 2. Prune the buffer. In C terminology: - - h[0] &= ~0x07; - h[31] &= 0x7F; - h[31] |= 0x40; - - 3. Interpret the buffer as the little-endian integer, forming a - secret scalar a. Perform a known-base-point scalar - multiplication a B. - - 4. The public key A is the encoding of the point aB. First encode - the y coordinate (in the range 0 <= y < p) as a little-endian - string of 32 octets. The most significant bit of the final octet - is always zero. To form the encoding of the point aB, copy the - least significant bit of the x coordinate to the most significant - bit of the final octet. The result is the public key. - -5.6. Sign - - - The imputs to the signing procedure is the secret key, a 32-octet - string, and a message M of arbitrary size. - - 1. Hash the secret key, 32-octets, using SHA-512. Let h denote the - resulting digest. Construct the secret scalar a from the first - half of the digest, and the corresponding public key A, as - described in the previous section. Let prefix denote the second - half of the hash digest, h[32],...,h[63]. - - 2. Compute SHA-512(prefix || M), where M is the message to be - signed. Interpret the 64-octet digest as a little-endian integer - r. - - 3. Compute the point rB. For efficiency, do this by first reducing - r modulo q, the group order of B. Let the string R be the - encoding of this point. - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 8] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - 4. Compute SHA512(R || A || M), and interpret the 64-octet digest as - a little-endian integer k. - - 5. Compute s = (r + k a) mod q. For efficiency, again reduce k - modulo q first. - - 6. Form the signature of the concatenation of R (32 octets) and the - little-endian encoding of s (32 octets, three most significant - bits of the final octets always zero). - -5.7. Verify - - - 1. To verify a signature on a message M, first split the signature - into two 32-octet halves. Decode the first half as a point R, - and the second half as an integer s, in the range 0 <= s < q. If - the decoding fails, the signature is invalid. - - 2. Compute SHA512(R || A || M), and interpret the 64-octet digest as - a little-endian integer k. - - 3. Check the group equation 8s B = 8 R + 8k A. It's sufficient, but - not required, to instead check s B = R + k A. - -5.8. Python illustration - - - The rest of this section describes how Ed25519 can be implemented in - Python (version 3.2 or later) for illustration. See appendix A for - the complete implementation and appendix B for a test-driver to run - it through some test vectors. - - First some preliminaries that will be needed. - - - - - - - - - - - - - - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 9] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - import hashlib - - def sha512(s): - return hashlib.sha512(s).digest() - - # Base field Z_p - p = 2**255 - 19 - - def modp_inv(x): - return pow(x, p-2, p) - - # Curve constant - d = -121665 * modp_inv(121666) % p - - # Group order - q = 2**252 + 27742317777372353535851937790883648493 - - def sha512_modq(s): - return int.from_bytes(sha512(s), "little") % q - - Then follows functions to perform point operations. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 10] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -# Points are represented as tuples (X, Y, Z, T) of extended coordinates, -# with x = X/Z, y = Y/Z, x*y = T/Z - -def point_add(P, Q): - A = (P[1]-P[0])*(Q[1]-Q[0]) % p - B = (P[1]+P[0])*(Q[1]+Q[0]) % p - C = 2 * P[3] * Q[3] * d % p - D = 2 * P[2] * Q[2] % p - E = B-A - F = D-C - G = D+C - H = B+A - return (E*F, G*H, F*G, E*H) - -# Computes Q = s * Q -def point_mul(s, P): - Q = (0, 1, 1, 0) # Neutral element - while s > 0: - # Is there any bit-set predicate? - if s & 1: - Q = point_add(Q, P) - P = point_add(P, P) - s >>= 1 - return Q - -def point_equal(P, Q): - # x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1 - if (P[0] * Q[2] - Q[0] * P[2]) % p != 0: - return False - if (P[1] * Q[2] - Q[1] * P[2]) % p != 0: - return False - return True - - Now follows functions for point compression. - - - - - - - - - - - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 11] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -# Square root of -1 -modp_sqrt_m1 = pow(2, (p-1) // 4, p) - -# Compute corresponding x coordinate, with low bit corresponding to sign, -# or return None on failure -def recover_x(y, sign): - x2 = (y*y-1) * modp_inv(d*y*y+1) - if x2 == 0: - if sign: - return None - else: - return 0 - - # Compute square root of x2 - x = pow(x2, (p+3) // 8, p) - if (x*x - x2) % p != 0: - x = x * modp_sqrt_m1 % p - if (x*x - x2) % p != 0: - return None - - if (x & 1) != sign: - x = p - x - return x - -# Base point -g_y = 4 * modp_inv(5) % p -g_x = recover_x(g_y, 0) -G = (g_x, g_y, 1, g_x * g_y % p) - -def point_compress(P): - zinv = modp_inv(P[2]) - x = P[0] * zinv % p - y = P[1] * zinv % p - return int.to_bytes(y | ((x & 1) << 255), 32, "little") - -def point_decompress(s): - if len(s) != 32: - raise Exception("Invalid input length for decompression") - y = int.from_bytes(s, "little") - sign = y >> 255 - y &= (1 << 255) - 1 - - x = recover_x(y, sign) - if x is None: - return None - else: - return (x, y, 1, x*y % p) - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 12] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - These are functions for manipulating the secret. - - def secret_expand(secret): - if len(secret) != 32: - raise Exception("Bad size of private key") - h = sha512(secret) - a = int.from_bytes(h[:32], "little") - a &= (1 << 254) - 8 - a |= (1 << 254) - return (a, h[32:]) - - def secret_to_public(secret): - (a, dummy) = secret_expand(secret) - return point_compress(point_mul(a, G)) - - The signature function works as below. - - def sign(secret, msg): - a, prefix = secret_expand(secret) - A = point_compress(point_mul(a, G)) - r = sha512_modq(prefix + msg) - R = point_mul(r, G) - Rs = point_compress(R) - h = sha512_modq(Rs + A + msg) - s = (r + h * a) % q - return Rs + int.to_bytes(s, 32, "little") - - And finally the verification function. - - def verify(public, msg, signature): - if len(public) != 32: - raise Exception("Bad public-key length") - if len(signature) != 64: - Exception("Bad signature length") - A = point_decompress(public) - if not A: - return False - Rs = signature[:32] - R = point_decompress(Rs) - if not R: - return False - s = int.from_bytes(signature[32:], "little") - h = sha512_modq(Rs + public + msg) - sB = point_mul(s, G) - hA = point_mul(h, A) - return point_equal(sB, point_add(R, hA)) - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 13] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -6. Test Vectors for Ed25519 - - - Below is a sequence of octets with test vectors for the the Ed25519 - signature algorithm. The octets are hex encoded and whitespace is - inserted for readability. Private keys are 64 bytes, public keys 32 - bytes, message of arbitrary length, and signatures are 64 bytes. The - test vectors are taken from [ED25519-TEST-VECTORS] (but we removed - the public key as a suffix of the secret key, and removed the message - from the signature) and [ED25519-LIBGCRYPT-TEST-VECTORS]. - - -----TEST 