Network Working Group S. Josefsson
Internet-Draft SJD AB
Intended status: Informational I. Liusvaara
Expires: June 11, 2016 Independent
December 9, 2015
Edwards-curve Digital Signature Algorithm (EdDSA)
draft-irtf-cfrg-eddsa-01
Abstract
The elliptic curve signature scheme Edwards-curve Digital Signature
Algorithm (EdDSA) is described. The algorithm is instantiated with
recommended parameters for the Curve25519 and Curve448 curves. An
example implementation and test vectors are provided.
NOTE: Anything not about Ed25519 in this document is premature and
there is at least one FIXME that makes some things unimplementable.
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Copyright (c) 2015 IETF Trust and the persons identified as the
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Notation and Conventions . . . . . . . . . . . . . . . . . . 4
3. EdDSA Algorithm . . . . . . . . . . . . . . . . . . . . . . . 5
3.1. Encoding . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Keys . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3. Sign . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4. Verify . . . . . . . . . . . . . . . . . . . . . . . . . 7
4. PureEdDSA, HashEdDSA and Naming . . . . . . . . . . . . . . . 7
5. EdDSA Instances . . . . . . . . . . . . . . . . . . . . . . . 8
5.1. Ed25519ph and Ed25519 . . . . . . . . . . . . . . . . . . 8
5.1.1. Modular arithmetic . . . . . . . . . . . . . . . . . 8
5.1.2. Encoding . . . . . . . . . . . . . . . . . . . . . . 9
5.1.3. Decoding . . . . . . . . . . . . . . . . . . . . . . 9
5.1.4. Point addition . . . . . . . . . . . . . . . . . . . 10
5.1.5. Key Generation . . . . . . . . . . . . . . . . . . . 10
5.1.6. Sign . . . . . . . . . . . . . . . . . . . . . . . . 11
5.1.7. Verify . . . . . . . . . . . . . . . . . . . . . . . 12
5.2. Ed448ph and Ed448 . . . . . . . . . . . . . . . . . . . . 12
5.2.1. Modular arithmetic . . . . . . . . . . . . . . . . . 12
5.2.2. Encoding . . . . . . . . . . . . . . . . . . . . . . 12
5.2.3. Decoding . . . . . . . . . . . . . . . . . . . . . . 13
5.2.4. Point addition . . . . . . . . . . . . . . . . . . . 13
5.2.5. Key Generation . . . . . . . . . . . . . . . . . . . 14
5.2.6. Sign . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2.7. Verify . . . . . . . . . . . . . . . . . . . . . . . 15
6. Ed25519 Python illustration . . . . . . . . . . . . . . . . . 15
7. Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . 20
7.1. Test Vectors for Ed25519 . . . . . . . . . . . . . . . . 20
7.2. Test Vectors for Ed25519ph . . . . . . . . . . . . . . . 23
7.3. Test Vectors for Ed448 . . . . . . . . . . . . . . . . . 23
7.4. Test Vectors for Ed448ph . . . . . . . . . . . . . . . . 23
8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 23
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 23
10. Security Considerations . . . . . . . . . . . . . . . . . . . 23
10.1. Side-channel leaks . . . . . . . . . . . . . . . . . . . 23
10.2. Mixing different prehashes . . . . . . . . . . . . . . . 24
10.3. Signing large amounts of data at once . . . . . . . . . 24
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 24
11.1. Normative References . . . . . . . . . . . . . . . . . . 25
11.2. Informative References . . . . . . . . . . . . . . . . . 25
Appendix A. Ed25519 Python Library . . . . . . . . . . . . . . . 26
Appendix B. Library driver . . . . . . . . . . . . . . . . . . . 29
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Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 30
1. 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 describe some recommended variants.
To facilitate adoption in the Internet community of EdDSA, this
document describe the signature scheme in an implementation-oriented
way, and provide sample code and test vectors.
