RFC 8032
Edwards-Curve Digital Signature Algorithm (EdDSA)
Edwards-Curve Digital Signature Algorithm (EdDSA) ✨
okay bestie, here's the title block tea 🍵
- Who wrote it: S. Josefsson (SJD AB) and I. Liusvaara (Independent) 💖
- Stream: Internet Research Task Force (IRTF) 🧚
- Category: Informational 🌸
- Number and ISSN: RFC 8032, ISSN 2070-1721 🎀
- Date: January 2017 📅
- Obsoletes / Updates: the header doesn't name anything it obsoletes or updates 🤷♀️
Abstract 🦋
okay so this doc describes an elliptic curve signature scheme called the Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm gets instantiated with recommended parameters for the edwards25519 and edwards448 curves 🔥. There's also an example implementation and test vectors, so you can check your work 💅.
Status of This Memo
this one is NOT a standards track thing 🙈 It's published for informational purposes, so it's more of a "here's the good stuff, go learn" kind of doc 📚.
It's a product of the Internet Research Task Force (IRTF), which publishes Internet-related research and development results. Those results might not be suitable for deployment, so keep that in mind 👀. This RFC represents the consensus of the Crypto Forum Research Group of the IRTF. Documents approved for publication by the IRSG are not a candidate for any level of Internet Standard, and you can read Section 2 of RFC 7841 for the details 💀.
If you want the current status, any errata, or a way to give feedback, check 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. Please read those carefully, because they spell out 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 💖
EdDSA is a variant of Schnorr's signature system that uses (possibly twisted) Edwards curves 🌊. It needs certain parameters to be instantiated, and this doc describes some recommended variants 🎀.
To help EdDSA get adopted across the Internet community, the doc describes the signature scheme in an implementation-oriented way. It also comes with sample code and test vectors 🧑💻. So basically, it's built for people who actually want to ship it 🔥.
The advantages with EdDSA are as follows ✨:
- EdDSA provides high performance on a variety of platforms 🚀;
- The use of a unique random number for each signature is not required 🙌;
- It is more resilient to side-channel attacks 🛡️;
- EdDSA uses small public keys (32 or 57 bytes) and signatures (64 or 114 bytes) for Ed25519 and Ed448, respectively 🦋;
- 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
- 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] give you the deeper background 📚. RFC 7748 [RFC7748] talks about specific curves, including Curve25519 [CURVE25519] and Ed448-Goldilocks [ED448] 🌸.
ok the security levels are the big deal 😭 Ed25519 is meant to run 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 what classical computers can do conclude that Ed25519 is perfectly safe 🫶. Ed448 is there for applications with relaxed performance requirements, and for anyone who wants to hedge against analytical attacks on elliptic curves 🧚.
2. Notation and Conventions 🎀
Here's the notation this doc uses all the way through 📝. Notice the table is kept exactly as written:
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).So the two domain helpers are dom2(x, y) and dom4(x, y) 🧁. For dom2, you get a blank octet string when signing or verifying Ed25519. Otherwise you get "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. That first string is 32 octets of ASCII, which is a lot of vibes in one place 🫠. For dom4, you get "SigEd448" || octet(x) || octet(OLEN(y)) || y with the same range rules, and "SigEd448" is 8 octets of ASCII 💜.
Parentheses are used to group expressions, so the description doesn't depend on a binding order between operators 🧩.
Bit strings get turned into octet strings by taking bits from left to right, packing them from the least significant bit of each octet up to the most significant bit, and moving to the next octet once each one fills up 🍓. Going from octet string back to bit string is the reverse of that. For example, a 16-bit bit string gets turned into two octets, x0 and x1 (in that order), like this:
b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 b15
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+b8Little-endian encoding into bits places bits from left to right and from least significant to most significant 🌈. Combine that with the bit-string-to-octet-string conversion above and you get little-endian encoding into octets. If the length isn't a multiple of 8, the most significant bits of the last octet just stay 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] ✨.
to be continued 🦋 (part 1 of 9)
3. EdDSA Algorithm ✨
okay so EdDSA is a digital signature system with 11 parameters 💅 that's the whole vibe. 🎀
You're not supposed to implement the generic EdDSA system directly, bestie. 🙈 Picking the parameters is critical for it to be secure and fast, so in practice you implement a specific parameter choice, like Ed25519 or Ed448. Sometimes you generalize it a little so the same code can cover both Ed25519 and Ed448. 🧚
So a super precise explanation of generic EdDSA isn't really useful for implementers. 🦋 It's here anyway for background and completeness, because it's giving thorough. 💜
The definitions of some parameters, like n and c, help explain some steps that aren't intuitive at first. ✨
This description closely follows [EDDSA2]. 🌸
EdDSA has 11 parameters:
- An odd prime power p. EdDSA uses an elliptic curve over the finite field GF(p). 🌈
- 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. 🍓
- A (b-1)-bit encoding of elements of the finite field GF(p). 🫶
- 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. 🔥
- 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. 👑
- 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. ✨
- 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. 🦋
- 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. 💖
- 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 }. 🌸
- 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. 🧚
- 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 * y2The 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. Literally no edge-case drama. 💅
There are more efficient formulas that are still complete. They 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). It's 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}. 🔥
Then 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, which gives an algorithm called PureEdDSA. Or it may be a collision-resistant hash function such as SHA-512, which gives an algorithm called HashEdDSA. 🌸
Choosing which variant to use depends on which property matters more: 1) collision resilience, or 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 they 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 one gets 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 Ed25519For Ed25519, dom2(f,c) is the empty string. The phflag value is irrelevant. The context (if present at all) MUST be empty. This makes the scheme one and the same as 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 means the input is hashed using SHA-512 before signing with Ed25519. 🦋
The value of context is set by the signer and verifier. It has a maximum of 255 octets, and the default is the empty string, except for Ed25519, which can't have context. It has to match octet by octet for verification to be successful. 🍓
The curve used is equivalent to Curve25519 [CURVE25519], under a change of coordinates. This 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). Then2^((p-1)/4) * xis 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. That means 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
okay bestie, decoding a point from a 32-octet string is a little more complicated 🫶 but we're gonna walk through it together, no stress 💅
- First, read the string as an integer in little-endian representation. Bit 255 of this number is the least significant bit of the x-coordinate, and we call this value x_0. To get the y-coordinate, just clear that bit 🧚 If the resulting value is >= p, decoding fails 💀
- To recover the x-coordinate, the curve equation gives us x^2 = (y^2 - 1) / (d y^2 + 1) (mod p). The denominator is never zero mod p, so we're safe ✨ Let u = y^2 - 1 and v = d y^2 + 1. To find the square root of (u/v), the first step is to compute the candidate root x = (u/v)^((p+3)/8). This trick is so clever 😭 It uses a single modular powering to handle 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)- Okay, now there are three cases 🎀
- If v x^2 = u (mod p), then x is a square root.
- If v x^2 = -u (mod p), set x <-- x * 2^((p-1)/4), and that gives you a square root.
