I'm working on modernizing Rosetta Code's infrastructure. Starting with communications. Please accept this time-limited open invite to RC's Slack.. --Michael Mol (talk) 20:59, 30 May 2020 (UTC)

Elliptic Curve Digital Signature Algorithm

From Rosetta Code
Task
Elliptic Curve Digital Signature Algorithm
You are encouraged to solve this task according to the task description, using any language you may know.
Elliptic curves.

An elliptic curve E over ℤp (p ≥ 5) is defined by an equation of the form y^2 = x^3 + ax + b, where a, b ∈ ℤp and the discriminant ≢ 0 (mod p), together with a special point 𝒪 called the point at infinity. The set E(ℤp) consists of all points (x, y), with x, y ∈ ℤp, which satisfy the above defining equation, together with 𝒪.

There is a rule for adding two points on an elliptic curve to give a third point. This addition operation and the set of points E(ℤp) form a group with identity 𝒪. It is this group that is used in the construction of elliptic curve cryptosystems.

The addition rule — which can be explained geometrically — is summarized as follows:

1. P + 𝒪 = 𝒪 + P = P for all P ∈ E(ℤp).

2. If P = (x, y) ∈ E(ℤp), then inverse -P = (x,-y), and P + (-P) = 𝒪.

3. Let P = (xP, yP) and Q = (xQ, yQ), both ∈ E(ℤp), where P ≠ -Q.
   Then R = P + Q = (xR, yR), where

   xR = λ^2 - xP - xQ
   yR = λ·(xP - xR) - yP,

   with

   λ = (yP - yQ) / (xP - xQ) if P ≠ Q,
       (3·xP·xP + a) / 2·yP  if P = Q (point doubling).

Remark: there already is a task page requesting “a simplified (without modular arithmetic) version of the elliptic curve arithmetic”. Here we do add modulo operations. If also the domain is changed from reals to rationals, the elliptic curves are no longer continuous but break up into a finite number of distinct points. In that form we use them to implement ECDSA:


Elliptic curve digital signature algorithm.

A digital signature is the electronic analogue of a hand-written signature that convinces the recipient that a message has been sent intact by the presumed sender. Anyone with access to the public key of the signer may verify this signature. Changing even a single bit of a signed message will cause the verification procedure to fail.

ECDSA key generation. Party A does the following:
1. Select an elliptic curve E defined over ℤp.
 The number of points in E(ℤp) should be divisible by a large prime r.
2. Select a base point G ∈ E(ℤp) of order r (which means that rG = 𝒪).
3. Select a random integer s in the interval [1, r - 1].
4. Compute W = sG.
 The public key is (E, G, r, W), the private key is s.

ECDSA signature computation. To sign a message m, A does the following:
1. Compute message representative f = H(m), using a cryptographic hash function.
 Note that f can be greater than r but not longer (measuring bits).
2. Select a random integer u in the interval [1, r - 1].
3. Compute V = uG = (xV, yV) and c ≡ xV mod r  (goto (2) if c = 0).
4. Compute d ≡ u^-1·(f + s·c) mod r  (goto (2) if d = 0).
 The signature for the message m is the pair of integers (c, d).

ECDSA signature verification. To verify A's signature, B should do the following:
1. Obtain an authentic copy of A's public key (E, G, r, W).
 Verify that c and d are integers in the interval [1, r - 1].
2. Compute f = H(m) and h ≡ d^-1 mod r.
3. Compute h1 ≡ f·h mod r and h2 ≡ c·h mod r.
4. Compute h1G + h2W = (x1, y1) and c1 ≡ x1 mod r.
 Accept the signature if and only if c1 = c.

To be cryptographically useful, the parameter r should have at least 250 bits. The basis for the security of elliptic curve cryptosystems is the intractability of the elliptic curve discrete logarithm problem (ECDLP) in a group of this size: given two points G, W ∈ E(ℤp), where W lies in the subgroup of order r generated by G, determine an integer k such that W = kG and 0 ≤ k < r.


Task.

The task is to write a toy version of the ECDSA, quasi the equal of a real-world implementation, but utilizing parameters that fit into standard arithmetic types. To keep things simple there's no need for key export or a hash function (just a sample hash value and a way to tamper with it). The program should be lenient where possible (for example: if it accepts a composite modulus N it will either function as expected, or demonstrate the principle of elliptic curve factorization) — but strict where required (a point G that is not on E will always cause failure).
Toy ECDSA is of course completely useless for its cryptographic purpose. If this bothers you, please add a multiple-precision version.


Reference.

