RFC 8032
Edwards-Curve Digital Signature Algorithm (EdDSA)
Internet Research Task Force (IRTF) S. Josefsson Request for Comments: 8032 SJD AB Category: Informational I. Liusvaara ISSN: 2070-1721 Independent January 2017 Edwards-Curve Digital Signature Algorithm (EdDSA)Abstract
This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm is instantiated with recommended parameters for the edwards25519 and edwards448 curves. An example implementation and test vectors are provided. Status of This Memo This document is not an Internet Standards Track specification; it is published for informational purposes. This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internet-related research and development activities. These results might not be suitable for deployment. This RFC represents the consensus of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not a candidate for any level of Internet Standard; see Section 2 of RFC 7841. Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at http://www.rfc-editor.org/info/rfc8032. Copyright Notice Copyright (c) 2017 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.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Notation and Conventions . . . . . . . . . . . . . . . . . . 4 3. EdDSA Algorithm . . . . . . . . . . . . . . . . . . . . . . . 5 3.1. Encoding . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2. Keys . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3. Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4. Verify . . . . . . . . . . . . . . . . . . . . . . . . . 8 4. PureEdDSA, HashEdDSA, and Naming . . . . . . . . . . . . . . 8 5. EdDSA Instances . . . . . . . . . . . . . . . . . . . . . . . 9 5.1. Ed25519ph, Ed25519ctx, and Ed25519 . . . . . . . . . . . 9 5.1.1. Modular Arithmetic . . . . . . . . . . . . . . . . . 10 5.1.2. Encoding . . . . . . . . . . . . . . . . . . . . . . 10 5.1.3. Decoding . . . . . . . . . . . . . . . . . . . . . . 11 5.1.4. Point Addition . . . . . . . . . . . . . . . . . . . 11 5.1.5. Key Generation . . . . . . . . . . . . . . . . . . . 13 5.1.6. Sign . . . . . . . . . . . . . . . . . . . . . . . . 13 5.1.7. Verify . . . . . . . . . . . . . . . . . . . . . . . 14 5.2. Ed448ph and Ed448 . . . . . . . . . . . . . . . . . . . . 15 5.2.1. Modular Arithmetic . . . . . . . . . . . . . . . . . 16 5.2.2. Encoding . . . . . . . . . . . . . . . . . . . . . . 16 5.2.3. Decoding . . . . . . . . . . . . . . . . . . . . . . 16 5.2.4. Point Addition . . . . . . . . . . . . . . . . . . . 17 5.2.5. Key Generation . . . . . . . . . . . . . . . . . . . 18 5.2.6. Sign . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2.7. Verify . . . . . . . . . . . . . . . . . . . . . . . 19 6. Ed25519 Python Illustration . . . . . . . . . . . . . . . . . 20 7. Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . 23 7.1. Test Vectors for Ed25519 . . . . . . . . . . . . . . . . 24 7.2. Test Vectors for Ed25519ctx . . . . . . . . . . . . . . . 27 7.3. Test Vectors for Ed25519ph . . . . . . . . . . . . . . . 30 7.4. Test Vectors for Ed448 . . . . . . . . . . . . . . . . . 30 7.5. Test Vectors for Ed448ph . . . . . . . . . . . . . . . . 38 8. Security Considerations . . . . . . . . . . . . . . . . . . . 40 8.1. Side-Channel Leaks . . . . . . . . . . . . . . . . . . . 40 8.2. Randomness Considerations . . . . . . . . . . . . . . . . 40 8.3. Use of Contexts . . . . . . . . . . . . . . . . . . . . . 41 8.4. Signature Malleability . . . . . . . . . . . . . . . . . 41 8.5. Choice of Signature Primitive . . . . . . . . . . . . . . 41 8.6. Mixing Different Prehashes . . . . . . . . . . . . . . . 42 8.7. Signing Large Amounts of Data at Once . . . . . . . . . . 42 8.8. Multiplication by Cofactor in Verification . . . . . . . 43 8.9. Use of SHAKE256 as a Hash Function . . . . . . . . . . . 43 9. References . . . . . . . . . . . . . . . . . . . . . . . . . 43 9.1. Normative References . . . . . . . . . . . . . . . . . . 43 9.2. Informative References . . . . . . . . . . . . . . . . . 44
Appendix A. Ed25519/Ed448 Python Library . . . . . . . . . . . . 46 Appendix B. Library Driver . . . . . . . . . . . . . . . . . . . 58 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 60 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 601. Introduction