- Otherwise, no square root exists modulo p, and decoding fails 🥀
- Last step! Use the x_0 bit to pick the right square root 👑 If x = 0 and x_0 = 1, decoding fails. Otherwise, if x_0 != x mod 2, set x <-- p - x. Then return the decoded point (x,y) 🦋
5.1.4. Point Addition
For point addition, the following method is recommended 💖 A point (x,y) gets represented in extended homogeneous coordinates (X, Y, Z, T), where x = X/Z, y = Y/Z, and x * y = T/Z. Very aesthetic, very organized 🌸
The neutral point is (0,1). In extended homogeneous coordinates, that's (0, Z, Z, 0) for any non-zero Z, so you get to pick your Z and it still slays ✨
The following formulas add two points, (x3,y3) = (x1,y1)+(x2,y2), on twisted Edwards curves with a=-1, square a, and non-square d. They're described in Section 3.1 of [Edwards-revisited] and in [EFD-TWISTED-ADD]. The best part is that they're complete, meaning they work for any pair of valid input points. Iconic behaviour 👑
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*GFor point doubling, (x3,y3) = (x1,y1)+(x1,y1), you could just plug equal points into the formulas above, since completeness makes that valid, and then notice that four multiplications turn into squares 🙈 But the formulas in Section 3.2 of [Edwards-revisited] and in [EFD-TWISTED-DBL] save a few smaller operations, so that's the move 💅
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*G5.1.5. Key Generation
The private key is 32 octets (256 bits, corresponding to b) of cryptographically secure random data 🔐 For a deep dive on randomness, check out [RFC4086] 🌈
The 32-byte public key is generated in these steps:
- Hash the 32-byte private key with SHA-512, storing the digest in a 64-octet large buffer called h. Only the lower 32 bytes are used to generate the public key 🍓
- Prune the buffer: clear the lowest three bits of the first octet, clear the highest bit of the last octet, and set the second highest bit of the last octet 🧁
- Interpret the buffer as a little-endian integer to form a secret scalar s. Then perform a fixed-base scalar multiplication [s]B 💜
- 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 [s]B, copy the least significant bit of the x-coordinate to the most significant bit of the final octet. And that's your public key! 🎀
5.1.6. Sign
The inputs to the signing procedure are the private key, a 32-octet string, and a message M of arbitrary size 📨 For Ed25519ctx and Ed25519ph, there's also a context C of at most 255 octets, plus a flag F: 0 for Ed25519ctx and 1 for Ed25519ph ✨
- Hash the private key (32 octets) with SHA-512. Call the resulting digest h. Build the secret scalar s from the first half of the digest, and build the corresponding public key A as described in the previous section. Let prefix be the second half of the hash digest, h[32],...,h[63] 🦋
- Compute SHA-512(dom2(F, C) || prefix || PH(M)), where M is the message being signed. Interpret the 64-octet digest as a little-endian integer r 🌸
- Compute the point [r]B. For efficiency, first reduce r modulo L, the group order of B. Let the string R be the encoding of this point 🧚
- Compute SHA512(dom2(F, C) || R || A || PH(M)), and interpret the 64-octet digest as a little-endian integer k 💖
- Compute S = (r + k * s) mod L. For efficiency, reduce k modulo L first again 🫶
- Form the signature by concatenating R (32 octets) with the little-endian encoding of S (32 octets; the three most significant bits of the final octet are always zero) 🍓
5.1.7. Verify
- To verify a signature on a message M using public key A, with F being 0 for Ed25519ctx and 1 for Ed25519ph (and C being the context if Ed25519ctx or Ed25519ph is used), 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 decoding fails (including S being out of range), the signature is invalid 🚫
- Compute SHA512(dom2(F, C) || R || A || PH(M)), and interpret the 64-octet digest as a little-endian integer k 🔥
- Check the group equation
[8][S]B = [8]R + [8][k]A'. It's sufficient, but not required, to check[S]B = R + [k]A'instead 🌈
5.2. Ed448ph and Ed448
Ed448 is EdDSA instantiated with the parameters below 🍓 Look at this table, it's so organized 🥹
+-----------+-------------------------------------------------------+
| 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 Ed448Ed448ph is basically the same, but with PH being SHAKE256(x, 64) and phflag being 1. In other words, the input gets hashed before signing with Ed448, with a hash constant modified 🦋
The context value is set by the signer and the verifier. It can be at most 255 octets, and the default is the empty string. It has to match octet by octet for verification to succeed 🫶
The curve is equivalent to Ed448-Goldilocks under a change of basepoint, which preserves the difficulty of the discrete logarithm. So it's giving security 👑✨
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, because that would need invalid input (which would've been caught earlier) or a calculation error 🙈
For point decoding (aka "decompression"), we need square roots modulo p. You can compute them by first finding the candidate root x = a^((p+1)/4) (mod p), and then checking whether x^2 = a. If it is, then x is the square root of a. If it isn't, then a doesn't have a square root 🌸
5.2.2. Encoding
All values are coded as octet strings, and integers use the little-endian convention. So 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 like this. 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 from a 57-octet string is a little more complicated 🥺 but you got this bestie 💪
- First, read the string as an integer in little-endian representation. Bit 455 of this number is the least significant bit of the x-coordinate, and we call this value x_0. To get the y-coordinate, clear this bit. If the resulting value is >= p, decoding fails 🚫
- To recover the x-coordinate, the curve equation gives us x^2 = (y^2 - 1) / (d y^2 - 1) (mod p). The denominator is never zero mod p. Let u = y^2 - 1 and v = d y^2 - 1. To find the square root of (u/v), the first step is to compute the candidate root x = (u/v)^((p+1)/4). This trick uses a single modular powering to handle 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)- If v * x^2 = u, the recovered x-coordinate is x. Otherwise, no square root exists, and decoding fails 💀
- Last step! Use the x_0 bit to pick the right square root 👑 If x = 0 and x_0 = 1, decoding fails. Otherwise, if x_0 != x mod 2, set x <-- p - x. Then return the decoded point (x,y) 🦋
5.2.4. Point Addition 💖
okay so for point addition, the method being recommended is this one 💅 A point (x,y) lives in projective coordinates (X, Y, Z), where x = X/Z and y = Y/Z.
the neutral point is (0,1), or equivalently (0, Z, Z) in projective coordinates for any non-zero Z 🦋
these formulas for adding two points, (x3,y3) = (x1,y1)+(x2,y2), live on the untwisted Edwards curve (aka a=1) with a non-square d. they're described in Section 4 of [Faster-ECC] and in [EFD-ADD] and they are giving complete energy, meaning 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*Gagain, similar to the other curve, you can get doubling formulas by swapping in equal points, which turns four multiplications into squares. but ngl that's not even close to optimal 🙈 the next formulas, described in Section 4 of [Faster-ECC] and in [EFD-DBL], save multiple multiplications and are the iconic move 👑
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 🔐 if you want the vibe check on randomness, see [RFC4086].
the 57-byte public key gets generated in these steps:
- 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.
- Prune the buffer: the two least significant bits of the first octet are cleared, all eight bits of the last octet are cleared, and the highest bit of the second to last octet is set. ✂️
- Interpret the buffer as the little-endian integer, forming a secret scalar s. Then perform a known-base-point scalar multiplication [s]B. 🧚
- 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 (0 for Ed448, 1 for Ed448ph), context C of at most 255 octets, and a message M of arbitrary size. main character energy 👑
- 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]. 🌈 - 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. 🎀 - 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. 🦋
- Compute
SHAKE256(dom4(F, C) || R || A || PH(M), 114), and interpret the 114-octet digest as a little-endian integer k. ✨ - Compute
S = (r + k * s) mod L. For efficiency, again reduce k modulo L first. 🔥 - Form the signature as 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 🔍
- 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. 💀 - Compute
SHAKE256(dom4(F, C) || R || A || PH(M), 114), and interpret the 114-octet digest as a little-endian integer k. 🌸 - 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'. 🧚♀️
6. Ed25519 Python Illustration 🐍
the rest of this section shows how Ed25519 can be implemented in Python (version 3.2 or later), just for illustration 💖 see Appendix A for the complete implementation and Appendix B for a test driver that runs it through some test vectors.
Heads up bestie: this code is NOT for production. It isn't proven to be correct for all inputs, and it doesn't protect against side-channel attacks. The whole point is to illustrate the algorithm so you can build your own implementation. 😭⚠️
## First, some preliminaries that will be needed.
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.
# 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, B = (P[1]-P[0]) * (Q[1]-Q[0]) % p, (P[1]+P[0]) * (Q[1]+Q[0]) % p;
C, D = 2 * P[3] * Q[3] * d % p, 2 * P[2] * Q[2] % p;
E, F, G, H = B-A, D-C, D+C, 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:
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.
# 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):
if y >= p:
return None
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)