Elliptic curves are in the IEEE Std 1363-2000 (Standard Specifications for Public-Key Cryptography), see:

7. Primitives based on the elliptic curve discrete logarithm problem (p. 27ff.)

7.1 The EC setting
7.1.2 EC domain parameters
7.1.3 EC key pairs

7.2 Primitives
7.2.7 ECSP-DSA (p. 35)
7.2.8 ECVP-DSA (p. 36)

Annex A. Number-theoretic background
A.9 Elliptic curves: overview (p. 115)
A.10 Elliptic curves: algorithms (p. 121)


C[edit]

Parallel to: FreeBASIC

 
/*
subject: Elliptic curve digital signature algorithm,
toy version for small modulus N.
tested : gcc 4.6.3, tcc 0.9.27
*/

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
 
// 64-bit integer type
typedef long long int dlong;
// rational ec point
typedef struct {
dlong x, y;
} epnt;
// elliptic curve parameters
typedef struct {
long a, b;
dlong N;
epnt G;
dlong r;
} curve;
// signature pair
typedef struct {
long a, b;
} pair;
 
// dlong for holding intermediate results,
// long variables in exgcd() for efficiency,
// maximum parameter size 2 * p.y (line 129)
// limits the modulus size to 30 bits.
 
// maximum modulus
const long mxN = 1073741789;
// max order G = mxN + 65536
const long mxr = 1073807325;
// symbolic infinity
const long inf = -2147483647;
 
// single global curve
curve e;
// point at infinity zerO
epnt zerO;
// impossible inverse mod N
int inverr;
 
 
// return mod(v^-1, u)
long exgcd (long v, long u)
{
register long q, t;
long r = 0, s = 1;
if (v < 0) v += u;
 
while (v) {
q = u / v;
t = u - q * v;
u = v; v = t;
t = r - q * s;
r = s; s = t;
}
if (u != 1) {
printf (" impossible inverse mod N, gcd = %d\n", u);
inverr = 1;
}
return r;
}
 
// return mod(a, N)
static inline dlong modn (dlong a)
{
a %= e.N;
if (a < 0) a += e.N;
return a;
}
 
// return mod(a, r)
dlong modr (dlong a)
{
a %= e.r;
if (a < 0) a += e.r;
return a;
}
 
 
// return the discriminant of E
long disc (void)
{
dlong c, a = e.a, b = e.b;
c = 4 * modn(a * modn(a * a));
return modn(-16 * (c + 27 * modn(b * b)));
}
 
// return 1 if P = zerO
int isO (epnt p)
{
return (p.x == inf) && (p.y == 0);
}
 
// return 1 if P is on curve E
int ison (epnt p)
{
long r, s;
if (! isO (p)) {
r = modn(e.b + p.x * modn(e.a + p.x * p.x));
s = modn(p.y * p.y);
}
return (r == s);
}
 
 
// full ec point addition
void padd (epnt *r, epnt p, epnt q)
{
dlong la, t;
 
if (isO(p)) {*r = q; return;}
if (isO(q)) {*r = p; return;}
 
if (p.x != q.x) { // R:= P + Q
t = p.y - q.y;
la = modn(t * exgcd(p.x - q.x, e.N));
}
else // P = Q, R := 2P
if ((p.y == q.y) && (p.y != 0)) {
t = modn(3 * modn(p.x * p.x) + e.a);
la = modn(t * exgcd (2 * p.y, e.N));
}
else
{*r = zerO; return;} // P = -Q, R := O
 
t = modn(la * la - p.x - q.x);
r->y = modn(la * (p.x - t) - p.y);
r->x = t; if (inverr) *r = zerO;
}
 
// R:= multiple kP
void pmul (epnt *r, epnt p, long k)
{
epnt s = zerO, q = p;
 
for (; k; k >>= 1) {
if (k & 1) padd(&s, s, q);
if (inverr) {s = zerO; break;}
padd(&q, q, q);
}
*r = s;
}
 
 
// print point P with prefix f
void pprint (char *f, epnt p)
{
dlong y = p.y;
 
if (isO (p))
printf ("%s (0)\n", f);
 
else {
if (y > e.N - y) y -= e.N;
printf ("%s (%lld, %lld)\n", f, p.x, y);
}
}
 
// initialize elliptic curve
int ellinit (long i[])
{
long a = i[0], b = i[1];
e.N = i[2]; inverr = 0;
 
if ((e.N < 5) || (e.N > mxN)) return 0;
 
e.a = modn(a);
e.b = modn(b);
e.G.x = modn(i[3]);
e.G.y = modn(i[4]);
e.r = i[5];
 
if ((e.r < 5) || (e.r > mxr)) return 0;
 
printf ("\nE: y^2 = x^3 + %dx + %d", a, b);
printf (" (mod %lld)\n", e.N);
pprint ("base point G", e.G);
printf ("order(G, E) = %lld\n", e.r);
 
return 1;
}
 
// pseudorandom number [0..1)
double rnd(void)
{
return rand() / ((double)RAND_MAX + 1);
}
 