The advantages with EdDSA include:
1. High-performance on a variety of platforms.
2. Does not require the use of a unique random number for each
signature.
3. More resilient to side-channel attacks.
4. Small public keys (32 or 57 bytes) and signatures (64 or 114
bytes).
5. The formulas are "strongly unified", i.e., they are valid for all
points on the curve, with no exceptions. This obviates the need
for EdDSA to perform expensive point validation on untrusted
public values.
6. Collision resilience, meaning that hash-function collisions do
not break this system. (Only holds for PureEdDSA.)
The original EdDSA paper [EDDSA] and the generalized version
described in "EdDSA for more curves" [EDDSA2] provide further
background. The [I-D.irtf-cfrg-curves] document discuss 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 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.
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2. Notation and Conventions
FIXME: make sure this is aligned with irtf-cfrg-curves
The following notation is used throughout the document:
GF(p) --- finite field with p elements
x^y --- x multiplied by itself y times
B --- generator of the group or subgroup of interest
[n]B --- B added to itself n times.
h_i --- the i'th bit of h
a || b --- (bit-)string a concatenated with (bit-)string b
a <= b --- a is less than or equal to b
i+j --- sum of i and j
i*j --- multiplication of i and j
i x j --- cartesian product of i and j
Use of parenthesis (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 and packing those from least significant bit of each octet
to most siginficant 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. E.g. 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 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 multiple of 8, the most
significant bits of last octet remain unused).
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3. EdDSA Algorithm
EdDSA is a digital signature system with eleven parameters.
The generic EdDSA digital signature system with its eleven input
parameters is not intended to be implemented directly. Chosing
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 re-
use 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 non-intuitive steps of the algorithm.
This description closely follows [EDDSA2].
EdDSA has eleven 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 multiple of 8, so public key and signature
lengths are 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 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 nonzero 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.
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8. 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 }.
9. 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.
10. 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 "strongly unified": they are valid for all points on the
curve, with no exceptions. In particular, the denominators are non-
zero for all input points.
There are more efficient formulas, which are still strongly unified,
which use homogeneous coordinates to avoid the expensive modulo p
inversions. See [Faster-ECC] and [Edwards-revisited].
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 secret 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
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multiple A = [s]B. The EdDSA public key is ENC(A). The bits h_b,
..., h_(2b-1) is used below during signing.
3.3. Sign
The EdDSA signature of a message M under a secret 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 secret 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) || P(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 is an element of E, and S is a member of the set {0, 1, ...,
l-1 }. Compute h = H(ENC(R) || ENC(A) || M) and check the group
equation [2^c * S] B = 2^c * R + [2^c * h] A in E. Verification is
rejected if parsing fails or 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.
Chosing 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
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because most of the time one can't process the message until
signature is validated, which needs a pass on the entire message.
This document specify parameters resulting in the HashEdDSA variants
Ed25519ph and Ed448ph, and the PureEdDSA variants Ed25519 and Ed448.
5. EdDSA Instances
This section instantiate the general EdDSA algorithm for the
Curve25519 and Ed448 curves, each for the PureEdDSA and HashEdDSA
variants. Thus four different parameter sets are described.
5.1. Ed25519ph and Ed25519
Ed25519 is PureEdDSA instantiated with: p as the prime 2^255-19,
b=256, the 255-bit encoding of GF(p) being the little-endian encoding
of {0, 1, ..., p-1}, H being SHA-512 [RFC4634], c being 3, n being
254, a being -1, d = -121665/121666 which is a member of GF(p), and B
is the unique point (x, 4/5) in E for which x is "positive", which
with the encoding used simply means that the least significant bit of
x is 0, l is the prime 2^252 +
27742317777372353535851937790883648493.
Ed25519ph is the same but with PH being SHA-512 instead, i.e., the
input is hashed using SHA-512 before signing with Ed25519.
Written out explicitly, B is the point (15112221349535400772501151409
588531511454012693041857206046113283949847762202, 4631683569492647816
9428394003475163141307993866256225615783033603165251855960).