## These are functions for manipulating the private key.
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")
if s >= q: return False
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))7. Test Vectors 🧪
this section has test vectors for Ed25519ph, Ed25519ctx, Ed448ph, Ed25519, and Ed448 🌈 each section is a sequence of test vectors, with the octets hex encoded and whitespace added so they're easier to read. Ed25519, Ed25519ctx, and Ed25519ph private and public keys are 32 octets, and their signatures are 64 octets. Ed448 and Ed448ph private and public keys are 57 octets, and their signatures are 114 octets. Messages can be any length. If the context is non-empty, it's given as 1-255 octets. the vibes are immaculate 💅✨
7.1. Test Vectors for Ed25519
okay bestie, now we get to the fun part 🧪✨ these are the answer-key vibes, and they come from [ED25519-TEST-VECTORS] (with a couple of tweaks: we dropped the public key that was tacked onto the end of the private key, and we took the message out of the signature 💅) plus [ED25519-LIBGCRYPT-TEST-VECTORS]. Every single hex string below is copied exactly as written 🫶
-----TEST 1
ALGORITHM:
Ed25519
SECRET KEY:
9d61b19deffd5a60ba844af492ec2cc4
4449c5697b326919703bac031cae7f60
PUBLIC KEY:
d75a980182b10ab7d54bfed3c964073a
0ee172f3daa62325af021a68f707511a
MESSAGE (length 0 bytes):
SIGNATURE:
e5564300c360ac729086e2cc806e828a
84877f1eb8e5d974d873e06522490155
5fb8821590a33bacc61e39701cf9b46b
d25bf5f0595bbe24655141438e7a100b
-----TEST 2
ALGORITHM:
Ed25519
SECRET KEY:
4ccd089b28ff96da9db6c346ec114e0f
5b8a319f35aba624da8cf6ed4fb8a6fb
PUBLIC KEY:
3d4017c3e843895a92b70aa74d1b7ebc
9c982ccf2ec4968cc0cd55f12af4660c
MESSAGE (length 1 byte):
72
SIGNATURE:
92a009a9f0d4cab8720e820b5f642540
a2b27b5416503f8fb3762223ebdb69da
085ac1e43e15996e458f3613d0f11d8c
387b2eaeb4302aeeb00d291612bb0c00
-----TEST 3
ALGORITHM:
Ed25519
SECRET KEY:
c5aa8df43f9f837bedb7442f31dcb7b1
66d38535076f094b85ce3a2e0b4458f7
PUBLIC KEY:
fc51cd8e6218a1a38da47ed00230f058
0816ed13ba3303ac5deb911548908025
MESSAGE (length 2 bytes):
af82
SIGNATURE:
6291d657deec24024827e69c3abe01a3
0ce548a284743a445e3680d7db5ac3ac
18ff9b538d16f290ae67f760984dc659
4a7c15e9716ed28dc027beceea1ec40a
-----TEST 1024
ALGORITHM:
Ed25519
SECRET KEY:
f5e5767cf153319517630f226876b86c
8160cc583bc013744c6bf255f5cc0ee5
PUBLIC KEY:
278117fc144c72340f67d0f2316e8386
ceffbf2b2428c9c51fef7c597f1d426e
MESSAGE (length 1023 bytes):
08b8b2b733424243760fe426a4b54908
632110a66c2f6591eabd3345e3e4eb98
fa6e264bf09efe12ee50f8f54e9f77b1
e355f6c50544e23fb1433ddf73be84d8
79de7c0046dc4996d9e773f4bc9efe57
38829adb26c81b37c93a1b270b20329d
658675fc6ea534e0810a4432826bf58c
941efb65d57a338bbd2e26640f89ffbc
1a858efcb8550ee3a5e1998bd177e93a
7363c344fe6b199ee5d02e82d522c4fe
ba15452f80288a821a579116ec6dad2b
3b310da903401aa62100ab5d1a36553e
06203b33890cc9b832f79ef80560ccb9
a39ce767967ed628c6ad573cb116dbef
efd75499da96bd68a8a97b928a8bbc10
3b6621fcde2beca1231d206be6cd9ec7
aff6f6c94fcd7204ed3455c68c83f4a4
1da4af2b74ef5c53f1d8ac70bdcb7ed1
85ce81bd84359d44254d95629e9855a9
4a7c1958d1f8ada5d0532ed8a5aa3fb2
d17ba70eb6248e594e1a2297acbbb39d
502f1a8c6eb6f1ce22b3de1a1f40cc24
554119a831a9aad6079cad88425de6bd
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
-----TEST SHA(abc)
ALGORITHM:
Ed25519
SECRET KEY:
833fe62409237b9d62ec77587520911e
9a759cec1d19755b7da901b96dca3d42
PUBLIC KEY:
ec172b93ad5e563bf4932c70e1245034
c35467ef2efd4d64ebf819683467e2bf
MESSAGE (length 64 bytes):
ddaf35a193617abacc417349ae204131
12e6fa4e89a97ea20a9eeee64b55d39a
2192992a274fc1a836ba3c23a3feebbd
454d4423643ce80e2a9ac94fa54ca49f
SIGNATURE:
dc2a4459e7369633a52b1bf277839a00
201009a3efbf3ecb69bea2186c26b589
09351fc9ac90b3ecfdfbc7c66431e030
3dca179c138ac17ad9bef1177331a704
-----7.2. Test Vectors for Ed25519ctx
now for the Ed25519ctx vectors 🌸 there are four of them, and they're labelled by their context strings (foo, bar, foo2, foo3). the CONTEXT: lines hold those values in hex, so 666f6f is foo and 626172 is bar 🦋
-----foo
ALGORITHM:
Ed25519ctx
SECRET KEY:
0305334e381af78f141cb666f6199f57
bc3495335a256a95bd2a55bf546663f6
PUBLIC KEY:
dfc9425e4f968f7f0c29f0259cf5f9ae
d6851c2bb4ad8bfb860cfee0ab248292
MESSAGE (length 16 bytes):
f726936d19c800494e3fdaff20b276a8
CONTEXT:
666f6f
SIGNATURE:
55a4cc2f70a54e04288c5f4cd1e45a7b
b520b36292911876cada7323198dd87a
8b36950b95130022907a7fb7c4e9b2d5
f6cca685a587b4b21f4b888e4e7edb0d
-----bar
ALGORITHM:
Ed25519ctx
SECRET KEY:
0305334e381af78f141cb666f6199f57
bc3495335a256a95bd2a55bf546663f6
PUBLIC KEY:
dfc9425e4f968f7f0c29f0259cf5f9ae
d6851c2bb4ad8bfb860cfee0ab248292
MESSAGE (length 16 bytes):
f726936d19c800494e3fdaff20b276a8
CONTEXT:
626172
SIGNATURE:
fc60d5872fc46b3aa69f8b5b4351d580
8f92bcc044606db097abab6dbcb1aee3
216c48e8b3b66431b5b186d1d28f8ee1
5a5ca2df6668346291c2043d4eb3e90d
-----foo2
ALGORITHM:
Ed25519ctx
SECRET KEY:
0305334e381af78f141cb666f6199f57
bc3495335a256a95bd2a55bf546663f6
PUBLIC KEY:
dfc9425e4f968f7f0c29f0259cf5f9ae
d6851c2bb4ad8bfb860cfee0ab248292
MESSAGE (length 16 bytes):
508e9e6882b979fea900f62adceaca35
CONTEXT:
666f6f
SIGNATURE:
8b70c1cc8310e1de20ac53ce28ae6e72
07f33c3295e03bb5c0732a1d20dc6490
8922a8b052cf99b7c4fe107a5abb5b2c
4085ae75890d02df26269d8945f84b0b
-----foo3
ALGORITHM:
Ed25519ctx
SECRET KEY:
ab9c2853ce297ddab85c993b3ae14bca
d39b2c682beabc27d6d4eb20711d6560
PUBLIC KEY:
0f1d1274943b91415889152e893d80e9
3275a1fc0b65fd71b4b0dda10ad7d772
MESSAGE (length 16 bytes):
f726936d19c800494e3fdaff20b276a8
CONTEXT:
666f6f
SIGNATURE:
21655b5f1aa965996b3f97b3c849eafb
a922a0a62992f73b3d1b73106a84ad85
e9b86a7b6005ea868337ff2d20a7f5fb
d4cd10b0be49a68da2b2e0dc0ad8960f
-----7.3. Test Vectors for Ed25519ph
last but not least, the Ed25519ph vector 👑 it's a single test, labelled TEST abc, and it reuses the same secret and public keys as the SHA(abc) case from 7.1. the message here is just the three bytes 616263 (that's abc in hex) 🍓
-----TEST abc
ALGORITHM:
Ed25519ph
SECRET KEY:
833fe62409237b9d62ec77587520911e
9a759cec1d19755b7da901b96dca3d42
PUBLIC KEY:
ec172b93ad5e563bf4932c70e1245034
c35467ef2efd4d64ebf819683467e2bf
MESSAGE (length 3 bytes):
616263
SIGNATURE:
98a70222f0b8121aa9d30f813d683f80
9e462b469c7ff87639499bb94e6dae41
31f85042463c2a355a2003d062adf5aa
a10b8c61e636062aaad11c2a26083406
-----7.4. Test Vectors for Ed448 💅
okay bestie, this is where the Ed448 test vectors live 🎀 Each one is a full set of inputs and outputs you can check your implementation against ✨
-----Blank
ALGORITHM:
Ed448
SECRET KEY:
6c82a562cb808d10d632be89c8513ebf
6c929f34ddfa8c9f63c9960ef6e348a3
528c8a3fcc2f044e39a3fc5b94492f8f
032e7549a20098f95b
PUBLIC KEY:
5fd7449b59b461fd2ce787ec616ad46a
1da1342485a70e1f8a0ea75d80e96778
edf124769b46c7061bd6783df1e50f6c
d1fa1abeafe8256180
MESSAGE (length 0 bytes):
SIGNATURE:
533a37f6bbe457251f023c0d88f976ae
2dfb504a843e34d2074fd823d41a591f
2b233f034f628281f2fd7a22ddd47d78
28c59bd0a21bfd3980ff0d2028d4b18a
9df63e006c5d1c2d345b925d8dc00b41
04852db99ac5c7cdda8530a113a0f4db
b61149f05a7363268c71d95808ff2e65
2600-----1 octet
ALGORITHM:
Ed448
SECRET KEY:
c4eab05d357007c632f3dbb48489924d
552b08fe0c353a0d4a1f00acda2c463a
fbea67c5e8d2877c5e3bc397a659949e
f8021e954e0a12274e
PUBLIC KEY:
43ba28f430cdff456ae531545f7ecd0a
c834a55d9358c0372bfa0c6c6798c086
6aea01eb00742802b8438ea4cb82169c
235160627b4c3a9480
MESSAGE (length 1 byte):
03
SIGNATURE:
26b8f91727bd62897af15e41eb43c377
efb9c610d48f2335cb0bd0087810f435
2541b143c4b981b7e18f62de8ccdf633
fc1bf037ab7cd779805e0dbcc0aae1cb
cee1afb2e027df36bc04dcecbf154336
c19f0af7e0a6472905e799f1953d2a0f
f3348ab21aa4adafd1d234441cf807c0
3a00-----1 octet (with context)