// signature primitive
pair signature (dlong s, long f)
{
long c, d, u, u1;
pair sg;
epnt V;
 
printf ("\nsignature computation\n");
do {
do {
u = 1 + (long)(rnd() * (e.r - 1));
pmul (&V, e.G, u);
c = modr(V.x);
}
while (c == 0);
 
u1 = exgcd (u, e.r);
d = modr(u1 * (f + modr(s * c)));
}
while (d == 0);
printf ("one-time u = %d\n", u);
pprint ("V = uG", V);
 
sg.a = c; sg.b = d;
return sg;
}
 
// verification primitive
int verify (epnt W, long f, pair sg)
{
long c = sg.a, d = sg.b;
long t, c1, h1, h2;
dlong h;
epnt V, V2;
 
// domain check
t = (c > 0) && (c < e.r);
t &= (d > 0) && (d < e.r);
if (! t) return 0;
 
printf ("\nsignature verification\n");
h = exgcd (d, e.r);
h1 = modr(f * h);
h2 = modr(c * h);
printf ("h1,h2 = %d, %d\n", h1,h2);
pmul (&V, e.G, h1);
pmul (&V2, W, h2);
pprint ("h1G", V);
pprint ("h2W", V2);
padd (&V, V, V2);
pprint ("+ =", V);
if (isO (V)) return 0;
c1 = modr(V.x);
printf ("c' = %d\n", c1);
 
return (c1 == c);
}
 
// digital signature on message hash f, error bit d
void ec_dsa (long f, long d)
{
long i, s, t;
pair sg;
epnt W;
 
// parameter check
t = (disc() == 0);
t |= isO (e.G);
pmul (&W, e.G, e.r);
t |= ! isO (W);
t |= ! ison (e.G);
if (t) goto errmsg;
 
printf ("\nkey generation\n");
s = 1 + (long)(rnd() * (e.r - 1));
pmul (&W, e.G, s);
printf ("private key s = %d\n", s);
pprint ("public key W = sG", W);
 
// next highest power of 2 - 1
t = e.r;
for (i = 1; i < 32; i <<= 1)
t |= t >> i;
while (f > t) f >>= 1;
printf ("\naligned hash %x\n", f);
 
sg = signature (s, f);
if (inverr) goto errmsg;
printf ("signature c,d = %d, %d\n", sg.a, sg.b);
 
if (d > 0) {
while (d > t) d >>= 1;
f ^= d;
printf ("\ncorrupted hash %x\n", f);
}
 
t = verify (W, f, sg);
if (inverr) goto errmsg;
if (t)
printf ("Valid\n_____\n");
else
printf ("invalid\n_______\n");
 
return;
 
errmsg:
printf ("invalid parameter set\n");
printf ("_____________________\n");
}
 
 
void main (void)
{
typedef long eparm[6];
long d, f;
zerO.x = inf; zerO.y = 0;
srand(time(NULL));
 
// Test vectors: elliptic curve domain parameters,
// short Weierstrass model y^2 = x^3 + ax + b (mod N)
eparm *sp, sets[10] = {
// a, b, modulus N, base point G, order(G, E), cofactor
{355, 671, 1073741789, 13693, 10088, 1073807281},
{ 0, 7, 67096021, 6580, 779, 16769911}, // 4
{ -3, 1, 877073, 0, 1, 878159},
{ 0, 14, 22651, 63, 30, 151}, // 151
{ 3, 2, 5, 2, 1, 5},
 
// ecdsa may fail if...
// the base point is of composite order
{ 0, 7, 67096021, 2402, 6067, 33539822}, // 2
// the given order is a multiple of the true order
{ 0, 7, 67096021, 6580, 779, 67079644}, // 1
// the modulus is not prime (deceptive example)
{ 0, 7, 877069, 3, 97123, 877069},
// fails if the modulus divides the discriminant
{ 39, 387, 22651, 95, 27, 22651},
};
// Digital signature on message hash f,
// set d > 0 to simulate corrupted data
f = 0x789abcde; d = 0;
 
for (sp = sets; ; sp++) {
if (ellinit (*sp))
ec_dsa (f, d);
 
else
break;
}
}
 
Output:

(tcc, srand(1); first set only)

E: y^2 = x^3 + 355x + 671 (mod 1073741789)
base point G (13693, 10088)
order(G, E) = 1073807281

key generation
private key s = 1343570
public key W = sG (817515107, -192163292)

aligned hash 789abcde

signature computation
one-time u = 605163545
V = uG (464115167, -267961770)
signature c,d = 464115167, 407284989

signature verification
h1,h2 = 871754294, 34741072
h1G (708182134, 29830217)
h2W (270156466, -328492261)
+ = (464115167, -267961770)
c' = 464115167
Valid
_____

FreeBASIC[edit]

Parallel to: C

 
'subject: Elliptic curve digital signature algorithm,
' toy version for small modulus N.
'tested : FreeBasic 1.05.0
 
'rational ec point
type epnt
as longint x, y
end type
'elliptic curve parameters
type curve
as long a, b
as longint N
as epnt G
as longint r
end type
'signature pair
type pair
as long a, b
end type
 
'longint for holding intermediate results,
'long variables in exgcd() for efficiency,
'maximum parameter size 2 * p.y (line 118)
'limits the modulus size to 30 bits.
 