The values for p, a, d, B and l follows from the "edwards25519"
values in [I-D.irtf-cfrg-curves].
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 advise 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).
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,
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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
nonzero 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 using the
following trick, to use a single modular powering for both the
inversion of v and the square root:
(p+3)/8 3 (p-5)/8
x = (u/v) = u v (u v^7) (mod p)
3. Again, there are three cases:
1. If v x^2 = u (mod p), x is a square root.
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2. If v x^2 = -u (mod p), set x <-- x 2^((p-1)/4), which is a
square root.
3. Otherwise, no square root exists modulo p, and decoding
fails.
4. Finally, use the x_0 bit to select the right square root. If x =
0, and x_0 = 1, decoding fails. Otherwise, if x_0 != x mod 2,
set x <-- p - x. Return the decoded point (x,y).
5.1.4. 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 following formulas for adding two points, (x3,y3) =
(x1,y1)+(x2,y2) are described in [Edwards-revisited], section 3.1.
They are strongly unified, i.e., they work for any pair of valid
input points.
A = (Y1-X1)*(Y2-X2)
B = (Y1+X1)*(Y2+X2)
C = T1*2*d*T2
D = Z1*2*Z2
E = B-A
F = D-C
G = D+C
H = B+A
X3 = E*F
Y3 = G*H
T3 = E*H
Z3 = F*G
5.1.5. Key Generation
The secret is 32 octets (256 bits, corresponding to b) of
cryptographically-secure random data. See [RFC4086] for a discussion
about randomness.
The 32-byte public key is generated by the following steps.
1. Hash the 32-byte secret using SHA-512, storing the digest in a
64-octet large buffer, denoted h. Only the lower 32 bytes are
used for generating the public key.
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2. Prune the buffer: The lowest 3 bits of the first octet are
cleared, the highest bit of the last octet is cleared, and second
highest bit of the last octet is set.
3. Interpret the buffer as the little-endian integer, forming a
secret scalar a. Perform a fixed-base scalar multiplication
[a]B.
4. The public key A is the encoding of the point [a]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 [a]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 secret key, a 32-octet
string, and a message M of arbitrary size.
1. Hash the secret key, 32-octets, using SHA-512. Let h denote the
resulting digest. Construct the secret scalar a from the first
half of the digest, and the corresponding public key A, as
described in the previous section. Let prefix denote the second
half of the hash digest, h[32],...,h[63].
2. Compute SHA-512(prefix || M), where M is the message to be
signed. Interpret the 64-octet digest as a little-endian integer
r.
3. Compute the point [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(R || A || M), and interpret the 64-octet digest as
a little-endian integer k.
5. Compute S = (r + k * a) 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, three most significant
bits of the final octet are always zero).
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5.1.7. Verify
1. To verify a signature on a message M, first split the signature
into two 32-octet halves. Decode the first half as a point R,
and the second half as an integer S, in the range 0 <= s < l. If
the decoding fails, the signature is invalid.
2. Compute SHA512(R || A || M), and interpret the 64-octet digest as
a little-endian integer k.
3. Check the group equation [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 PureEdDSA instantiated with p as the prime 2^448 - 2^224 -
1, b=456, the 455-bit encoding of GF(2^448-2^224-1) is the usual
little-endian encoding of {0, 1, ..., 2^448 - 2^224 - 2}, H is
[FIXME: needs 912-bit hash], c being 2, n being 448, a being 1, d
being - 39081, B is (X(P), Y(P)), and l is the prime 2^446 -
13818066809895115352007386748515426880336692474882178609894547503885.
Ed448ph is the same but with P being SHA-512 instead, i.e., the input
is hashed using SHA-512 before signing with Ed448.
The values of p, a, d, X(p), Y(p), and l are taken from curve named
"edwards448" in [I-D.irtf-cfrg-curves].