ALGORITHM:
Ed448
SECRET KEY:
c4eab05d357007c632f3dbb48489924d
552b08fe0c353a0d4a1f00acda2c463a
fbea67c5e8d2877c5e3bc397a659949e
f8021e954e0a12274e
PUBLIC KEY:
43ba28f430cdff456ae531545f7ecd0a
c834a55d9358c0372bfa0c6c6798c086
6aea01eb00742802b8438ea4cb82169c
235160627b4c3a9480
MESSAGE (length 1 byte):
03
CONTEXT:
666f6f
SIGNATURE:
d4f8f6131770dd46f40867d6fd5d5055
de43541f8c5e35abbcd001b32a89f7d2
151f7647f11d8ca2ae279fb842d60721
7fce6e042f6815ea000c85741de5c8da
1144a6a1aba7f96de42505d7a7298524
fda538fccbbb754f578c1cad10d54d0d
5428407e85dcbc98a49155c13764e66c
3c00-----11 octets
ALGORITHM:
Ed448
SECRET KEY:
cd23d24f714274e744343237b93290f5
11f6425f98e64459ff203e8985083ffd
f60500553abc0e05cd02184bdb89c4cc
d67e187951267eb328
PUBLIC KEY:
dcea9e78f35a1bf3499a831b10b86c90
aac01cd84b67a0109b55a36e9328b1e3
65fce161d71ce7131a543ea4cb5f7e9f
1d8b00696447001400
MESSAGE (length 11 bytes):
0c3e544074ec63b0265e0c
SIGNATURE:
1f0a8888ce25e8d458a21130879b840a
9089d999aaba039eaf3e3afa090a09d3
89dba82c4ff2ae8ac5cdfb7c55e94d5d
961a29fe0109941e00b8dbdeea6d3b05
1068df7254c0cdc129cbe62db2dc957d
bb47b51fd3f213fb8698f064774250a5
028961c9bf8ffd973fe5d5c206492b14
0e00-----12 octets
ALGORITHM:
Ed448
SECRET KEY:
258cdd4ada32ed9c9ff54e63756ae582
fb8fab2ac721f2c8e676a72768513d93
9f63dddb55609133f29adf86ec9929dc
cb52c1c5fd2ff7e21b
PUBLIC KEY:
3ba16da0c6f2cc1f30187740756f5e79
8d6bc5fc015d7c63cc9510ee3fd44adc
24d8e968b6e46e6f94d19b945361726b
d75e149ef09817f580
MESSAGE (length 12 bytes):
64a65f3cdedcdd66811e2915
SIGNATURE:
7eeeab7c4e50fb799b418ee5e3197ff6
bf15d43a14c34389b59dd1a7b1b85b4a
e90438aca634bea45e3a2695f1270f07
fdcdf7c62b8efeaf00b45c2c96ba457e
b1a8bf075a3db28e5c24f6b923ed4ad7
47c3c9e03c7079efb87cb110d3a99861
e72003cbae6d6b8b827e4e6c143064ff
3c00-----13 octets
ALGORITHM:
Ed448
SECRET KEY:
7ef4e84544236752fbb56b8f31a23a10
e42814f5f55ca037cdcc11c64c9a3b29
49c1bb60700314611732a6c2fea98eeb
c0266a11a93970100e
PUBLIC KEY:
b3da079b0aa493a5772029f0467baebe
e5a8112d9d3a22532361da294f7bb381
5c5dc59e176b4d9f381ca0938e13c6c0
7b174be65dfa578e80
MESSAGE (length 13 bytes):
64a65f3cdedcdd66811e2915e7
SIGNATURE:
6a12066f55331b6c22acd5d5bfc5d712
28fbda80ae8dec26bdd306743c5027cb
4890810c162c027468675ecf645a8317
6c0d7323a2ccde2d80efe5a1268e8aca
1d6fbc194d3f77c44986eb4ab4177919
ad8bec33eb47bbb5fc6e28196fd1caf5
6b4e7e0ba5519234d047155ac727a105
3100-----64 octets
ALGORITHM:
Ed448
SECRET KEY:
d65df341ad13e008567688baedda8e9d
cdc17dc024974ea5b4227b6530e339bf
f21f99e68ca6968f3cca6dfe0fb9f4fa
b4fa135d5542ea3f01
PUBLIC KEY:
df9705f58edbab802c7f8363cfe5560a
b1c6132c20a9f1dd163483a26f8ac53a
39d6808bf4a1dfbd261b099bb03b3fb5
0906cb28bd8a081f00
MESSAGE (length 64 bytes):
bd0f6a3747cd561bdddf4640a332461a
4a30a12a434cd0bf40d766d9c6d458e5
512204a30c17d1f50b5079631f64eb31
12182da3005835461113718d1a5ef944
SIGNATURE:
554bc2480860b49eab8532d2a533b7d5
78ef473eeb58c98bb2d0e1ce488a98b1
8dfde9b9b90775e67f47d4a1c3482058
efc9f40d2ca033a0801b63d45b3b722e
f552bad3b4ccb667da350192b61c508c
f7b6b5adadc2c8d9a446ef003fb05cba
5f30e88e36ec2703b349ca229c267083
3900-----256 octets
ALGORITHM:
Ed448
SECRET KEY:
2ec5fe3c17045abdb136a5e6a913e32a
b75ae68b53d2fc149b77e504132d3756
9b7e766ba74a19bd6162343a21c8590a
a9cebca9014c636df5
PUBLIC KEY:
79756f014dcfe2079f5dd9e718be4171
e2ef2486a08f25186f6bff43a9936b9b
fe12402b08ae65798a3d81e22e9ec80e
7690862ef3d4ed3a00
MESSAGE (length 256 bytes):
15777532b0bdd0d1389f636c5f6b9ba7
34c90af572877e2d272dd078aa1e567c
fa80e12928bb542330e8409f31745041
07ecd5efac61ae7504dabe2a602ede89
e5cca6257a7c77e27a702b3ae39fc769
fc54f2395ae6a1178cab4738e543072f
c1c177fe71e92e25bf03e4ecb72f47b6
4d0465aaea4c7fad372536c8ba516a60
39c3c2a39f0e4d832be432dfa9a706a6
e5c7e19f397964ca4258002f7c0541b5
90316dbc5622b6b2a6fe7a4abffd9610
5eca76ea7b98816af0748c10df048ce0
12d901015a51f189f3888145c03650aa
23ce894c3bd889e030d565071c59f409
a9981b51878fd6fc110624dcbcde0bf7
a69ccce38fabdf86f3bef6044819de11
SIGNATURE:
c650ddbb0601c19ca11439e1640dd931
f43c518ea5bea70d3dcde5f4191fe53f
00cf966546b72bcc7d58be2b9badef28
743954e3a44a23f880e8d4f1cfce2d7a
61452d26da05896f0a50da66a239a8a1
88b6d825b3305ad77b73fbac0836ecc6
0987fd08527c1a8e80d5823e65cafe2a
3d00-----1023 octets
ALGORITHM:
Ed448
SECRET KEY:
872d093780f5d3730df7c212664b37b8
a0f24f56810daa8382cd4fa3f77634ec
44dc54f1c2ed9bea86fafb7632d8be19
9ea165f5ad55dd9ce8
PUBLIC KEY:
a81b2e8a70a5ac94ffdbcc9badfc3feb
0801f258578bb114ad44ece1ec0e799d
a08effb81c5d685c0c56f64eecaef8cd
f11cc38737838cf400
MESSAGE (length 1023 bytes):
6ddf802e1aae4986935f7f981ba3f035
1d6273c0a0c22c9c0e8339168e675412
a3debfaf435ed651558007db4384b650
fcc07e3b586a27a4f7a00ac8a6fec2cd
86ae4bf1570c41e6a40c931db27b2faa
15a8cedd52cff7362c4e6e23daec0fbc
3a79b6806e316efcc7b68119bf46bc76
a26067a53f296dafdbdc11c77f7777e9
72660cf4b6a9b369a6665f02e0cc9b6e
dfad136b4fabe723d2813db3136cfde9
b6d044322fee2947952e031b73ab5c60
3349b307bdc27bc6cb8b8bbd7bd32321
9b8033a581b59eadebb09b3c4f3d2277
d4f0343624acc817804728b25ab79717
2b4c5c21a22f9c7839d64300232eb66e
53f31c723fa37fe387c7d3e50bdf9813
a30e5bb12cf4cd930c40cfb4e1fc6225
92a49588794494d56d24ea4b40c89fc0
596cc9ebb961c8cb10adde976a5d602b
1c3f85b9b9a001ed3c6a4d3b1437f520
96cd1956d042a597d561a596ecd3d173
5a8d570ea0ec27225a2c4aaff26306d1
526c1af3ca6d9cf5a2c98f47e1c46db9
a33234cfd4d81f2c98538a09ebe76998
d0d8fd25997c7d255c6d66ece6fa56f1
1144950f027795e653008f4bd7ca2dee
85d8e90f3dc315130ce2a00375a318c7
c3d97be2c8ce5b6db41a6254ff264fa6
155baee3b0773c0f497c573f19bb4f42
40281f0b1f4f7be857a4e59d416c06b4
c50fa09e1810ddc6b1467baeac5a3668
d11b6ecaa901440016f389f80acc4db9
77025e7f5924388c7e340a732e554440
e76570f8dd71b7d640b3450d1fd5f041
0a18f9a3494f707c717b79b4bf75c984
00b096b21653b5d217cf3565c9597456
f70703497a078763829bc01bb1cbc8fa
04eadc9a6e3f6699587a9e75c94e5bab
0036e0b2e711392cff0047d0d6b05bd2
a588bc109718954259f1d86678a579a3
120f19cfb2963f177aeb70f2d4844826
262e51b80271272068ef5b3856fa8535
aa2a88b2d41f2a0e2fda7624c2850272
ac4a2f561f8f2f7a318bfd5caf969614
9e4ac824ad3460538fdc25421beec2cc
6818162d06bbed0c40a387192349db67
a118bada6cd5ab0140ee273204f628aa
d1c135f770279a651e24d8c14d75a605
9d76b96a6fd857def5e0b354b27ab937
a5815d16b5fae407ff18222c6d1ed263
be68c95f32d908bd895cd76207ae7264
87567f9a67dad79abec316f683b17f2d
02bf07e0ac8b5bc6162cf94697b3c27c
d1fea49b27f23ba2901871962506520c
392da8b6ad0d99f7013fbc06c2c17a56
9500c8a7696481c1cd33e9b14e40b82e
79a5f5db82571ba97bae3ad3e0479515
bb0e2b0f3bfcd1fd33034efc6245eddd
7ee2086ddae2600d8ca73e214e8c2b0b
db2b047c6a464a562ed77b73d2d841c4
b34973551257713b753632efba348169
abc90a68f42611a40126d7cb21b58695
568186f7e569d2ff0f9e745d0487dd2e
b997cafc5abf9dd102e62ff66cba87
SIGNATURE:
e301345a41a39a4d72fff8df69c98075
a0cc082b802fc9b2b6bc503f926b65bd
df7f4c8f1cb49f6396afc8a70abe6d8a
ef0db478d4c6b2970076c6a0484fe76d
76b3a97625d79f1ce240e7c576750d29
5528286f719b413de9ada3e8eb78ed57
3603ce30d8bb761785dc30dbc320869e
1a00
-----7.5. Test Vectors for Ed448ph 🌈
and now the Ed448ph ones 🦋 These come in a plain version and a with-context version, same as before ✨
-----TEST abc
ALGORITHM:
Ed448ph
SECRET KEY:
833fe62409237b9d62ec77587520911e
9a759cec1d19755b7da901b96dca3d42
ef7822e0d5104127dc05d6dbefde69e3
ab2cec7c867c6e2c49
PUBLIC KEY:
259b71c19f83ef77a7abd26524cbdb31
61b590a48f7d17de3ee0ba9c52beb743
c09428a131d6b1b57303d90d8132c276
d5ed3d5d01c0f53880
MESSAGE (length 3 bytes):
616263
SIGNATURE:
822f6901f7480f3d5f562c592994d969
3602875614483256505600bbc281ae38
1f54d6bce2ea911574932f52a4e6cadd
78769375ec3ffd1b801a0d9b3f4030cd
433964b6457ea39476511214f97469b5
7dd32dbc560a9a94d00bff07620464a3
ad203df7dc7ce360c3cd3696d9d9fab9
0f00-----TEST abc (with context)