'maximum modulus
const mxN = 1073741789
'max order G = mxN + 65536
const mxr = 1073807325
'symbolic infinity
const inf = -2147483647
 
'single global curve
dim shared as curve e
'point at infinity zerO
dim shared as epnt zerO
'impossible inverse mod N
dim shared as byte inverr
 
 
'return mod(v^-1, u)
Function exgcd (byval v as long, byval u as long) as long
dim as long q, t
dim as long r = 0, s = 1
if v < 0 then v += u
 
while v
q = u \ v
t = u - q * v
u = v: v = t
t = r - q * s
r = s: s = t
wend
 
if u <> 1 then
print " impossible inverse mod N, gcd ="; u
inverr = -1
end if
 
exgcd = r
End Function
 
'return mod(a, N)
Function modn (byval a as longint) as longint
a mod= e.N
if a < 0 then a += e.N
modn = a
End Function
 
'return mod(a, r)
Function modr (byval a as longint) as longint
a mod= e.r
if a < 0 then a += e.r
modr = a
End Function
 
 
'return the discriminant of E
Function disc as long
dim as longint c, a = e.a, b = e.b
c = 4 * modn(a * modn(a * a))
disc = modn(-16 * (c + 27 * modn(b * b)))
End Function
 
'return -1 if P = zerO
Function isO (byref p as epnt) as byte
isO = (p.x = inf and p.y = 0)
End Function
 
'return -1 if P is on curve E
Function ison (byref p as epnt) as byte
dim as long r, s
if not isO (p) then
r = modn(e.b + p.x * modn(e.a + p.x * p.x))
s = modn(p.y * p.y)
end if
ison = (r = s)
End Function
 
 
'full ec point addition
Sub padd (byref r as epnt, byref p as epnt, byref q as epnt)
dim as longint la, t
 
if isO (p) then r = q: exit sub
if isO (q) then r = p: exit sub
 
if p.x <> q.x then ' R := P + Q
t = p.y - q.y
la = modn(t * exgcd (p.x - q.x, e.N))
 
else ' P = Q, R := 2P
if (p.y = q.y) and (p.y <> 0) then
t = modn(3 * modn(p.x * p.x) + e.a)
la = modn(t * exgcd (2 * p.y, e.N))
 
else
r = zerO: exit sub ' P = -Q, R := O
end if
end if
 
t = modn(la * la - p.x - q.x)
r.y = modn(la * (p.x - t) - p.y)
r.x = t: if inverr then r = zerO
End Sub
 
'R:= multiple kP
Sub pmul (byref r as epnt, byref p as epnt, byval k as long)
dim as epnt s = zerO, q = p
 
while k
if k and 1 then padd (s, s, q)
if inverr then s = zerO: exit while
k shr= 1: padd (q, q, q)
wend
r = s
End Sub
 
 
'print point P with prefix f
Sub pprint (byref f as string, byref p as epnt)
dim as longint y = p.y
 
if isO (p) then
print f;" (0)"
 
else
if y > e.N - y then y -= e.N
print f;" (";str(p.x);",";y;")"
 
end if
End Sub
 
'initialize elliptic curve
Function ellinit (i() as long) as byte
dim as long a = i(0), b = i(1)
ellinit = 0: inverr = 0
e.N = i(2)
 
if (e.N < 5) or (e.N > mxN) then exit function
 
e.a = modn(a)
e.b = modn(b)
e.G.x = modn(i(3))
e.G.y = modn(i(4))
e.r = i(5)
 
if (e.r < 5) or (e.r > mxr) then exit function
 
print : ? "E: y^2 = x^3 + ";str(a);"x +";b;
print " (mod ";str(e.N);")"
pprint ("base point G", e.G)
print "order(G, E) ="; e.r
 
ellinit = -1
End Function
 
 
'signature primitive
Function signature (byval s as longint, byval f as long) as pair
dim as long c, d, u, u1
dim as pair sg
dim as epnt V
 
print : ? "signature computation"
do
do
u = 1 + int(rnd * (e.r - 1))
pmul (V, e.G, u)
c = modr(V.x)
loop while c = 0
 
u1 = exgcd (u, e.r)
d = modr(u1 * (f + modr(s * c)))
loop while d = 0
print "one-time u ="; u
pprint ("V = uG", V)
 
sg.a = c: sg.b = d
signature = sg
End Function
 
'verification primitive
Function verify (byref W as epnt, byval f as long, byref sg as pair) as byte
dim as long c = sg.a, d = sg.b
dim as long t, c1, h1, h2
dim as longint h
dim as epnt V, V2
verify = 0
 