The curve is equivalent to Ed448-Goldilocks under change of
basepoint, which preserves difficulty of the discrete logarithm.
5.2.1. Modular arithmetic
For advise 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).
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
square root of a.
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].
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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
nonzero 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 following formulas for adding two points, (x3,y3) =
(x1,y1)+(x2,y2) are described in [FIXME: Add reference]. They are
strongly unified, i.e., they work for any pair of valid input points.
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A = Z1*Z2
B = A^2
C = X1*X2
D = Y1*Y2
E = d*C*D
F = B-E
G = B+E
H = (X1+X2)*(Y1+Y2)
X3 = A*G*(H-C-D)
Y3 = A*G*(D-C)
Z3 = F*G
5.2.5. Key Generation
The secret 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 secret using FIXME-HASH, 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 8 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 a. Perform a known-base-point scalar
multiplication [a]B.
4. The public key A is the encoding of the point [a]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 [a]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 secret key, a 32-octet
string, and a message M of arbitrary size.
1. Hash the secret key, 57-octets, using FIXME-HASH. Let h denote
the resulting digest. Construct the secret scalar a from the
first half of the digest, and the corresponding public key A, as
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described in the previous section. Let prefix denote the second
half of the hash digest, h[57],...,h[113].
2. Compute FIXME-HASH(prefix || M), where M is the message to be
signed. 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 FIXME-HASH(R || A || M), and interpret the 114- octet
digest as a little-endian integer k.
5. Compute S = (r + k * a) 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, ten most significant bits
of the final octets always zero).
5.2.7. Verify
1. To verify a signature on a message M, 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. If
the decoding fails, the signature is invalid.
2. Compute FIXME-HASH(R || A || M), 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.
6. Ed25519 Python illustration
The rest of this section describes how Ed25519 can be implemented in
Python (version 3.2 or later) for illustration. See appendix A for
the complete implementation and appendix B for a test-driver to run
it through some test vectors.
Note that this code is not intended for production as it is not
proven to be correct for all inputs, nor does it protect against
side-channel attacks. The purpose is to illustrate the algorithm to
help implementers with their own implementation.
First some preliminaries that will be needed.
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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.
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# Points are represented as tuples (X, Y, Z, T) of extended coordinates,
# with x = X/Z, y = Y/Z, x*y = T/Z
def point_add(P, Q):
A = (P[1]-P[0])*(Q[1]-Q[0]) % p
B = (P[1]+P[0])*(Q[1]+Q[0]) % p
C = 2 * P[3] * Q[3] * d % p
D = 2 * P[2] * Q[2] % p
E = B-A
F = D-C
G = D+C
H = B+A
return (E*F, G*H, F*G, E*H)
# Computes Q = s * Q
def point_mul(s, P):
Q = (0, 1, 1, 0) # Neutral element
while s > 0:
# Is there any bit-set predicate?
if s & 1:
Q = point_add(Q, P)
P = point_add(P, P)
s >>= 1
return Q
def point_equal(P, Q):
# x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1
if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
return False
if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
return False
return True
Now follows functions for point compression.
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# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)
# Compute corresponding x coordinate, with low bit corresponding to sign,
# or return None on failure
def recover_x(y, sign):
x2 = (y*y-1) * modp_inv(d*y*y+1)
if x2 == 0:
if sign:
return None
else:
return 0
# Compute square root of x2
x = pow(x2, (p+3) // 8, p)
if (x*x - x2) % p != 0:
x = x * modp_sqrt_m1 % p
if (x*x - x2) % p != 0:
return None
if (x & 1) != sign:
x = p - x
return x
# Base point
g_y = 4 * modp_inv(5) % p
g_x = recover_x(g_y, 0)
G = (g_x, g_y, 1, g_x * g_y % p)
def point_compress(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
return int.to_bytes(y | ((x & 1) << 255), 32, "little")
def point_decompress(s):
if len(s) != 32:
raise Exception("Invalid input length for decompression")
y = int.from_bytes(s, "little")
sign = y >> 255
y &= (1 << 255) - 1
x = recover_x(y, sign)
if x is None:
return None
else:
return (x, y, 1, x*y % p)
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These are functions for manipulating the secret.