ALGORITHM:
Ed448ph
SECRET KEY:
833fe62409237b9d62ec77587520911e
9a759cec1d19755b7da901b96dca3d42
ef7822e0d5104127dc05d6dbefde69e3
ab2cec7c867c6e2c49
PUBLIC KEY:
259b71c19f83ef77a7abd26524cbdb31
61b590a48f7d17de3ee0ba9c52beb743
c09428a131d6b1b57303d90d8132c276
d5ed3d5d01c0f53880
MESSAGE (length 3 bytes):
616263
CONTEXT:
666f6f
SIGNATURE:
c32299d46ec8ff02b54540982814dce9
a05812f81962b649d528095916a2aa48
1065b1580423ef927ecf0af5888f90da
0f6a9a85ad5dc3f280d91224ba9911a3
653d00e484e2ce232521481c8658df30
4bb7745a73514cdb9bf3e15784ab7128
4f8d0704a608c54a6b62d97beb511d13
2100
-----8. Security Considerations 🔐
okay this part is SERIOUS bestie 😭 but it's also really important, so let's go 👑
8.1. Side-Channel Leaks 👀
For implementations doing signatures, keeping the private key secret is everything 🔑 One way to guard against some side-channel attacks is to make sure the implementation runs exactly the same sequence of instructions and does exactly the same memory accesses, no matter what the value of the private key is 🛡️
To make an implementation side-channel silent like that, the modulo p arithmetic must not use any data-dependent branches, like ones tied to carry propagation 🙈 Side-channel silent point addition is pretty straightforward, thanks to the unified formulas ✨
Scalar multiplication (multiplying a point by an integer) takes some extra effort to make side-channel silent 💅 One simple approach is a side-channel silent conditional assignment, used together with the binary algorithm to look at one bit of the integer at a time 🧠
Compared to other signature schemes, avoiding data-dependent branches is easier, since side-channel silent modulo p arithmetic is easier (with recommended curves) and the formulas are complete instead of having a bunch of special cases 🫶
Note that the example implementations in this document do not attempt to be side-channel silent. 🚨
8.2. Randomness Considerations 🎲
okay so EdDSA signatures are deterministic 🔐 which is literally the best news for us, because it protects against attacks that come from signing with bad randomness. depending on the algorithm, those attacks can go all the way up to full private key compromise 💀 and it is surprisingly hard to get good-quality random numbers, so there have been a bunch of security fails over this. yikes.
obviously private key generation needs randomness 🎀 but the private key gets hashed before it's used, so a few missing bits of entropy isn't a total disaster. phew 🫶
the basic signature verification is also deterministic ✨ however, some speedups that verify multiple signatures at once do need random numbers.
8.3. Use of Contexts 🧩
contexts are so useful bc they keep uses of the protocol separate from different protocols (which is very hard to do reliably otherwise) and also separate different uses inside the same protocol 🦋 but the following SHOULD be kept in mind when you use this facility:
- The context SHOULD be a constant string specified by the protocol using it. It SHOULD NOT incorporate variable elements from the message itself.
- Contexts SHOULD NOT be used opportunistically, bc that kind of use is very error prone 🙈 If contexts are used, one SHOULD require that all signature schemes available for use in that purpose support contexts.
- Contexts are an extra input that percolate out of APIs 🫠 so even if the signature scheme supports contexts, those may not be available for use. it gets worse bc a lot of the time the application isn't calling the signing and verification functions directly, it's doing it through some other protocol. main character problem 😭
8.4. Signature Malleability 🪞
some systems assume signatures are not malleable. that means that given a valid signature for some message under some key, the attacker can't produce another valid signature for the same message and key 🔒
ed25519 and ed448 signatures are not malleable 💅 because of the verification check that the decoded S is smaller than l. without that check, you could add a multiple of l into the scalar part and still pass signature verification, and then you'd get malleable signatures. so that check is 🫶 non-negotiable.
8.5. Choice of Signature Primitive 💎
Ed25519 and Ed25519ph have a nominal strength of 128 bits, whereas Ed448 and Ed448ph have the strength of 224. the lower strength is sufficient for the foreseeable future, but the higher level brings some defense against possible future cryptographic advances 🛡️ both are demolished by quantum computers just about the same 💀
the Ed25519ph and Ed448ph variants are prehashed. this is mainly useful for interoperation with legacy APIs, bc in most cases either the amount of data signed isn't large, or the protocol is in a position to do digesting in ways better than just prehashing (e.g., tree hashing or splitting the data) 🌸 prehashing also makes the functions way more vulnerable to weaknesses in the hash functions used. these variants SHOULD NOT be used. 🚫
Ed25519ctx and Ed448 have contexts. however, this is balanced by the problems noted in Section 8.3 about contexts. 🎀
on the implementation front, Ed25519 is widely implemented and has lots of high-quality implementations ⭐ the others have much worse support 😬
in summary, if a high 128-bit security level is enough, use of Ed25519 is RECOMMENDED. otherwise, Ed448 is RECOMMENDED. 👑
8.6. Mixing Different Prehashes 🌈
the schemes in this document are designed to be resistant to mixing prehashes 🧁 that means it's infeasible to find a message that verifies using the same signature under another scheme, even if the original signed message was chosen. so you can use the same key pair for Ed25519, Ed25519ctx, and Ed25519ph, and correspondingly with Ed448 and Ed448ph 🔑✨
the "SigEd25519 no Ed25519 collisions" constant is chosen to be a textual string so that it does not decode as a point 🧚 bc the inner hash input in the Ed25519 signature always starts with a valid point, there's no way a trivial collision can be constructed. in the case of seed hash, trivial collisions are so unlikely, even with an attacker choosing all inputs, that it's way more probable that something else goes catastrophically wrong 🙃
8.7. Signing Large Amounts of Data at Once 📦
avoid signing large amounts of data at once (where "large" depends on the expected verifier) 🚫 in particular, unless the underlying protocol doesn't require it, the receiver MUST buffer the entire message (or enough information to reconstruct it, e.g., a compressed or encrypted version) to be verified. no skipping this one, bestie 🫶
this is needed bc most of the time it's unsafe to process unverified data, and verifying the signature makes a pass through the whole message, so you end up with at least two passes through it 🔁
as an API consideration, this means any Initialize Update Finalize (IFU) verification interface is prone to misuse 😬
it is a bad idea to modify Ed25519 or Ed448 signing so you can create valid Ed25519/Ed448 signatures using an IUF interface with only constant buffering. pretty much any error in that would cause a catastrophic security failure 💀 do not do this 🙅♀️
8.8. Multiplication by Cofactor in Verification 🔢
the given verification formulas for both Ed25519 and Ed448 multiply points by the cofactor. while this is not strictly necessary for security (in fact, any signature that meets the non-multiplied equation will satisfy the multiplied one), in some applications it's undesirable for implementations to disagree about the exact set of valid signatures 🕵️♀️ those disagreements could open up, e.g., fingerprinting attacks.