The Edwards-curve Digital Signature Algorithm (EdDSA) is a variant of Schnorr's signature system with (possibly twisted) Edwards curves. EdDSA needs to be instantiated with certain parameters, and this document describes some recommended variants. To facilitate adoption of EdDSA in the Internet community, this document describes the signature scheme in an implementation-oriented way and provides sample code and test vectors. The advantages with EdDSA are as follows: 1. EdDSA provides high performance on a variety of platforms; 2. The use of a unique random number for each signature is not required; 3. It is more resilient to side-channel attacks; 4. EdDSA uses small public keys (32 or 57 bytes) and signatures (64 or 114 bytes) for Ed25519 and Ed448, respectively; 5. The formulas are "complete", 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; and 6. EdDSA provides collision resilience, meaning that hash-function collisions do not break this system (only holds for PureEdDSA). The original EdDSA paper [EDDSA] and the generalized version described in "EdDSA for more curves" [EDDSA2] provide further background. RFC 7748 [RFC7748] discusses specific curves, including Curve25519 [CURVE25519] and Ed448-Goldilocks [ED448]. Ed25519 is intended to operate at around the 128-bit security level and Ed448 at around the 224-bit security level. A sufficiently large quantum computer would be able to break both. Reasonable projections of the abilities of classical computers conclude that Ed25519 is perfectly safe. Ed448 is provided for those applications with relaxed performance requirements and where there is a desire to hedge against analytical attacks on elliptic curves.
2. Notation and Conventions
The following notation is used throughout the document: p Denotes the prime number defining the underlying field 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]X X added to itself n times h[i] The i'th octet of octet string h_i The i'th bit of h a || b (bit-)string a concatenated with (bit-)string b a <= b a is less than or equal to b a >= b a is greater than or equal to b i+j Sum of i and j i*j Multiplication of i and j i-j Subtraction of j from i i/j Division of i by j i x j Cartesian product of i and j (u,v) Elliptic curve point with x-coordinate u and y-coordinate v SHAKE256(x, y) The y first octets of SHAKE256 [FIPS202] output for input x OCTET(x) The octet with value x OLEN(x) The number of octets in string x
dom2(x, y) The blank octet string when signing or verifying
Ed25519. Otherwise, the octet string: "SigEd25519 no
Ed25519 collisions" || octet(x) || octet(OLEN(y)) ||
y, where x is in range 0-255 and y is an octet string
of at most 255 octets. "SigEd25519 no Ed25519
collisions" is in ASCII (32 octets).
dom4(x, y) The octet string "SigEd448" || octet(x) ||
octet(OLEN(y)) || y, where x is in range 0-255 and y
is an octet string of at most 255 octets. "SigEd448"
is in ASCII (8 octets).
Parentheses (i.e., '(' and ')') are used to group expressions, in
order to avoid having the description depend on a binding order
between operators.
Bit strings are converted to octet strings by taking bits from left
to right, packing those from the least significant bit of each octet
to the most significant bit, and moving to the next octet when each
octet fills up. The conversion from octet string to bit string is
the reverse of this process; for example, the 16-bit bit string
b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15
is converted into two octets x0 and x1 (in this order) as
x0 = b7*128+b6*64+b5*32+b4*16+b3*8+b2*4+b1*2+b0
x1 = b15*128+b14*64+b13*32+b12*16+b11*8+b10*4+b9*2+b8
Little-endian encoding into bits places bits from left to right and
from least significant to most significant. If combined with
bit-string-to-octet-string conversion defined above, this results in
little-endian encoding into octets (if length is not a multiple of 8,
the most significant bits of the last octet remain unused).
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].
3. EdDSA Algorithm
EdDSA is a digital signature system with 11 parameters.