def secret_expand(secret):
if len(secret) != 32:
raise Exception("Bad size of private key")
h = sha512(secret)
a = int.from_bytes(h[:32], "little")
a &= (1 << 254) - 8
a |= (1 << 254)
return (a, h[32:])
def secret_to_public(secret):
(a, dummy) = secret_expand(secret)
return point_compress(point_mul(a, G))
The signature function works as below.
def sign(secret, msg):
a, prefix = secret_expand(secret)
A = point_compress(point_mul(a, G))
r = sha512_modq(prefix + msg)
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + msg)
s = (r + h * a) % q
return Rs + int.to_bytes(s, 32, "little")
And finally the verification function.
def verify(public, msg, signature):
if len(public) != 32:
raise Exception("Bad public-key length")
if len(signature) != 64:
Exception("Bad signature length")
A = point_decompress(public)
if not A:
return False
Rs = signature[:32]
R = point_decompress(Rs)
if not R:
return False
s = int.from_bytes(signature[32:], "little")
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))
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7. Test Vectors
This section contains test vectors for Ed25519ph, Ed448ph, Ed25519
and Ed448.
Each section contains sequence of test vectors. The octets are hex
encoded and whitespace is inserted for readability Ed25519 and
Ed25519ph private and public keys 32 octets, signatures are 64
octets. Ed448 and Ed448ph private and public keys are 57 octets,
signatures are 114 octets. Messages are of arbitrary length.
7.1. Test Vectors for Ed25519
These test vectors are taken from [ED25519-TEST-VECTORS] (but we
removed the public key as a suffix of the secret key, and removed the
message from the signature) and [ED25519-LIBGCRYPT-TEST-VECTORS].
-----TEST 1
SECRET KEY:
9d61b19deffd5a60ba844af492ec2cc4
4449c5697b326919703bac031cae7f60
PUBLIC KEY:
d75a980182b10ab7d54bfed3c964073a
0ee172f3daa62325af021a68f707511a
MESSAGE (length 0 bytes):
SIGNATURE:
e5564300c360ac729086e2cc806e828a
84877f1eb8e5d974d873e06522490155
5fb8821590a33bacc61e39701cf9b46b
d25bf5f0595bbe24655141438e7a100b
-----TEST 2
SECRET KEY:
4ccd089b28ff96da9db6c346ec114e0f
5b8a319f35aba624da8cf6ed4fb8a6fb
PUBLIC KEY:
3d4017c3e843895a92b70aa74d1b7ebc
9c982ccf2ec4968cc0cd55f12af4660c
MESSAGE (length 1 byte):
72
SIGNATURE:
92a009a9f0d4cab8720e820b5f642540
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a2b27b5416503f8fb3762223ebdb69da
085ac1e43e15996e458f3613d0f11d8c
387b2eaeb4302aeeb00d291612bb0c00
-----TEST 3
SECRET KEY:
c5aa8df43f9f837bedb7442f31dcb7b1
66d38535076f094b85ce3a2e0b4458f7
PUBLIC KEY:
fc51cd8e6218a1a38da47ed00230f058
0816ed13ba3303ac5deb911548908025
MESSAGE (length 2 bytes):
af82
SIGNATURE:
6291d657deec24024827e69c3abe01a3
0ce548a284743a445e3680d7db5ac3ac
18ff9b538d16f290ae67f760984dc659
4a7c15e9716ed28dc027beceea1ec40a
-----TEST 1024
SECRET KEY:
f5e5767cf153319517630f226876b86c
8160cc583bc013744c6bf255f5cc0ee5
PUBLIC KEY:
278117fc144c72340f67d0f2316e8386
ceffbf2b2428c9c51fef7c597f1d426e
MESSAGE (length 1023 bytes):
08b8b2b733424243760fe426a4b54908
632110a66c2f6591eabd3345e3e4eb98
fa6e264bf09efe12ee50f8f54e9f77b1
e355f6c50544e23fb1433ddf73be84d8
79de7c0046dc4996d9e773f4bc9efe57
38829adb26c81b37c93a1b270b20329d
658675fc6ea534e0810a4432826bf58c
941efb65d57a338bbd2e26640f89ffbc
1a858efcb8550ee3a5e1998bd177e93a
7363c344fe6b199ee5d02e82d522c4fe
ba15452f80288a821a579116ec6dad2b
3b310da903401aa62100ab5d1a36553e
06203b33890cc9b832f79ef80560ccb9
a39ce767967ed628c6ad573cb116dbef