1 - SECRET KEY: - 9d61b19deffd5a60ba844af492ec2cc4 - 4449c5697b326919703bac031cae7f60 - - PUBLIC KEY: - d75a980182b10ab7d54bfed3c964073a - 0ee172f3daa62325af021a68f707511a - - MESSAGE (length 0 bytes): - - SIGNATURE: - e5564300c360ac729086e2cc806e828a - 84877f1eb8e5d974d873e06522490155 - 5fb8821590a33bacc61e39701cf9b46b - d25bf5f0595bbe24655141438e7a100b - - -----TEST 2 - SECRET KEY: - 4ccd089b28ff96da9db6c346ec114e0f - 5b8a319f35aba624da8cf6ed4fb8a6fb - - PUBLIC KEY: - 3d4017c3e843895a92b70aa74d1b7ebc - 9c982ccf2ec4968cc0cd55f12af4660c - - MESSAGE (length 1 byte): - 72 - - SIGNATURE: - 92a009a9f0d4cab8720e820b5f642540 - a2b27b5416503f8fb3762223ebdb69da - 085ac1e43e15996e458f3613d0f11d8c - 387b2eaeb4302aeeb00d291612bb0c00 - - -----TEST 3 - SECRET KEY: - c5aa8df43f9f837bedb7442f31dcb7b1 - - - -Josefsson & Moeller Expires August 26, 2015 [Page 14] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - 66d38535076f094b85ce3a2e0b4458f7 - - PUBLIC KEY: - fc51cd8e6218a1a38da47ed00230f058 - 0816ed13ba3303ac5deb911548908025 - - MESSAGE (length 2 bytes): - af82 - - SIGNATURE: - 6291d657deec24024827e69c3abe01a3 - 0ce548a284743a445e3680d7db5ac3ac - 18ff9b538d16f290ae67f760984dc659 - 4a7c15e9716ed28dc027beceea1ec40a - - -----TEST 1024 - SECRET KEY: - f5e5767cf153319517630f226876b86c - 8160cc583bc013744c6bf255f5cc0ee5 - - PUBLIC KEY: - 278117fc144c72340f67d0f2316e8386 - ceffbf2b2428c9c51fef7c597f1d426e - - MESSAGE: - 08b8b2b733424243760fe426a4b54908 - 632110a66c2f6591eabd3345e3e4eb98 - fa6e264bf09efe12ee50f8f54e9f77b1 - e355f6c50544e23fb1433ddf73be84d8 - 79de7c0046dc4996d9e773f4bc9efe57 - 38829adb26c81b37c93a1b270b20329d - 658675fc6ea534e0810a4432826bf58c - 941efb65d57a338bbd2e26640f89ffbc - 1a858efcb8550ee3a5e1998bd177e93a - 7363c344fe6b199ee5d02e82d522c4fe - ba15452f80288a821a579116ec6dad2b - 3b310da903401aa62100ab5d1a36553e - 06203b33890cc9b832f79ef80560ccb9 - a39ce767967ed628c6ad573cb116dbef - efd75499da96bd68a8a97b928a8bbc10 - 3b6621fcde2beca1231d206be6cd9ec7 - aff6f6c94fcd7204ed3455c68c83f4a4 - 1da4af2b74ef5c53f1d8ac70bdcb7ed1 - 85ce81bd84359d44254d95629e9855a9 - 4a7c1958d1f8ada5d0532ed8a5aa3fb2 - d17ba70eb6248e594e1a2297acbbb39d - 502f1a8c6eb6f1ce22b3de1a1f40cc24 - 554119a831a9aad6079cad88425de6bd - - - -Josefsson & Moeller Expires August 26, 2015 [Page 15] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - e1a9187ebb6092cf67bf2b13fd65f270 - 88d78b7e883c8759d2c4f5c65adb7553 - 878ad575f9fad878e80a0c9ba63bcbcc - 2732e69485bbc9c90bfbd62481d9089b - eccf80cfe2df16a2cf65bd92dd597b07 - 07e0917af48bbb75fed413d238f5555a - 7a569d80c3414a8d0859dc65a46128ba - b27af87a71314f318c782b23ebfe808b - 82b0ce26401d2e22f04d83d1255dc51a - ddd3b75a2b1ae0784504df543af8969b - e3ea7082ff7fc9888c144da2af58429e - c96031dbcad3dad9af0dcbaaaf268cb8 - fcffead94f3c7ca495e056a9b47acdb7 - 51fb73e666c6c655ade8297297d07ad1 - ba5e43f1bca32301651339e22904cc8c - 42f58c30c04aafdb038dda0847dd988d - cda6f3bfd15c4b4c4525004aa06eeff8 - ca61783aacec57fb3d1f92b0fe2fd1a8 - 5f6724517b65e614ad6808d6f6ee34df - f7310fdc82aebfd904b01e1dc54b2927 - 094b2db68d6f903b68401adebf5a7e08 - d78ff4ef5d63653a65040cf9bfd4aca7 - 984a74d37145986780fc0b16ac451649 - de6188a7dbdf191f64b5fc5e2ab47b57 - f7f7276cd419c17a3ca8e1b939ae49e4 - 88acba6b965610b5480109c8b17b80e1 - b7b750dfc7598d5d5011fd2dcc5600a3 - 2ef5b52a1ecc820e308aa342721aac09 - 43bf6686b64b2579376504ccc493d97e - 6aed3fb0f9cd71a43dd497f01f17c0e2 - cb3797aa2a2f256656168e6c496afc5f - b93246f6b1116398a346f1a641f3b041 - e989f7914f90cc2c7fff357876e506b5 - 0d334ba77c225bc307ba537152f3f161 - 0e4eafe595f6d9d90d11faa933a15ef1 - 