8.9. Use of SHAKE256 as a Hash Function 🦋
Ed448 uses SHAKE256 as a hash function, even though SHAKE256 is specifically defined not to be a hash function 🤨
the first potentially troublesome property is that shorter outputs are prefixes of longer ones. this is acceptable bc output lengths are fixed 👍
the second potentially troublesome property is failing to meet standard hash security notions (especially with preimages). however, the estimated 256-bit security level against collisions and preimages is sufficient to pair with a 224-bit level elliptic curve 💖
9. References 📚
9.1. Normative References
- [FIPS202] National Institute of Standards and Technology, "SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions", FIPS PUB 202, August 2015,
http://dx.doi.org/10.6028/NIST.FIPS.202. - [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997,
http://www.rfc-editor.org/info/rfc2119. - [RFC6234] Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms (SHA and SHA-based HMAC and HKDF)", RFC 6234, DOI 10.17487/RFC6234, May 2011,
http://www.rfc-editor.org/info/rfc6234. - [RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016,
http://www.rfc-editor.org/info/rfc7748.
9.2. Informative References
- [CURVE25519] Bernstein, D., "Curve25519: new Diffie-Hellman speed records", DOI 10.1007/11745853_14, February 2006,
http://cr.yp.to/ecdh.html. - [ED25519-LIBGCRYPT-TEST-VECTORS] Koch, W., "Ed25519 Libgcrypt test vectors", July 2014,
http://git.gnupg.org/cgi-bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=tests/t-ed25519.inp;h=e13566f826321eece65e02c593bc7d885b3dbe23;hb=refs/heads/master. - [ED25519-TEST-VECTORS] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. Yang, "Ed25519 test vectors", July 2011,
http://ed25519.cr.yp.to/python/sign.input. - [ED448] Hamburg, M., "Ed448-Goldilocks, a new elliptic curve", June 2015,
http://eprint.iacr.org/2015/625. - [EDDSA] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B. Yang, "High-speed high-security signatures", DOI 10.1007/978-3-642-23951-9_9, September 2011,
http://ed25519.cr.yp.to/ed25519-20110926.pdf. - [EDDSA2] Bernstein, D., Josefsson, S., Lange, T., Schwabe, P., and B. Yang, "EdDSA for more curves", July 2015,
http://ed25519.cr.yp.to/eddsa-20150704.pdf. - [Edwards-revisited] Hisil, H., Wong, K., Carter, G., and E. Dawson, "Twisted Edwards Curves Revisited", DOI 10.1007/978-3-540-89255-7_20, December 2008,
http://eprint.iacr.org/2008/522. - [EFD-ADD] Bernstein, D. and T. Lange, "Projective coordinates for Edwards curves", The 'add-2007-bl' addition formulas, 2007,
http://www.hyperelliptic.org/EFD/g1p/auto-edwards-projective.html#addition-add-2007-bl. - [EFD-DBL] Bernstein, D. and T. Lange, "Projective coordinates for Edwards curves", The 'dbl-2007-bl' doubling formulas, 2007,
http://www.hyperelliptic.org/EFD/g1p/auto-edwards-projective.html#doubling-dbl-2007-bl. - [EFD-TWISTED-ADD] Hisil, H., Wong, K., Carter, G., and E. Dawson, "Extended coordinates with a=-1 for twisted Edwards curves", The 'add-2008-hwcd-3' addition formulas, December 2008,
http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3. - [EFD-TWISTED-DBL] Hisil, H., Wong, K., Carter, G., and E. Dawson, "Extended coordinates with a=-1 for twisted Edwards curves", The 'dbl-2008-hwcd' doubling formulas, December 2008,
http://www.hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#doubling-dbl-2008-hwcd. - [Faster-ECC] Bernstein, D. and T. Lange, "Faster addition and doubling on elliptic curves", DOI 10.1007/978-3-540-76900-2_3, July 2007,
http://eprint.iacr.org/2007/286. - [RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker, "Randomness Requirements for Security", BCP 106, RFC 4086, DOI 10.17487/RFC4086, June 2005,
http://www.rfc-editor.org/info/rfc4086.
Appendix A. Ed25519/Ed448 Python Library 🐍
below is an example implementation of Ed25519/Ed448 written in Python ✨ version 3.2 or higher is required.
Note: This code is not intended for production. Although it should produce correct results for every input, it is slow and makes no attempt to avoid side-channel attacks. ⚠️
import hashlib;
import os;
#Compute candidate square root of x modulo p, with p = 3 (mod 4).
def sqrt4k3(x,p): return pow(x,(p + 1)//4,p)
#Compute candidate square root of x modulo p, with p = 5 (mod 8).
def sqrt8k5(x,p):
y = pow(x,(p+3)//8,p)
#If the square root exists, it is either y or y*2^(p-1)/4.
if (y * y) % p == x % p: return y
else:
z = pow(2,(p - 1)//4,p)
return (y * z) % p
#Decode a hexadecimal string representation of the integer.
def hexi(s): return int.from_bytes(bytes.fromhex(s),byteorder="big")
#Rotate a word x by b places to the left.
def rol(x,b): return ((x << b) | (x >> (64 - b))) & (2**64-1)
#From little endian.
def from_le(s): return int.from_bytes(s, byteorder="little")
#Do the SHA-3 state transform on state s.okay bestie the SHA-3 state transform is up next and it's giving pure main character energy 💅✨
def sha3_transform(s):
ROTATIONS = [0,1,62,28,27,36,44,6,55,20,3,10,43,25,39,41,45,15,\
21,8,18,2,61,56,14]
PERMUTATION = [1,6,9,22,14,20,2,12,13,19,23,15,4,24,21,8,16,5,3,\
18,17,11,7,10]
RC = [0x0000000000000001,0x0000000000008082,0x800000000000808a,\
0x8000000080008000,0x000000000000808b,0x0000000080000001,\
0x8000000080008081,0x8000000000008009,0x000000000000008a,\
0x0000000000000088,0x0000000080008009,0x000000008000000a,\
0x000000008000808b,0x800000000000008b,0x8000000000008089,\
0x8000000000008003,0x8000000000008002,0x8000000000000080,\
0x000000000000800a,0x800000008000000a,0x8000000080008081,\
0x8000000000008080,0x0000000080000001,0x8000000080008008]
for rnd in range(0,24):
#AddColumnParity (Theta)
c = [0]*5;
d = [0]*5;
for i in range(0,25): c[i%5]^=s[i]
for i in range(0,5): d[i]=c[(i+4)%5]^rol(c[(i+1)%5],1)
for i in range(0,25): s[i]^=d[i%5]
#RotateWords (Rho)
for i in range(0,25): s[i]=rol(s[i],ROTATIONS[i])
#PermuteWords (Pi)
t = s[PERMUTATION[0]]
for i in range(0,len(PERMUTATION)-1):
s[PERMUTATION[i]]=s[PERMUTATION[i+1]]
s[PERMUTATION[-1]]=t;
#NonlinearMixRows (Chi)
for i in range(0,25,5):
t=[s[i],s[i+1],s[i+2],s[i+3],s[i+4],s[i],s[i+1]]
for j in range(0,5): s[i+j]=t[j]^((~t[j+1])&(t[j+2]))
#AddRoundConstant (Iota)
s[0]^=RC[rnd]next up are the little helpers that shuffle bytes and words around, and they're honestly so iconic 🦋🎀
#Reinterpret octet array b to word array and XOR it to state s.
def reinterpret_to_words_and_xor(s,b):
for j in range(0,len(b)//8):
s[j]^=from_le(b[8*j:][:8])
#Reinterpret word array w to octet array and return it.
def reinterpret_to_octets(w):
mp=bytearray()
for j in range(0,len(w)):
mp+=w[j].to_bytes(8,byteorder="little")
return mpand now the generic sponge construction, which is where all the pieces come together 🌈💖
#(semi-)generic SHA-3 implementation
def sha3_raw(msg,r_w,o_p,e_b):
r_b=8*r_w
s=[0]*25
#Handle whole blocks.
idx=0
blocks=len(msg)//r_b
for i in range(0,blocks):
reinterpret_to_words_and_xor(s,msg[idx:][:r_b])
idx+=r_b
sha3_transform(s)
#Handle last block padding.
m=bytearray(msg[idx:])
m.append(o_p)
while len(m) < r_b: m.append(0)
m[len(m)-1]|=128
#Handle padded last block.
reinterpret_to_words_and_xor(s,m)
sha3_transform(s)
#Output.
out = bytearray()
while len(out)<e_b:
out+=reinterpret_to_octets(s[:r_w])
sha3_transform(s)
return out[:e_b]
#Implementation of SHAKE256 functions.
def shake256(msg,olen): return sha3_raw(msg,17,31,olen)okay the Field class is where the prime-field maths lives, bestie, and it's giving elegant 👑🧚
#A (prime) field element.
class Field:
#Construct number x (mod p).
def __init__(self,x,p):
self.__x=x%p
self.__p=p
#Check that fields of self and y are the same.
def __check_fields(self,y):
if type(y) is not Field or self.__p!=y.__p:
raise ValueError("Fields don't match")
#Field addition. The fields must match.
def __add__(self,y):
self.__check_fields(y)
return Field(self.__x+y.__x,self.__p)
#Field subtraction. The fields must match.
def __sub__(self,y):
self.__check_fields(y)
return Field(self.__p+self.__x-y.__x,self.__p)
#Field negation.
def __neg__(self):
return Field(self.__p-self.__x,self.__p)
#Field multiplication. The fields must match.
def __mul__(self,y):
self.__check_fields(y)
return Field(self.__x*y.__x,self.__p)
#Field division. The fields must match.
def __truediv__(self,y):
return self*y.inv()
#Field inverse (inverse of 0 is 0).
def inv(self):
return Field(pow(self.__x,self.__p-2,self.__p),self.__p)
#Field square root. Returns none if square root does not exist.