efd75499da96bd68a8a97b928a8bbc10
3b6621fcde2beca1231d206be6cd9ec7
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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
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SIGNATURE:
0aab4c900501b3e24d7cdf4663326a3a
87df5e4843b2cbdb67cbf6e460fec350
aa5371b1508f9f4528ecea23c436d94b
5e8fcd4f681e30a6ac00a9704a188a03
-----
7.2. Test Vectors for Ed25519ph
TODO
7.3. Test Vectors for Ed448
TODO
7.4. Test Vectors for Ed448ph
TODO
8. Acknowledgements
EdDSA and Ed25519 was initially described in a paper due to Daniel J.
Bernstein, Niels Duif, Tanja Lange, Peter Schwabe and Bo-Yin Yang.
The Ed448 curves is due to Mike Hamburg.
This draft is based on an earlier draft co-authored by Niels Moeller.
Feedback on this document was received 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 3
separate implementations (one based on TweetNaCl and 2 different
implementations based on code from SUPERCOP).
9. IANA Considerations
None.
10. Security Considerations
10.1. Side-channel leaks
For implementations performing signatures, secrecy of the key is
fundamental. It is possible to protect against some side-channel
attacks by ensuring that the implementation executes exactly the same
sequence of instructions and performs exactly the same memory
accesses, for any value of the secret key.
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To make an implementation side-channel silent in this way, the modulo
p arithmetic must not use any data-dependent branches, e.g., related
to carry propagation. Side channel-silent point addition is
straight-forward, thanks to the unified formulas.
Scalar multiplication, multiplying a point by an integer, needs some
additional effort to implement in a side-channel silent manner. One
simple approach is to implement a side-channel silent conditional
assignment, and use together with the binary algorithm to examine one
bit of the integer at a time.
Note that the example implementations in this document do not attempt
to be side-channel silent.
10.2. Mixing different prehashes
Using the same key with different prehashes is a bad idea. The most
obvious problem is that identity transform can produce message values
colliding with hashes of interesting messages (or vice versa).
Additionally, even if it is infeasible to find collisions in two hash
functions, there is nothing guaranteeing that finding collisions
between hashes is infeasible.
For these reasons, the same private key MUST NOT be used for multiple
algorithms (including prehashes) without protocol preventing messages
output by different prehash algorithms from colliding.
10.3. Signing large amounts of data at once
Avoid signing large amounts of data at once (where "large" depends on
expected verifier). In practicular, unless the underlying protocol
does not require it, the receiver MUST buffer the entire message (or
enough information to reconstruct it, e.g. compressed or encrypted
version) to be verified.
This is needed becuse most of the time, it is unsafe to process
unverified data, and verifying the signature makes a pass through
whole message, causing ultimately at least two passes through.
As API consideration, this means that any IUF verification interface
is prone to misuse.
11. References
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11.1. Normative References
[RFC4634] Eastlake, D. and T. Hansen, "US Secure Hash Algorithms
(SHA and HMAC-SHA)", RFC 4634, July 2006.