369546868a7f3a45a96768d40fd9d034 - 12c091c6315cf4fde7cb68606937380d - b2eaaa707b4c4185c32eddcdd306705e - 4dc1ffc872eeee475a64dfac86aba41c - 0618983f8741c5ef68d3a101e8a3b8ca - c60c905c15fc910840b94c00a0b9d0 - - SIGNATURE: - 0aab4c900501b3e24d7cdf4663326a3a - 87df5e4843b2cbdb67cbf6e460fec350 - aa5371b1508f9f4528ecea23c436d94b - 5e8fcd4f681e30a6ac00a9704a188a03 - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 16] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - -----TEST 1A - -----An additional test with the data from test 1 but using an - -----uncompressed public key. - SECRET KEY: - 9d61b19deffd5a60ba844af492ec2cc4 - 4449c5697b326919703bac031cae7f60 - - PUBLIC KEY: - 0455d0e09a2b9d34292297e08d60d0f6 - 20c513d47253187c24b12786bd777645 - ce1a5107f7681a02af2523a6daf372e1 - 0e3a0764c9d3fe4bd5b70ab18201985a - d7 - - MSG (length 0 bytes): - - SIGNATURE: - e5564300c360ac729086e2cc806e828a - 84877f1eb8e5d974d873e06522490155 - 5fb8821590a33bacc61e39701cf9b46b - d25bf5f0595bbe24655141438e7a100b - - -----TEST 1B - -----An additional test with the data from test 1 but using an - -----compressed prefix. - SECRET KEY: - 9d61b19deffd5a60ba844af492ec2cc4 - 4449c5697b326919703bac031cae7f60 - - PUBLIC KEY: - 40d75a980182b10ab7d54bfed3c96407 - 3a0ee172f3daa62325af021a68f70751 - 1a - - MESSAGE (length 0 bytes): - - SIGNATURE: - e5564300c360ac729086e2cc806e828a - 84877f1eb8e5d974d873e06522490155 - 5fb8821590a33bacc61e39701cf9b46b - d25bf5f0595bbe24655141438e7a100b - ----- - -7. Acknowledgements - - - Feedback on this document was received from Werner Koch and Damien - Miller. - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 17] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -8. IANA Considerations - - - None. - -9. Security Considerations - - -9.1. Side-channel leaks - - - For implementations performing signatures, secrecy of the key is - fundamental. It is possible to protect against some side-channel - attacks by ensuring that the implementation executes exactly the same - sequence of instructions and performs exactly the same memory - accesses, for any value of the secret key. - - To make an implementation side-channel silent in this way, the modulo - p arithmetic must not use any data-dependent branches, e.g., related - to carry propagation. Side channel-silent point addition is - straight-forward, thanks to the unified formulas. - - Scalar multiplication, multiplying a point by an integer, needs some - additional effort to implement in a side-channel silent manner. One - simple approach is to implement a side-channel silent conditional - assignment, and use together with the binary algorithm to examine one - bit of the integer at a time. - - Note that the example implementation in this document does not - attempt to be side-channel silent. - -10. References - - -10.1. Normative References - - - [RFC4634] Eastlake, D. and T. Hansen, "US Secure Hash Algorithms - (SHA and HMAC-SHA)", RFC 4634, July 2006. - - [I-D.irtf-cfrg-curves] - Langley, A., Salz, R., and S. Turner, "Elliptic Curves for - Security", draft-irtf-cfrg-curves-01 (work in progress), - January 2015. - -10.2. Informative References - - - [RFC4086] Eastlake, D., Schiller, J., and S. Crocker, "Randomness - Requirements for Security", BCP 106, RFC 4086, June 2005. - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 18] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - [EDDSA] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. - Yang, "High-speed high-security signatures", WWW - http://ed25519.cr.yp.to/ed25519-20110926.pdf, September - 2011. - - [Faster-ECC] - Bernstein, D. and