'domain check
t = (c > 0) and (c < e.r)
t and= (d > 0) and (d < e.r)
if not t then exit function
 
print : ? "signature verification"
h = exgcd (d, e.r)
h1 = modr(f * h)
h2 = modr(c * h)
print "h1,h2 ="; h1;",";h2
pmul (V, e.G, h1)
pmul (V2, W, h2)
pprint ("h1G", V)
pprint ("h2W", V2)
padd (V, V, V2)
pprint ("+ =", V)
if isO (V) then exit function
c1 = modr(V.x)
print "c' ="; c1
 
verify = (c1 = c)
End Function
 
'digital signature on message hash f, error bit d
Sub ec_dsa (byval f as long, byval d as long)
dim as long i, s, t
dim as pair sg
dim as epnt W
 
'parameter check
t = (disc = 0)
t or= isO (e.G)
pmul (W, e.G, e.r)
t or= not isO (W)
t or= not ison (e.G)
if t then goto errmsg
 
print : ? "key generation"
s = 1 + int(rnd * (e.r - 1))
pmul (W, e.G, s)
print "private key s ="; s
pprint ("public key W = sG", W)
 
'next highest power of 2 - 1
t = e.r: i = 1
while i < 32
t or= t shr i: i shl= 1
wend
while f > t
f shr= 1: wend
print : ? "aligned hash "; hex(f)
 
sg = signature (s, f)
if inverr then goto errmsg
print "signature c,d ="; sg.a;",";sg.b
 
if d > 0 then
while d > t
d shr= 1: wend
f xor= d
print : ? "corrupted hash "; hex(f)
end if
 
t = verify (W, f, sg)
if inverr then goto errmsg
if t then
print "Valid" : ? "_____"
else
print "invalid" : ? "_______"
end if
 
exit sub
 
errmsg:
print "invalid parameter set"
print "_____________________"
End Sub
 
 
'main
dim as long d, f, t, eparm(5)
zerO.x = inf: zerO.y = 0
randomize timer
 
'Test vectors: elliptic curve domain parameters,
'short Weierstrass model y^2 = x^3 + ax + b (mod N)
 
' a, b, modulus N, base point G, order(G, E), cofactor
data 355, 671, 1073741789, 13693, 10088, 1073807281
data 0, 7, 67096021, 6580, 779, 16769911 ' 4
data -3, 1, 877073, 0, 1, 878159
data 0, 14, 22651, 63, 30, 151 ' 151
data 3, 2, 5, 2, 1, 5
 
'ecdsa may fail if...
'the base point is of composite order
data 0, 7, 67096021, 2402, 6067, 33539822 ' 2
'the given order is a multiple of the true order
data 0, 7, 67096021, 6580, 779, 67079644 ' 1
'the modulus is not prime (deceptive example)
data 0, 7, 877069, 3, 97123, 877069
'fails if the modulus divides the discriminant
data 39, 387, 22651, 95, 27, 22651
data 0, 0, 0
 
'Digital signature on message hash f,
'set d > 0 to simulate corrupted data
f = &h789ABCDE : d = 0
 
do
for t = 0 to 5
read eparm(t): next
 
if ellinit (eparm()) then
ec_dsa (f, d)
 
else
exit do
 
end if
loop
 
system
 
Output:

(randomize 1, first set only)

E: y^2 = x^3 + 355x + 671 (mod 1073741789)
base point G (13693, 10088)
order(G, E) = 1073807281

key generation
private key s = 509100772
public key W = sG (992563138, 238074938)

aligned hash 789ABCDE

signature computation
one-time u = 571533488
V = uG (896670665, 183547995)
signature c,d = 896670665, 728505276

signature verification
h1,h2 = 667118700, 709185150
h1G (315367421, 343743703)
h2W (1040319975,-262613483)
+ = (896670665, 183547995)
c' = 896670665
Valid
_____

Go[edit]

Since Go has an ECDSA package in its standard library which uses 'big integers', we use that rather than translating one of the reference implementations for a 'toy' version into Go.

package main
 
import (
"crypto/ecdsa"
"crypto/elliptic"
"crypto/rand"
"crypto/sha256"
"encoding/binary"
"fmt"
"log"
)
 
func check(err error) {
if err != nil {
log.Fatal(err)
}
}
 
func main() {
priv, err := ecdsa.GenerateKey(elliptic.P256(), rand.Reader)
check(err)
fmt.Println("Private key:\nD:", priv.D)
pub := priv.Public().(*ecdsa.PublicKey)
fmt.Println("\nPublic key:")
fmt.Println("X:", pub.X)
fmt.Println("Y:", pub.Y)
 
msg := "Rosetta Code"
fmt.Println("\nMessage:", msg)
hash := sha256.Sum256([]byte(msg)) // as [32]byte
hexHash := fmt.Sprintf("0x%x", binary.BigEndian.Uint32(hash[:]))
fmt.Println("Hash  :", hexHash)
 
r, s, err := ecdsa.Sign(rand.Reader, priv, hash[:])
check(err)
fmt.Println("\nSignature:")
fmt.Println("R:", r)
fmt.Println("S:", s)
 
valid := ecdsa.Verify(&priv.PublicKey, hash[:], r, s)
fmt.Println("\nSignature verified:", valid)
}
Output:

Sample run:

Private key:
D: 25700608762903774973512323993645267346590725880891580901973011512673451968935

Public key:
X: 37298454876588653961191059192981094503652951300904260069480867699946371240473
Y: 69073688506493709421315518164229531832022167466292360349457318041854718641652

Message: Rosetta Code
Hash   : 0xe6f9ed0d

Signature:
R: 91827099055706804696234859308003894767808769875556550819128270941615405955877
S: 20295707309473352071389945163735458699476300346398176659149368970668313772860

Signature verified: true

Julia[edit]

module ToyECDSA
 
using SHA
 
import Base.in, Base.==, Base.+, Base.*
 
export ECDSA_Key, ECDSA_Public_Key, genkey, ECDSA_sign, isverifiedECDSA
 
# T will be BigInt in most applications
struct CurveFP{T}
p::T
a::T
b::T
CurveFP(p, a::T, b::T) where T <: Number = new{T}(p, a, b)
end
 
struct PointEC{T}
curve::CurveFP{T}
x::T
y::T
order::Union{Number, Nothing}
function PointEC(curve, x::T, y::T, order=nothing) where T <: Number
@assert((x, y) in curve)
new{T}(curve, x, y, order)
end
end
 
struct PointINF end
const INFINITY = PointINF()
 
function ==(point_a::PointEC, point_b::PointEC)
if point_a.curve == point_b.curve && point_a.x == point_b.x && point_a.y == point_b.y
return true
end
return false
end
 
function ==(curve_a::CurveFP, curve_b::CurveFP)
if curve_a.a == curve_b.a && curve_a.b == curve_b.b && curve_a.p == curve_b.p
return true
end
return false
end
 
+(point_a::PointINF, point_b::PointINF) = point_b
+(point_a::PointINF, point_b::PointEC) = point_b
+(point_a::PointEC, point_b::PointINF) = point_a
 
function +(point_a::PointEC, point_b::PointEC)
@assert(point_a.curve == point_b.curve)
if point_a.x == point_b.x
if (point_a.y + point_b.y) % point_a.curve.p == 0
return INFINITY
else
return double(point_a)
end
end
p = point_a.curve.p
λ = (point_a.y - point_b.y) * invmod(point_a.x - point_b.x, p)
xr = mod(λ * λ - point_a.x - point_b.x, p)
yr = mod(λ * (point_a.x - xr) - point_a.y, p)
return PointEC(point_a.curve, xr, yr, point_a.order)
end
 
*(point_a::PointINF, int_b::Number) = INFINITY
*(int_b::Number, point_a::PointINF) = INFINITY
*(int_b::Number, point_a::PointEC) = point_a * int_b
 
function *(point_a::PointEC, int_b::Number)
leftmost_bit(x) = big"2"^Int(trunc(log(2, x)))
if point_a.order != nothing
int_b %= point_a.order
end
if int_b == 0
return INFINITY
end
int_3b = 3 * int_b
negative_a = PointEC(point_a.curve, point_a.x, -point_a.y, point_a.order)
i = BigInt(leftmost_bit(int_3b) ÷ 2)
result = point_a
while i > 1
result = double(result)
if (int_3b & i) != 0 && (int_b & i) == 0
result += point_a
end
if (int_3b & i) == 0 && (int_b & i) != 0
result += negative_a
end
i ÷= 2
end
return result
end
 
in(z::Tuple, curve::CurveFP) = (z[2]^2 - (z[1]^3 + curve.a*z[1] + curve.b)) % curve.p == 0
in(x::Number,y::Number, curve::CurveFP) = (y^2 -(x^3 + curve.a*x + curve.b)) % curve.p == 0
in(p::PointEC, curve::CurveFP) = (p.y^2 - (p.x^3 + curve.a * p.x + curve.b)) % curve.p == 0
in(point::PointINF, curve::CurveFP) = true
 
double(point_a::PointINF) = INFINITY
 
function double(point_a::PointEC)
a, p = point_a.curve.a, point_a.curve.p
l = mod((3 * point_a.x^2 + a) * invmod(2 * point_a.y, p), p)
x3 = mod(l^2 - 2 * point_a.x, p)
y3 = mod(l * (point_a.x - x3) - point_a.y, p)
return PointEC(point_a.curve, x3, y3, point_a.order)
end
 
const secp256k1 = ( # use the Bitcoin ECDSA curve
p = big"0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F",
a = big"0x0",
b = big"0x7",
r = big"0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141",
Gx = big"0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798",
Gy = big"0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8",
)
@assert((secp256k1.Gy^2 - secp256k1.Gx^3 - 7) % secp256k1.p == 0)
 
function generatemultiplier(stdcurve)
return foldl((x, y) -> 16 * BigInt(x) + y, rand(0:16, ndigits(stdcurve.r - 1, base=16)))
end
 