#Note: not presently implemented for p mod 8 = 1 case.
def sqrt(self):
#Compute candidate square root.
if self.__p%4==3: y=sqrt4k3(self.__x,self.__p)
elif self.__p%8==5: y=sqrt8k5(self.__x,self.__p)
else: raise NotImplementedError("sqrt(_,8k+1)")
_y=Field(y,self.__p);
#Check square root candidate valid.
return _y if _y*_y==self else None
#Make the field element with the same field as this, but
#with a different value.
def make(self,ival): return Field(ival,self.__p)
#Is the field element the additive identity?
def iszero(self): return self.__x==0
#Are field elements equal?
def __eq__(self,y): return self.__x==y.__x and self.__p==y.__p
#Are field elements not equal?
def __ne__(self,y): return not (self==y)
#Serialize number to b-1 bits.
def tobytes(self,b):
return self.__x.to_bytes(b//8,byteorder="little")
#Unserialize number from bits.
def frombytes(self,x,b):
rv=from_le(x)%(2**(b-1))
return Field(rv,self.__p) if rv<self.__p else None
#Compute sign of number, 0 or 1. The sign function
#has the following property:
#sign(x) = 1 - sign(-x) if x != 0.
def sign(self): return self.__x%2and now EdwardsPoint, the curve-point class, which is honestly where things get spicy 🔥💅
#A point on (twisted) Edwards curve.
class EdwardsPoint:
#base_field = None
#x = None
#y = None
#z = None
def initpoint(self, x, y):
self.x=x
self.y=y
self.z=self.base_field.make(1)
def decode_base(self,s,b):
#Check that point encoding is the correct length.
if len(s)!=b//8: return (None,None)
#Extract signbit.
xs=s[(b-1)//8]>>((b-1)&7)
#Decode y. If this fails, fail.
y = self.base_field.frombytes(s,b)
if y is None: return (None,None)
#Try to recover x. If it does not exist, or if zero and xs
#are wrong, fail.
x=self.solve_x2(y).sqrt()
if x is None or (x.iszero() and xs!=x.sign()):
return (None,None)
#If sign of x isn't correct, flip it.
if x.sign()!=xs: x=-x
# Return the constructed point.
return (x,y)
def encode_base(self,b):
xp,yp=self.x/self.z,self.y/self.z
#Encode y.
s=bytearray(yp.tobytes(b))
#Add sign bit of x to encoding.
if xp.sign()!=0: s[(b-1)//8]|=1<<(b-1)%8
return s
def __mul__(self,x):
r=self.zero_elem()
s=self
while x > 0:
if (x%2)>0:
r=r+s
s=s.double()
x=x//2
return r
#Check that two points are equal.
def __eq__(self,y):
#Need to check x1/z1 == x2/z2 and similarly for y, so cross
#multiply to eliminate divisions.
xn1=self.x*y.z
xn2=y.x*self.z
yn1=self.y*y.z
yn2=y.y*self.z
return xn1==xn2 and yn1==yn2
#Check if two points are not equal.
def __ne__(self,y): return not (self==y)okay the Edwards25519 class is the one I'm most obsessed with bestie, the base point, the constants, the whole vibe 😭🌸
#A point on Edwards25519.
class Edwards25519Point(EdwardsPoint):
#Create a new point on the curve.
base_field=Field(1,2**255-19)
d=-base_field.make(121665)/base_field.make(121666)
f0=base_field.make(0)
f1=base_field.make(1)
xb=base_field.make(hexi("216936D3CD6E53FEC0A4E231FDD6DC5C692CC76"+\
"09525A7B2C9562D608F25D51A"))
yb=base_field.make(hexi("666666666666666666666666666666666666666"+\
"6666666666666666666666658"))
#The standard base point.
@staticmethod
def stdbase():
return Edwards25519Point(Edwards25519Point.xb,\
Edwards25519Point.yb)
def __init__(self,x,y):
#Check the point is actually on the curve.
if y*y-x*x!=self.f1+self.d*x*x*y*y:
raise ValueError("Invalid point")
self.initpoint(x, y)
self.t=x*y
#Decode a point representation.
def decode(self,s):
x,y=self.decode_base(s,256);
return Edwards25519Point(x, y) if x is not None else None
#Encode a point representation.
def encode(self):
return self.encode_base(256)
#Construct a neutral point on this curve.
def zero_elem(self):
return Edwards25519Point(self.f0,self.f1)
#Solve for x^2.
def solve_x2(self,y):
return ((y*y-self.f1)/(self.d*y*y+self.f1))
#Point addition.
def __add__(self,y):
#The formulas are from EFD.
tmp=self.zero_elem()
zcp=self.z*y.z
A=(self.y-self.x)*(y.y-y.x)
B=(self.y+self.x)*(y.y+y.x)
C=(self.d+self.d)*self.t*y.t
D=zcp+zcp
E,H=B-A,B+A
F,G=D-C,D+C
tmp.x,tmp.y,tmp.z,tmp.t=E*F,G*H,F*G,E*H
return tmp
#Point doubling.
def double(self):
#The formulas are from EFD (with assumption a=-1 propagated).
tmp=self.zero_elem()
A=self.x*self.x
B=self.y*self.y
Ch=self.z*self.z
C=Ch+Ch
H=A+B
xys=self.x+self.y
E=H-xys*xys
G=A-B
F=C+G
tmp.x,tmp.y,tmp.z,tmp.t=E*F,G*H,F*G,E*H
return tmp
#Order of basepoint.
def l(self):
return hexi("1000000000000000000000000000000014def9dea2f79cd"+\
"65812631a5cf5d3ed")
#The logarithm of cofactor.