T. Lange, "Faster addition and doubling - on elliptic curves", WWW http://eprint.iacr.org/2007/286, - July 2007. - - [Edwards-revisited] - Hisil, H., Wong, K., Carter, G., and E. Dawson, "Twisted - Edwards Curves Revisited", WWW - http://eprint.iacr.org/2008/522, December 2008. - - [CURVE25519] - Bernstein, D., "Curve25519: new Diffie-Hellman speed - records", WWW http://cr.yp.to/ecdh.html, February 2006. - - [ED25519-TEST-VECTORS] - Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. - Yang, "Ed25519 test vectors", WWW - http://ed25519.cr.yp.to/python/sign.input, July 2011. - - [ED25519-LIBGCRYPT-TEST-VECTORS] - Koch, W., "Ed25519 Libgcrypt test vectors", WWW - http://git.gnupg.org/cgi- - bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=tests/t-ed25519.in - p;h=e13566f826321eece65e02c593bc7d885b3dbe23;hb=refs/ - heads/master, July 2014. - -Appendix A. Ed25519 Python Library - - - Below is an example implementation of Ed25519 written in Python, - version 3.2 or higher is required. - -# Loosely based on the public domain code at -# http://ed25519.cr.yp.to/software.html -# -# Needs python-3.2 - -import hashlib - - -def sha512(s): - return hashlib.sha512(s).digest() - -# Base field Z_p - - - -Josefsson & Moeller Expires August 26, 2015 [Page 19] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - -p = 2**255 - 19 - - -def modp_inv(x): - return pow(x, p-2, p) - -# Curve constant -d = -121665 * modp_inv(121666) % p - -# Group order -q = 2**252 + 27742317777372353535851937790883648493 - - -def sha512_modq(s): - return int.from_bytes(sha512(s), "little") % q - -# Points are represented as tuples (X, Y, Z, T) of extended coordinates, -# with x = X/Z, y = Y/Z, x*y = T/Z - - -def point_add(P, Q): - A = (P[1]-P[0])*(Q[1]-Q[0]) % p - B = (P[1]+P[0])*(Q[1]+Q[0]) % p - C = 2 * P[3] * Q[3] * d % p - D = 2 * P[2] * Q[2] % p - E = B-A - F = D-C - G = D+C - H = B+A - return (E*F, G*H, F*G, E*H) - - -# Computes Q = s * Q -def point_mul(s, P): - Q = (0, 1, 1, 0) # Neutral element - while s > 0: - # Is there any bit-set predicate? - if s & 1: - Q = point_add(Q, P) - P = point_add(P, P) - s >>= 1 - return Q - - -def point_equal(P, Q): - # x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1 - if (P[0] * Q[2] - Q[0] * P[2]) % p != 0: - return False - - - -Josefsson & Moeller Expires August 26, 2015 [Page 20] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - if (P[1] * Q[2] - Q[1] * P[2]) % p != 0: - return False - return True - -# Square root of -1 -modp_sqrt_m1 = pow(2, (p-1) // 4, p) - - -# Compute corresponding x coordinate, with low bit corresponding to sign, -# or return None on failure -def recover_x(y, sign): - x2 = (y*y-1) * modp_inv(d*y*y+1) - if x2 == 0: - if sign: - return None - else: - return 0 - - # Compute square root of x2 - x = pow(x2, (p+3) // 8, p) - if (x*x - x2) % p != 0: - x = x * modp_sqrt_m1 % p - if (x*x - x2) % p != 0: - return None - - if (x & 1) != sign: - x = p - x - return x - -# Base point -g_y = 4 * modp_inv(5) % p -g_x = recover_x(g_y, 0) -G = (g_x, g_y, 1, g_x * g_y % p) - - -def point_compress(P): - zinv = modp_inv(P[2]) - x = P[0] * zinv % p - y = P[1] * zinv % p - return int.to_bytes(y | ((x & 1) << 255), 32, "little") - - -def point_decompress(s): - if len(s) != 32: - raise Exception("Invalid input length for decompression") - y = int.from_bytes(s, "little") - sign = y >> 255 - y &= (1 << 255) - 1 - - - -Josefsson & Moeller Expires August 26, 2015 [Page 21] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - x = recover_x(y, sign) - if x is None: - return None - else: - return (x, y, 1, x*y % p) - - -def