The generic EdDSA digital signature system with its 11 input
parameters is not intended to be implemented directly. Choosing
parameters is critical for secure and efficient operation. Instead,
you would implement a particular parameter choice for EdDSA (such as
Ed25519 or Ed448), sometimes slightly generalized to achieve code reuse to cover Ed25519 and Ed448. Therefore, a precise explanation of the generic EdDSA is thus not particularly useful for implementers. For background and completeness, a succinct description of the generic EdDSA algorithm is given here. The definition of some parameters, such as n and c, may help to explain some steps of the algorithm that are not intuitive. This description closely follows [EDDSA2]. EdDSA has 11 parameters: 1. An odd prime power p. EdDSA uses an elliptic curve over the finite field GF(p). 2. An integer b with 2^(b-1) > p. EdDSA public keys have exactly b bits, and EdDSA signatures have exactly 2*b bits. b is recommended to be a multiple of 8, so public key and signature lengths are an integral number of octets. 3. A (b-1)-bit encoding of elements of the finite field GF(p). 4. A cryptographic hash function H producing 2*b-bit output. Conservative hash functions (i.e., hash functions where it is infeasible to create collisions) are recommended and do not have much impact on the total cost of EdDSA. 5. An integer c that is 2 or 3. Secret EdDSA scalars are multiples of 2^c. The integer c is the base-2 logarithm of the so-called cofactor. 6. An integer n with c <= n < b. Secret EdDSA scalars have exactly n + 1 bits, with the top bit (the 2^n position) always set and the bottom c bits always cleared. 7. A non-square element d of GF(p). The usual recommendation is to take it as the value nearest to zero that gives an acceptable curve. 8. A non-zero square element a of GF(p). The usual recommendation for best performance is a = -1 if p mod 4 = 1, and a = 1 if p mod 4 = 3. 9. An element B != (0,1) of the set E = { (x,y) is a member of GF(p) x GF(p) such that a * x^2 + y^2 = 1 + d * x^2 * y^2 }.
10. An odd prime L such that [L]B = 0 and 2^c * L = #E. The number
#E (the number of points on the curve) is part of the standard
data provided for an elliptic curve E, or it can be computed as
cofactor * order.
11. A "prehash" function PH. PureEdDSA means EdDSA where PH is the
identity function, i.e., PH(M) = M. HashEdDSA means EdDSA where
PH generates a short output, no matter how long the message is;
for example, PH(M) = SHA-512(M).
Points on the curve form a group under addition, (x3, y3) = (x1, y1)
+ (x2, y2), with the formulas
x1 * y2 + x2 * y1 y1 * y2 - a * 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 many other curves used for cryptographic applications, these
formulas are "complete"; 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 complete, that use
homogeneous coordinates to avoid the expensive modulo p inversions.
See [Faster-ECC] and [Edwards-revisited].
3.1. Encoding
An integer 0 < S < L - 1 is encoded in little-endian form as a b-bit
string ENC(S).
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.
The encoding of GF(p) is used to define "negative" elements of GF(p):
specifically, x is negative if the (b-1)-bit encoding of x is
lexicographically larger than the (b-1)-bit encoding of -x.
3.2. Keys
An EdDSA private key is a b-bit string k. Let the hash H(k) =
(h_0, h_1, ..., h_(2b-1)) determine an integer s, which is 2^n plus
the sum of m = 2^i * h_i for all integer i, c <= i < n. Let s
determine the multiple A = [s]B. The EdDSA public key is ENC(A).
The bits h_b, ..., h_(2b-1) are used below during signing.
3.3. Sign
The EdDSA signature of a message M under a private key k is defined as the PureEdDSA signature of PH(M). In other words, EdDSA simply uses PureEdDSA to sign PH(M). The PureEdDSA signature of a message M under a private key k is the 2*b-bit string ENC(R) || ENC(S). R and S are derived as follows. First define r = H(h_b || ... || h_(2b-1) || M) interpreting 2*b-bit strings in little-endian form as integers in {0, 1, ..., 2^(2*b) - 1}. Let R = [r]B and S = (r + H(ENC(R) || ENC(A) || PH(M)) * s) mod L. The s used here is from the previous section.3.4. Verify
To verify a PureEdDSA 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 are elements 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 [2^c * S] B = 2^c * R + [2^c * h] A in E. The signature is rejected if parsing fails (including S being out of range) or if the group equation does not hold. EdDSA verification for a message M is defined as PureEdDSA verification for PH(M).4. PureEdDSA, HashEdDSA, and Naming
One of the parameters of the EdDSA algorithm is the "prehash" function. This may be the identity function, resulting in an algorithm called PureEdDSA, or a collision-resistant hash function such as SHA-512, resulting in an algorithm called HashEdDSA. Choosing which variant to use depends on which property is deemed to be more important between 1) collision resilience and 2) a single- pass interface for creating signatures. The collision resilience property means EdDSA is secure even if it is feasible to compute collisions for the hash function. The single-pass interface property means that only one pass over the input message is required to create a signature. PureEdDSA requires two passes over the input. Many existing APIs, protocols, and environments assume digital signature algorithms only need one pass over the input and may have API or bandwidth concerns supporting anything else. Note that single-pass verification is not possible with most uses of signatures, no matter which signature algorithm is chosen. This is because most of the time, one can't process the message until the signature is validated, which needs a pass on the entire message.