[I-D.irtf-cfrg-curves]
Langley, A. and M. Hamburg, "Elliptic Curves for
Security", draft-irtf-cfrg-curves-10 (work in progress),
October 2015.
11.2. Informative References
[RFC4086] Eastlake, D., Schiller, J., and S. Crocker, "Randomness
Requirements for Security", BCP 106, RFC 4086, June 2005.
[EDDSA] Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
Yang, "High-speed high-security signatures", WWW
http://ed25519.cr.yp.to/ed25519-20110926.pdf, September
2011.
[EDDSA2] Bernstein, D., Josefsson, S., Lange, T., Schwabe, P., and
B. Yang, "EdDSA for more curves", WWW
http://ed25519.cr.yp.to/eddsa-20150704.pdf, July 2015.
[Faster-ECC]
Bernstein, D. and T. Lange, "Faster addition and doubling
on elliptic curves", WWW http://eprint.iacr.org/2007/286,
July 2007.
[Edwards-revisited]
Hisil, H., Wong, K., Carter, G., and E. Dawson, "Twisted
Edwards Curves Revisited", WWW
http://eprint.iacr.org/2008/522, December 2008.
[CURVE25519]
Bernstein, D., "Curve25519: new Diffie-Hellman speed
records", WWW http://cr.yp.to/ecdh.html, February 2006.
[ED448] Hamburg, M., "Ed448-Goldilocks, a new elliptic curve", WWW
http://eprint.iacr.org/2015/625, June 2015.
[ED25519-TEST-VECTORS]
Bernstein, D., Duif, N., Lange, T., Schwabe, P., and B.
Yang, "Ed25519 test vectors", WWW
http://ed25519.cr.yp.to/python/sign.input, July 2011.
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[ED25519-LIBGCRYPT-TEST-VECTORS]
Koch, W., "Ed25519 Libgcrypt test vectors", WWW
http://git.gnupg.org/cgi-
bin/gitweb.cgi?p=libgcrypt.git;a=blob;f=tests/t-ed25519.in
p;h=e13566f826321eece65e02c593bc7d885b3dbe23;hb=refs/
heads/master, July 2014.
Appendix A. Ed25519 Python Library
Below is an example implementation of Ed25519 written in Python,
version 3.2 or higher is required.
# Loosely based on the public domain code at
# http://ed25519.cr.yp.to/software.html
#
# Needs python-3.2
import hashlib
def sha512(s):
return hashlib.sha512(s).digest()
# Base field Z_p
p = 2**255 - 19
def modp_inv(x):
return pow(x, p-2, p)
# Curve constant
d = -121665 * modp_inv(121666) % p
# Group order
q = 2**252 + 27742317777372353535851937790883648493
def sha512_modq(s):
return int.from_bytes(sha512(s), "little") % q
# Points are represented as tuples (X, Y, Z, T) of extended coordinates,
# with x = X/Z, y = Y/Z, x*y = T/Z
def point_add(P, Q):
A = (P[1]-P[0])*(Q[1]-Q[0]) % p
B = (P[1]+P[0])*(Q[1]+Q[0]) % p
C = 2 * P[3] * Q[3] * d % p
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D = 2 * P[2] * Q[2] % p
E = B-A
F = D-C
G = D+C
H = B+A
return (E*F, G*H, F*G, E*H)
# Computes Q = s * Q
def point_mul(s, P):
Q = (0, 1, 1, 0) # Neutral element
while s > 0:
# Is there any bit-set predicate?
if s & 1:
Q = point_add(Q, P)
P = point_add(P, P)
s >>= 1
return Q
def point_equal(P, Q):
# x1 / z1 == x2 / z2 <==> x1 * z2 == x2 * z1
if (P[0] * Q[2] - Q[0] * P[2]) % p != 0:
return False
if (P[1] * Q[2] - Q[1] * P[2]) % p != 0:
return False
return True
# Square root of -1
modp_sqrt_m1 = pow(2, (p-1) // 4, p)
# Compute corresponding x coordinate, with low bit corresponding to sign,
# or return None on failure
def recover_x(y, sign):
x2 = (y*y-1) * modp_inv(d*y*y+1)
if x2 == 0:
if sign:
return None
else:
return 0
# Compute square root of x2
x = pow(x2, (p+3) // 8, p)
if (x*x - x2) % p != 0:
x = x * modp_sqrt_m1 % p
if (x*x - x2) % p != 0:
return None
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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)
def secret_expand(secret):
if len(secret) != 32:
raise Exception("Bad size of private key")
h = sha512(secret)
a = int.from_bytes(h[:32], "little")
a &= (1 << 254) - 8
a |= (1 << 254)
return (a, h[32:])
def secret_to_public(secret):
(a, dummy) = secret_expand(secret)
return point_compress(point_mul(a, G))
def sign(secret, msg):
a, prefix = secret_expand(secret)
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A = point_compress(point_mul(a, G))
r = sha512_modq(prefix + msg)
R = point_mul(r, G)
Rs = point_compress(R)
h = sha512_modq(Rs + A + msg)
s = (r + h * a) % q
return Rs + int.to_bytes(s, 32, "little")
def verify(public, msg, signature):
if len(public) != 32:
raise Exception("Bad public-key length")
if len(signature) != 64:
Exception("Bad signature length")
A = point_decompress(public)
if not A:
return False
Rs = signature[:32]
R = point_decompress(Rs)
if not R:
return False
s = int.from_bytes(signature[32:], "little")
h = sha512_modq(Rs + public + msg)
sB = point_mul(s, G)
hA = point_mul(h, A)
return point_equal(sB, point_add(R, hA))
Appendix B. Library driver
Below is a command-line tool that uses the library above to perform
computations, for interactive use or for self-checking.
import sys
import binascii
from ed25519 import *
def point_valid(P):
zinv = modp_inv(P[2])
x = P[0] * zinv % p
y = P[1] * zinv % p
assert (x*y - P[3]*zinv) % p == 0
return (-x*x + y*y - 1 - d*x*x*y*y) % p == 0
assert point_valid(G)
Z = (0, 1, 1, 0)
assert point_valid(Z)
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assert point_equal(Z, point_add(Z, Z))
assert point_equal(G, point_add(Z, G))
assert point_equal(Z, point_mul(0, G))
assert point_equal(G, point_mul(1, G))
assert point_equal(point_add(G, G), point_mul(2, G))
for i in range(0, 100):
assert point_valid(point_mul(i, G))
assert point_equal(Z, point_mul(q, G))
def munge_string(s, pos, change):
return (s[:pos] +
int.to_bytes(s[pos] ^ change, 1, "little") +
s[pos+1:])
# Read a file in the format of
# http://ed25519.cr.yp.to/python/sign.input
lineno = 0
while True:
line = sys.stdin.readline()
if not line:
break
lineno = lineno + 1
print(lineno)
fields = line.split(":")
secret = (binascii.unhexlify(fields[0]))[:32]
public = binascii.unhexlify(fields[1])
msg = binascii.unhexlify(fields[2])
signature = binascii.unhexlify(fields[3])[:64]
assert public == secret_to_public(secret)
assert signature == sign(secret, msg)
assert verify(public, msg, signature)
if len(msg) == 0:
bad_msg = b"x"
else:
bad_msg = munge_string(msg, len(msg) // 3, 4)
assert not verify(public, bad_msg, signature)
bad_signature = munge_string(signature, 20, 8)
assert not verify(public, msg, bad_signature)
bad_signature = munge_string(signature, 40, 16)
assert not verify(public, msg, bad_signature)
Authors' Addresses
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Simon Josefsson
SJD AB
Email: simon@josefsson.org
URI: http://josefsson.org/
Ilari Liusvaara
Independent
Email: ilariliusvaara@welho.com
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