secret_expand(secret): - if len(secret) != 32: - raise Exception("Bad size of private key") - h = sha512(secret) - a = int.from_bytes(h[:32], "little") - a &= (1 << 254) - 8 - a |= (1 << 254) - return (a, h[32:]) - - -def secret_to_public(secret): - (a, dummy) = secret_expand(secret) - return point_compress(point_mul(a, G)) - - -def sign(secret, msg): - a, prefix = secret_expand(secret) - A = point_compress(point_mul(a, G)) - r = sha512_modq(prefix + msg) - R = point_mul(r, G) - Rs = point_compress(R) - h = sha512_modq(Rs + A + msg) - s = (r + h * a) % q - return Rs + int.to_bytes(s, 32, "little") - - -def verify(public, msg, signature): - if len(public) != 32: - raise Exception("Bad public-key length") - if len(signature) != 64: - Exception("Bad signature length") - A = point_decompress(public) - if not A: - return False - Rs = signature[:32] - R = point_decompress(Rs) - if not R: - return False - s = int.from_bytes(signature[32:], "little") - h = sha512_modq(Rs + public + msg) - sB = point_mul(s, G) - - - -Josefsson & Moeller Expires August 26, 2015 [Page 22] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - hA = point_mul(h, A) - return point_equal(sB, point_add(R, hA)) - -Appendix B. Library driver - - - Below is a command-line tool that uses the library above to perform - computations, for interactive use or for self-checking. - - import sys - import binascii - - from ed25519 import * - - def point_valid(P): - zinv = modp_inv(P[2]) - x = P[0] * zinv % p - y = P[1] * zinv % p - assert (x*y - P[3]*zinv) % p == 0 - return (-x*x + y*y - 1 - d*x*x*y*y) % p == 0 - - assert point_valid(G) - Z = (0, 1, 1, 0) - assert point_valid(Z) - - assert point_equal(Z, point_add(Z, Z)) - assert point_equal(G, point_add(Z, G)) - assert point_equal(Z, point_mul(0, G)) - assert point_equal(G, point_mul(1, G)) - assert point_equal(point_add(G, G), point_mul(2, G)) - for i in range(0, 100): - assert point_valid(point_mul(i, G)) - assert point_equal(Z, point_mul(q, G)) - - def munge_string(s, pos, change): - return (s[:pos] + - int.to_bytes(s[pos] ^ change, 1, "little") + - s[pos+1:]) - - # Read a file in the format of - # http://ed25519.cr.yp.to/python/sign.input - lineno = 0 - while True: - line = sys.stdin.readline() - if not line: - break - lineno = lineno + 1 - print(lineno) - fields = line.split(":") - - - -Josefsson & Moeller Expires August 26, 2015 [Page 23] - - -Internet-Draft EdDSA & Ed25519 February 2015 - - - secret = (binascii.unhexlify(fields[0]))[:32] - public = binascii.unhexlify(fields[1]) - msg = binascii.unhexlify(fields[2]) - signature = binascii.unhexlify(fields[3])[:64] - - assert public == secret_to_public(secret) - assert signature == sign(secret, msg) - assert verify(public, msg, signature) - if len(msg) == 0: - bad_msg = b"x" - else: - bad_msg = munge_string(msg, len(msg) // 3, 4) - assert not verify(public, bad_msg, signature) - bad_signature = munge_string(signature, 20, 8) - assert not verify(public, msg, bad_signature) - bad_signature = munge_string(signature, 40, 16) - assert not verify(public, msg, bad_signature) - -Authors' Addresses - - Simon Josefsson - SJD AB - - Email: simon@josefsson.org - URI: http://josefsson.org/ - - - Niels Moeller - - Email: nisse@lysator.liu.se - - - - - - - - - - - - - - - - - - - - - -Josefsson & Moeller Expires August 26, 2015 [Page 24] - - - -Html markup produced by rfcmarkup 1.113, available from https://tools.ietf.org/tools/rfcmarkup/ +Note: This draft is now superseded by https://datatracker.ietf.org/doc/rfc8032/ +(review of the differences is left as an exercise for the reader) |