This document specifies parameters resulting in the HashEdDSA variants Ed25519ph and Ed448ph and the PureEdDSA variants Ed25519 and Ed448.5. EdDSA Instances
This section instantiates the general EdDSA algorithm for the edwards25519 and edwards448 curves, each for the PureEdDSA and HashEdDSA variants (plus a contextualized extension of the Ed25519 scheme). Thus, five different parameter sets are described.5.1. Ed25519ph, Ed25519ctx, and Ed25519
Ed25519 is EdDSA instantiated with: +-----------+-------------------------------------------------------+ | Parameter | Value | +-----------+-------------------------------------------------------+ | p | p of edwards25519 in [RFC7748] (i.e., 2^255 - 19) | | b | 256 | | encoding | 255-bit little-endian encoding of {0, 1, ..., p-1} | | of GF(p) | | | H(x) | SHA-512(dom2(phflag,context)||x) [RFC6234] | | c | base 2 logarithm of cofactor of edwards25519 in | | | [RFC7748] (i.e., 3) | | n | 254 | | d | d of edwards25519 in [RFC7748] (i.e., -121665/121666 | | | = 370957059346694393431380835087545651895421138798432 | | | 19016388785533085940283555) | | a | -1 | | B | (X(P),Y(P)) of edwards25519 in [RFC7748] (i.e., (1511 | | | 22213495354007725011514095885315114540126930418572060 | | | 46113283949847762202, 4631683569492647816942839400347 | | | 5163141307993866256225615783033603165251855960)) | | L | order of edwards25519 in [RFC7748] (i.e., | | | 2^252+27742317777372353535851937790883648493). | | PH(x) | x (i.e., the identity function) | +-----------+-------------------------------------------------------+ Table 1: Parameters of Ed25519 For Ed25519, dom2(f,c) is the empty string. The phflag value is irrelevant. The context (if present at all) MUST be empty. This causes the scheme to be one and the same with the Ed25519 scheme published earlier. For Ed25519ctx, phflag=0. The context input SHOULD NOT be empty.
For Ed25519ph, phflag=1 and PH is SHA512 instead. That is, the input is hashed using SHA-512 before signing with Ed25519. Value of context is set by the signer and verifier (maximum of 255 octets; the default is empty string, except for Ed25519, which can't have context) and has to match octet by octet for verification to be successful. 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 Curve25519.5.1.1. Modular Arithmetic
For advice on how to implement arithmetic modulo p = 2^255 - 19 efficiently and securely, see Curve25519 [CURVE25519]. For inversion modulo p, it is recommended to use the identity x^-1 = x^(p-2) (mod p). Inverting zero should never happen, as it would require invalid input, which would have been detected before, or would be a calculation error. 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.5.1.2. 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 coordinates 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.
5.1.3. 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). The denominator is always non-zero mod 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 root x = (u/v)^((p+3)/8). This can be done with the following trick, using 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 for 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).5.1.4. 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 neutral point is (0,1), or equivalently in extended homogeneous coordinates (0, Z, Z, 0) for any non-zero Z.
The following formulas for adding two points, (x3,y3) = (x1,y1)+(x2,y2), on twisted Edwards curves with a=-1, square a, and non-square d are described in Section 3.1 of [Edwards-revisited] and in [EFD-TWISTED-ADD]. They are complete, 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 For point doubling, (x3,y3) = (x1,y1)+(x1,y1), one could just substitute equal points in the above (because of completeness, such substitution is valid) and observe that four multiplications turn into squares. However, using the formulas described in Section 3.2 of [Edwards-revisited] and in [EFD-TWISTED-DBL] saves a few smaller operations. A = X1^2 B = Y1^2 C = 2*Z1^2 H = A+B E = H-(X1+Y1)^2 G = A-B F = C+G X3 = E*F Y3 = G*H T3 = E*H Z3 = F*G
5.1.5. Key Generation
The private key 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 private key 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: The lowest three bits of the first octet are cleared, the highest bit of the last octet is cleared, and the second highest bit of the last octet is set. 3. Interpret the buffer as the little-endian integer, forming a secret scalar s. Perform a fixed-base scalar multiplication [s]B. 4. The public key A is the encoding of the point [s]B. 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 [s]B, copy the least significant bit of the x coordinate to the most significant bit of the final octet. The result is the public key.5.1.6. Sign
The inputs to the signing procedure is the private key, a 32-octet string, and a message M of arbitrary size. For Ed25519ctx and Ed25519ph, there is additionally a context C of at most 255 octets and a flag F, 0 for Ed25519ctx and 1 for Ed25519ph. 1. Hash the private key, 32 octets, using SHA-512. Let h denote the resulting digest. Construct the secret scalar s 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(dom2(F, C) || prefix || PH(M)), where M is the message to be signed. Interpret the 64-octet digest as a little- endian integer r. 3. Compute the point [r]B. For efficiency, do this by first reducing r modulo L, the group order of B. Let the string R be the encoding of this point.