struct ECDSA_Key
E::CurveFP
secret::BigInt
G::PointEC
r::BigInt
W::PointEC
end
 
struct ECDSA_Public_Key
E::CurveFP
G::PointEC
r::BigInt
W::PointEC
end
 
function genkey(curve=secp256k1) # default: use Bitcoin standard EC curve secp256k1
E = CurveFP(curve.p, curve.a, curve.b)
s = generatemultiplier(curve)
G = PointEC(E, curve.Gx, curve.Gy, curve.r)
W = s * G
return ECDSA_Key(E, s, G, curve.r, W)
end
 
aspublickey(k::ECDSA_Key) = ECDSA_Public_Key(k.E, k.G, k.r, k.W)
privatekey(k::ECDSA_Key) = k.secret
 
function ECDSA_sign(m::String, key::ECDSA_Key, digestfunction=sha256)
r, f = key.r, digestfunction(codeunits(m)) # f = H(m)
# order of curve points length must be >= sha digest length (in bytes)
@assert(ndigits(r, base=16) >= length(f))
c, d, bindigest = BigInt(0), BigInt(0), foldl((x, y) -> 16 * BigInt(x) + y, f)
while c == 0 || d == 0
u = generatemultiplier(secp256k1)
V = u * key.G
c = mod(V.x, r)
d = mod((invmod(u, r) * (bindigest + key.secret * c)), r)
end
return aspublickey(key), c, d
end
 
function isverifiedECDSA(m::String, publickey, c, d, digestfunction=sha256)
if 1 <= c < publickey.r && 1 <= d < publickey.r
r, f = publickey.r, digestfunction(codeunits(m))
h, bindigest = invmod(d, r), foldl((x, y) -> 16 * BigInt(x) + y, f)
h1, h2 = mod(bindigest * h, r), mod(c * h, r)
verifierpoint = h1 * publickey.G + h2 * publickey.W
return mod(verifierpoint.x, r) == c
end
return false
end
 
end # module
 
using .ToyECDSA
 
const key = genkey()
const msg = "Bill says this is an elliptic curve digital signature algorithm."
const altered = "Bill says this isn't an elliptic curve digital signature algorithm."
 
publickey, c, d = ECDSA_sign(msg, key)
 
println("ECDSA of message <$msg> verified: ", isverifiedECDSA(msg, publickey, c, d))
 
println("ECDSA of message <$altered> verified: ", isverifiedECDSA(altered, publickey, c, d))
 
Output:
ECDSA of message <Bill says this is an elliptic curve digital signature algorithm.> verified: true
ECDSA of message <Bill says this isn't an elliptic curve digital signature algorithm.> verified: false

Nim[edit]

Translation of: C
import math, random, strformat
 
const
MaxN = 1073741789 # Maximum modulus.
MaxR = MaxN + 65536 # Maximum order "g".
Infinity = int64.high # Symbolic infinity.
 
type
 
Point = tuple[x, y: int64]
 
Curve = object
a, b: int64
n: int64
g: Point
r: int64
 
Pair = tuple[a, b: int64]
 
Parameters = tuple[a, b, n, gx, gy, r: int]
 
const ZerO: Point = (Infinity, 0i64)
 
type
InversionError = object of ValueError
InvalidParamError = object of ValueError
 
 
template `%`(a, n: int64): int64 =
## To simplify the writing.
floorMod(a, n)
 
 
proc exgcd(v, u: int64): int64 =
## Return 1/v mod u.
var u = u
var v = v
if v < 0: v += u
 
var r = 0i64
var s = 1i64
while v != 0:
let q = u div v
u = u mod v
swap u, v
r -= q * s
swap r, s
 
if u != 1:
raise newException(InversionError, "impossible inverse mod N, gcd = " & $u)
result = r
 
 
func discr(e: Curve): int64 =
## Return the discriminant of "e".
let c = e.a * e.a % e.n * e.a % e.n * 4
result = -16 * (e.b * e.b % e.n * 27 + c) % e.n
 
 
func isO(p: Point): bool =
## Return true if "p" is zero.
p.x == Infinity and p.y == 0
 
 
func isOn(e: Curve; p: Point): bool =
## Return true if "p" is on curve "e".
if p.isO: return true
let r = ((e.a + p.x * p.x) % e.n * p.x + e.b) % e.n
let s = p.y * p.y % e.n
result = r == s
 
 
proc add(e: Curve; p, q: Point): Point =
## Full Point addition.
 
if p.isO: return q
if q.isO: return p
 
var la: int64
if p.x != q.x:
la = (p.y - q.y) * exgcd(p.x - q.x, e.n) % e.n
elif p.y == q.y and p.y != 0:
la = (p.x * p.x % e.n * 3 + e.a) % e.n * exgcd(2 * p.y, e.n) % e.n
else:
return ZerO
 
result.x = (la * la - p.x - q.x) % e.n
result.y = (la * (p.x - result.x) - p.y) % e.n
 
 
proc mul(e: Curve; p: Point; k: int64): Point =
## Return "kp".
var q = p
var k = k
result = ZerO
 
while k != 0:
if (k and 1) != 0:
result = e.add(result, q)
q = e.add(q, q)
k = k shr 1
 
 
proc print(e: Curve; prefix: string; p: Point) =
## Print a point with a prefix.
var y = p.y
if p.isO:
echo prefix, " (0)"
else:
if y > e.n - y: y -= e.n
echo prefix, &" ({p.x}, {y})"
 
 
proc initCurve(params: Parameters): Curve =
## Initialize the curve.
 