def c(self): return 3
#The highest set bit
def n(self): return 254
#The coding length
def b(self): return 256
#Validity check (for debugging)
def is_valid_point(self):
x,y,z,t=self.x,self.y,self.z,self.t
x2=x*x
y2=y*y
z2=z*z
lhs=(y2-x2)*z2
rhs=z2*z2+self.d*x2*y2
assert(lhs == rhs)
assert(t*z == x*y)and that's where this bit leaves off, bestie, the Edwards448 part is coming up next and I'm already so excited 🌈✨💜
#A point on Edwards448.class Edwards448Point(EdwardsPoint): #Create a new point on the curve. base_field=Field(1,2448-2224-1) d=base_field.make(-39081) f0=base_field.make(0) f1=base_field.make(1) xb=base_field.make(hexi("4F1970C66BED0DED221D15A622BF36DA9E14657"+\ "0470F1767EA6DE324A3D3A46412AE1AF72AB66511433B80E18B00938E26"+\ "26A82BC70CC05E")) yb=base_field.make(hexi("693F46716EB6BC248876203756C9C7624BEA737"+\ "36CA3984087789C1E05A0C2D73AD3FF1CE67C39C4FDBD132C4ED7C8AD98"+\ "08795BF230FA14")) #The standard base point. @staticmethod def stdbase(): return Edwards448Point(Edwards448Point.xb,Edwards448Point.yb) def __init__(self,x,y): #Check that the point is actually on the curve. if y*y+x*x!=self.f1+self.d*x*x*y*y: raise ValueError("Invalid point") self.initpoint(x, y) #Decode a point representation. def decode(self,s): x,y=self.decode_base(s,456); return Edwards448Point(x, y) if x is not None else None #Encode a point representation. def encode(self): return self.encode_base(456) #Construct a neutral point on this curve. def zero_elem(self): return Edwards448Point(self.f0,self.f1) #Solve for x^2. def solve_x2(self,y): return ((y*y-self.f1)/(self.d*y*y-self.f1))
#Point addition. def __add__(self,y): #The formulas are from EFD. tmp=self.zero_elem() xcp,ycp,zcp=self.x*y.x,self.y*y.y,self.z*y.z B=zcp*zcp E=self.d*xcp*ycp F,G=B-E,B+E tmp.x=zcp*F*((self.x+self.y)(y.x+y.y)-xcp-ycp) tmp.y,tmp.z=zcpG*(ycp-xcp),F*G return tmp #Point doubling. def double(self): #The formulas are from EFD. tmp=self.zero_elem() x1s,y1s,z1s=self.x*self.x,self.y*self.y,self.z*self.z xys=self.x+self.y F=x1s+y1s J=F-(z1s+z1s) tmp.x,tmp.y,tmp.z=(xys*xys-x1s-y1s)J,F(x1s-y1s),F*J return tmp #Order of basepoint. def l(self): return hexi("3ffffffffffffffffffffffffffffffffffffffffffffff"+\ "fffffffff7cca23e9c44edb49aed63690216cc2728dc58f552378c2"+\ "92ab5844f3") #The logarithm of cofactor. def c(self): return 2 #The highest set bit. def n(self): return 447 #The coding length. def b(self): return 456 #Validity check (for debugging). def is_valid_point(self): x,y,z=self.x,self.y,self.z x2=x*x y2=y*y z2=z*z lhs=(x2+y2)*z2 rhs=z2*z2+self.d*x2*y2 assert(lhs == rhs)
#Simple self-check. def curve_self_check(point): p=point q=point.zero_elem() z=q l=p.l()+1 p.is_valid_point() q.is_valid_point() for i in range(0,point.b()): if (l>>i)&1 != 0: q=q+p q.is_valid_point() p=p.double() p.is_valid_point() assert q.encode() == point.encode() assert q.encode() != p.encode() assert q.encode() != z.encode()
#Simple self-check. def self_check_curves(): curve_self_check(Edwards25519Point.stdbase()) curve_self_check(Edwards448Point.stdbase())
#PureEdDSA scheme. #Limitation: only b mod 8 = 0 is handled. class PureEdDSA: #Create a new object. def __init__(self,properties): self.B=properties["B"] self.H=properties["H"] self.l=self.B.l() self.n=self.B.n() self.b=self.B.b() self.c=self.B.c() #Clamp a private scalar. def __clamp(self,a): _a = bytearray(a) for i in range(0,self.c): _a[i//8]&=~(1<<(i%8)) _a[self.n//8]|=1<<(self.n%8) for i in range(self.n+1,self.b): _a[i//8]&=~(1<<(i%8)) return _a #Generate a key. If privkey is None, a random one is generated. #In any case, the (privkey, pubkey) pair is returned. def keygen(self,privkey): #If no private key data is given, generate random. if privkey is None: privkey=os.urandom(self.b//8) #Expand key. khash=self.H(privkey,None,None) a=from_le(self.__clamp(khash[:self.b//8])) #Return the key pair (public key is A=Enc(aB). return privkey,(self.B*a).encode() #Sign with key pair. def sign(self,privkey,pubkey,msg,ctx,hflag): #Expand key. khash=self.H(privkey,None,None) a=from_le(self.__clamp(khash[:self.b//8])) seed=khash[self.b//8:] #Calculate r and R (R only used in encoded form). r=from_le(self.H(seed+msg,ctx,hflag))%self.l R=(self.B*r).encode() #Calculate h. h=from_le(self.H(R+pubkey+msg,ctx,hflag))%self.l #Calculate s. S=((r+h*a)%self.l).to_bytes(self.b//8,byteorder="little") #The final signature is a concatenation of R and S. return R+S #Verify signature with public key. def verify(self,pubkey,msg,sig,ctx,hflag): #Sanity-check sizes. if len(sig)!=self.b//4: return False if len(pubkey)!=self.b//8: return False #Split signature into R and S, and parse. Rraw,Sraw=sig[:self.b//8],sig[self.b//8:] R,S=self.B.decode(Rraw),from_le(Sraw) #Parse public key. A=self.B.decode(pubkey) #Check parse results. if (R is None) or (A is None) or S>=self.l: return False #Calculate h. h=from_le(self.H(Rraw+pubkey+msg,ctx,hflag))%self.l #Calculate left and right sides of check eq. rhs=R+(A*h) lhs=self.B*S for i in range(0, self.c): lhs = lhs.double() rhs = rhs.double() #Check eq. holds? return lhs==rhs
def Ed25519_inthash(data,ctx,hflag): if (ctx is not None and len(ctx) > 0) or hflag: raise ValueError("Contexts/hashes not supported") return hashlib.sha512(data).digest()
#The base PureEdDSA schemes. pEd25519=PureEdDSA({\ "B":Edwards25519Point.stdbase(),\ "H":Ed25519_inthash\ })
def Ed25519ctx_inthash(data,ctx,hflag): dompfx = b"" PREFIX=b"SigEd25519 no Ed25519 collisions" if ctx is not None: if len(ctx) > 255: raise ValueError("Context too big") dompfx=PREFIX+bytes([1 if hflag else 0,len(ctx)])+ctx return hashlib.sha512(dompfx+data).digest()
pEd25519ctx=PureEdDSA({\ "B":Edwards25519Point.stdbase(),\ "H":Ed25519ctx_inthash\ })
def Ed448_inthash(data,ctx,hflag): dompfx = b"" if ctx is not None: if len(ctx) > 255: raise ValueError("Context too big") dompfx=b"SigEd448"+bytes([1 if hflag else 0,len(ctx)])+ctx return shake256(dompfx+data,114)
pEd448 = PureEdDSA({\ "B":Edwards448Point.stdbase(),\ "H":Ed448_inthash\ })
#EdDSA scheme. class EdDSA: #Create a new scheme object, with the specified PureEdDSA base #scheme and specified prehash. def __init__(self,pure_scheme,prehash): self.__pflag = True self.__pure=pure_scheme self.__prehash=prehash if self.__prehash is None: self.__prehash = lambda x,y:x self.__pflag = False # Generate a key. If privkey is none, it generates a random # privkey key, otherwise it uses a specified private key. # Returns pair (privkey, pubkey). def keygen(self,privkey): return self.__pure.keygen(privkey) # Sign message msg using specified key pair. def sign(self,privkey,pubkey,msg,ctx=None): if ctx is None: ctx=b""; return self.__pure.sign(privkey,pubkey,self.__prehash(msg,ctx),\ ctx,self.__pflag) # Verify signature sig on message msg using public key pubkey. def verify(self,pubkey,msg,sig,ctx=None): if ctx is None: ctx=b""; return self.__pure.verify(pubkey,self.__prehash(msg,ctx),sig,\ ctx,self.__pflag)
def Ed448ph_prehash(data,ctx): return shake256(data,64)
#Our signature schemes. Ed25519 = EdDSA(pEd25519,None) Ed25519ctx = EdDSA(pEd25519ctx,None) Ed25519ph = EdDSA(pEd25519ctx,lambda x,y:hashlib.sha512(x).digest()) Ed448 = EdDSA(pEd448,None) Ed448ph = EdDSA(pEd448,Ed448ph_prehash)
def eddsa_obj(name): if name == "Ed25519": return Ed25519 if name == "Ed25519ctx": return Ed25519ctx if name == "Ed25519ph": return Ed25519ph if name == "Ed448": return Ed448 if name == "Ed448ph": return Ed448ph raise NotImplementedError("Algorithm not implemented")
Appendix B. Library Driver ✨
okay so this last bit is a little command-line tool that uses the whole library we just built 💅 it's for poking at things interactively or for running the self-checks, and it's giving very much "trust but verify" energy 🔥
import sys
import binascii
from eddsa2 import Ed25519
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(":")
secret = (binascii.unhexlify(fields[0]))[:32]
public = binascii.unhexlify(fields[1])
msg = binascii.unhexlify(fields[2])
signature = binascii.unhexlify(fields[3])[:64]
privkey,pubkey = Ed25519.keygen(secret)
assert public == pubkey
assert signature == Ed25519.sign(privkey, pubkey, msg)
assert Ed25519.verify(public, msg, signature)
if len(msg) == 0:
bad_msg = b"x"
else:
bad_msg = munge_string(msg, len(msg) // 3, 4)
assert not Ed25519.verify(public,bad_msg,signature)
assert not Ed25519.verify(public, msg, munge_string(signature,20,8))
assert not Ed25519.verify(public,msg,munge_string(signature,40,16))Acknowledgements 🎀
EdDSA and Ed25519 were first described in a paper by Daniel J. Bernstein, Niels Duif, Tanja Lange, Peter Schwabe, and Bo-Yin Yang 💖 The Ed448 curve comes from Mike Hamburg 🌈
An earlier draft of this document was coauthored by Niels Moeller 🦋
Feedback on this document came from Werner Koch, Damien Miller, Bob Bradley, Franck Rondepierre, Alexey Melnikov, Kenny Paterson, and Robert Edmonds 🫶
The Ed25519 test vectors were double checked by Bob Bradley using three separate implementations 👑 (one based on TweetNaCl and two different implementations based on code from SUPERCOP). the way he went above and beyond 😭🔥
Authors' Addresses 🌸
Simon Josefsson SJD AB
Email: simon@josefsson.org URI: http://josefsson.org/
Ilari Liusvaara Independent
Email: ilariliusvaara@welho.com 🧚
📦 where this page comes from
- ietf: rfc8032 2017-01
- 💖 girlified by claude-haiku-5-5 on : a language model's rewrite of the original, under its terms