4. Compute SHA512(dom2(F, C) || R || A || PH(M)), and interpret the
64-octet digest as a little-endian integer k.
5. Compute S = (r + k * s) mod L. For efficiency, again reduce k
modulo L first.
6. Form the signature of the concatenation of R (32 octets) and the
little-endian encoding of S (32 octets; the three most
significant bits of the final octet are always zero).
5.1.7. Verify
1. To verify a signature on a message M using public key A, with F
being 0 for Ed25519ctx, 1 for Ed25519ph, and if Ed25519ctx or
Ed25519ph is being used, C being the context, 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 < L. Decode the public key A as point A'. If any of the
decodings fail (including S being out of range), the signature is
invalid.
2. Compute SHA512(dom2(F, C) || R || A || PH(M)), and interpret the
64-octet digest as a little-endian integer k.
3. Check the group equation [8][S]B = [8]R + [8][k]A'. It's
sufficient, but not required, to instead check [S]B = R + [k]A'.
5.2. Ed448ph and Ed448
Ed448 is EdDSA instantiated with: +-----------+-------------------------------------------------------+ | Parameter | Value | +-----------+-------------------------------------------------------+ | p | p of edwards448 in [RFC7748] (i.e., 2^448 - 2^224 - | | | 1) | | b | 456 | | encoding | 455-bit little-endian encoding of {0, 1, ..., p-1} | | of GF(p) | | | H(x) | SHAKE256(dom4(phflag,context)||x, 114) | | phflag | 0 | | c | base 2 logarithm of cofactor of edwards448 in | | | [RFC7748] (i.e., 2) | | n | 447 | | d | d of edwards448 in [RFC7748] (i.e., -39081) | | a | 1 | | B | (X(P),Y(P)) of edwards448 in [RFC7748] (i.e., (224580 | | | 04029592430018760433409989603624678964163256413424612 | | | 54616869504154674060329090291928693579532825780320751 | | | 46446173674602635247710, 2988192100784814926760179304 | | | 43930673437544040154080242095928241372331506189835876 | | | 00353687865541878473398230323350346250053154506283266 | | | 0)) | | L | order of edwards448 in [RFC7748] (i.e., 2^446 - 13818 | | | 06680989511535200738674851542688033669247488217860989 | | | 4547503885). | | PH(x) | x (i.e., the identity function) | +-----------+-------------------------------------------------------+ Table 2: Parameters of Ed448 Ed448ph is the same but with PH being SHAKE256(x, 64) and phflag being 1, i.e., the input is hashed before signing with Ed448 with a hash constant modified. Value of context is set by signer and verifier (maximum of 255 octets; the default is empty string) and has to match octet by octet for verification to be successful. The curve is equivalent to Ed448-Goldilocks under change of the basepoint, which preserves difficulty of the discrete logarithm.
5.2.1. Modular Arithmetic
For advice on how to implement arithmetic modulo p = 2^448 - 2^224 - 1 efficiently and securely, see [ED448]. For inversion modulo p, it is recommended to use the identity x^-1 = x^(p-2) (mod p). Inverting zero should never happen, as it would require invalid input, which would have been detected before, or would be a calculation error. For point decoding or "decompression", square roots modulo p are needed. They can be computed by first computing candidate root x = a ^ (p+1)/4 (mod p) and then checking if x^2 = a. If it is, then x is the square root of a; if it isn't, then a does not have a square root.5.2.2. Encoding
All values are coded as octet strings, and integers are coded using little-endian convention, i.e., a 57-octet string h h[0],...h[56] represents the integer h[0] + 2^8 * h[1] + ... + 2^448 * h[56]. A curve point (x,y), with coordinates in the range 0 <= x,y < p, is coded as follows. First, encode the y-coordinate as a little-endian string of 57 octets. 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.5.2.3. Decoding
Decoding a point, given as a 57-octet string, is a little more complicated. 1. First, interpret the string as an integer in little-endian representation. Bit 455 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). The denominator is always non-zero mod 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 root x = (u/v)^((p+1)/4). 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+1)/4 3 (p-3)/4 x = (u/v) = u v (u^5 v^3) (mod p)
3. If v * x^2 = u, the recovered x-coordinate is x. Otherwise, no
square root exists, and the 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).
5.2.4. Point Addition
For point addition, the following method is recommended. A point
(x,y) is represented in projective coordinates (X, Y, Z), with
x = X/Z, y = Y/Z.
The neutral point is (0,1), or equivalently in projective coordinates
(0, Z, Z) for any non-zero Z.
The following formulas for adding two points, (x3,y3) =
(x1,y1)+(x2,y2) on untwisted Edwards curve (i.e., a=1) with non-
square d, are described in Section 4 of [Faster-ECC] and in
[EFD-ADD]. They are complete, i.e., they work for any pair of valid
input points.
A = Z1*Z2
B = A^2
C = X1*X2
D = Y1*Y2
E = d*C*D
F = B-E
G = B+E
H = (X1+Y1)*(X2+Y2)
X3 = A*F*(H-C-D)
Y3 = A*G*(D-C)
Z3 = F*G
Again, similar to the other curve, doubling formulas can be obtained by substituting equal points, turning four multiplications into squares. However, this is not even nearly optimal; the following formulas described in Section 4 of [Faster-ECC] and in [EFD-DBL] save multiple multiplications. B = (X1+Y1)^2 C = X1^2 D = Y1^2 E = C+D H = Z1^2 J = E-2*H X3 = (B-E)*J Y3 = E*(C-D) Z3 = E*J5.2.5. Key Generation
The private key is 57 octets (456 bits, corresponding to b) of cryptographically secure random data. See [RFC4086] for a discussion about randomness. The 57-byte public key is generated by the following steps: 1. Hash the 57-byte private key using SHAKE256(x, 114), storing the digest in a 114-octet large buffer, denoted h. Only the lower 57 bytes are used for generating the public key. 2. Prune the buffer: The two least significant bits of the first octet are cleared, all eight bits the last octet are cleared, and the highest bit of the second to last octet is set. 3. Interpret the buffer as the little-endian integer, forming a secret scalar s. Perform a known-base-point scalar multiplication [s]B. 4. The public key A is the encoding of the point [s]B. First encode the y-coordinate (in the range 0 <= y < p) as a little-endian string of 57 octets. The most significant bit of the final octet is always zero. To form the encoding of the point [s]B, copy the least significant bit of the x coordinate to the most significant bit of the final octet. The result is the public key.
5.2.6. Sign
The inputs to the signing procedure is the private key, a 57-octet string, a flag F, which is 0 for Ed448, 1 for Ed448ph, context C of at most 255 octets, and a message M of arbitrary size. 1. Hash the private key, 57 octets, using SHAKE256(x, 114). Let h denote the resulting digest. Construct the secret scalar s 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[57],...,h[113]. 2. Compute SHAKE256(dom4(F, C) || prefix || PH(M), 114), where M is the message to be signed, F is 1 for Ed448ph, 0 for Ed448, and C is the context to use. Interpret the 114-octet digest as a little-endian integer r. 3. Compute the point [r]B. For efficiency, do this by first reducing r modulo L, the group order of B. Let the string R be the encoding of this point. 4. Compute SHAKE256(dom4(F, C) || R || A || PH(M), 114), and interpret the 114-octet digest as a little-endian integer k. 5. Compute S = (r + k * s) mod L. For efficiency, again reduce k modulo L first. 6. Form the signature of the concatenation of R (57 octets) and the little-endian encoding of S (57 octets; the ten most significant bits of the final octets are always zero).5.2.7. Verify
1. To verify a signature on a message M using context C and public key A, with F being 0 for Ed448 and 1 for Ed448ph, first split the signature into two 57-octet halves. Decode the first half as a point R, and the second half as an integer S, in the range 0 <= s < L. Decode the public key A as point A'. If any of the decodings fail (including S being out of range), the signature is invalid. 2. Compute SHAKE256(dom4(F, C) || R || A || PH(M), 114), and interpret the 114-octet digest as a little-endian integer k. 3. Check the group equation [4][S]B = [4]R + [4][k]A'. It's sufficient, but not required, to instead check [S]B = R + [k]A'.
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