result.n = params.n
if result.n notin 5..MaxN:
raise newException(ValueError, "invalid value for N: " & $result.n)
result.a = params.a.int64 % result.n
result.b = params.b.int64 % result.n
result.g.x = params.gx.int64 % result.n
result.g.y = params.gy.int64 % result.n
result.r = params.r
 
if result.r notin 5..MaxR:
raise newException(ValueError, "invalid value for r: " & $result.r)
 
echo &"\nE: y^2 = x^3 + {result.a}x + {result.b} (mod {result.n})"
result.print("base point G", result.g)
echo &"order(G, E) = {result.r}"
 
 
proc rnd(): float =
## Return a pseudorandom number in range [0..1[.
while true:
result = rand(1.0)
if result != 1: break
 
proc signature(e: Curve; s, f: int64): Pair =
## Compute the signature.
 
var
c, d = 0i64
u: int64
v: Point
 
echo "Signature computation"
 
while true:
while true:
u = 1 + int64(rnd() * float(e.r - 1))
v = e.mul(e.g, u)
c = v.x % e.r
if c != 0: break
d = exgcd(u, e.r) * (f + s * c % e.r) % e.r
if d != 0: break
 
echo "one-time u = ", u
e.print("V = uG", v)
 
result = (c, d)
 
 
proc verify(e: Curve; w: Point; f: int64; sg: Pair): bool =
## Verify a signature.
 
# Domain check.
if sg.a notin 1..<e.r or sg.b notin 1..<e.r:
return false
 
echo "\nsignature verification"
let h = exgcd(sg.b, e.r)
let h1 = f * h % e.r
let h2 = sg.a * h % e.r
echo &"h1, h2 = {h1}, {h2}"
var v = e.mul(e.g, h1)
let v2 = e.mul(w, h2)
e.print("h1G", v)
e.print("h2W", v2)
v = e.add(v, v2)
e.print("+ =", v)
if v.isO: return false
let c1 = v.x % e.r
echo "c’ = ", c1
result = c1 == sg.a
 
 
proc ecdsa(e: Curve; f: int64; d: int) =
## Build digital signature for message hash "f" with error bit "d".
 
# Parameter check.
var w = e.mul(e.g, e.r)
if e.discr() == 0 or e.g.isO or not w.isO or not e.isOn(e.g):
raise newException(InvalidParamError, "invalid parameter set")
 
echo "\nkey generation"
let s = 1 + int64(rnd() * float(e.r - 1))
w = e.mul(e.g, s)
echo "private key s = ", s
e.print("public key W = sG", w)
 
# Find next highest power of two minus one.
var t = e.r
var i = 1
while i < 64:
t = t or t shr i
i = i shl 1
var f = f
while f > t: f = f shr 1
echo &"\naligned hash {f:x}"
 
let sg = e.signature(s, f)
echo &"signature c, d = {sg.a}, {sg.b}"
 
var d = d
if d > 0:
while d > t: d = d shr 1
f = f xor d
echo &"\ncorrupted hash {f:x}"
 
echo if e.verify(w, f, sg): "Valid" else: "Invalid"
 
when isMainModule:
 
randomize()
 
# Test vectors: elliptic curve domain parameters,
# short Weierstrass model y^2 = x^3 + ax + b (mod N)
 
const Sets = [
# a, b, modulus N, base point G, order(G, E), cofactor
(355, 671, 1073741789, 13693, 10088, 1073807281),
( 0, 7, 67096021, 6580, 779, 16769911),
( -3, 1, 877073, 0, 1, 878159),
( 0, 14, 22651, 63, 30, 151),
( 3, 2, 5, 2, 1, 5),
 
# ECDSA may fail if...
# the base point is of composite order
( 0, 7, 67096021, 2402, 6067, 33539822),
# the given order is of composite order
( 0, 7, 67096021, 6580, 779, 67079644),
# the modulus is not prime (deceptive example)
( 0, 7, 877069, 3, 97123, 877069),
# fails if the modulus divides the discriminant
( 39, 387, 22651, 95, 27, 22651)
]
 
# Digital signature on message hash f,
# set d > 0 to simulate corrupted data.
let f = 0x789abcdei64
let d = 0
 
for s in Sets:
let e = initCurve(s)
try:
e.ecdsa(f, d)
except ValueError:
echo getCurrentExceptionMsg()
echo "——————————————"
Output:

First set only.

E: y^2 = x^3 + 355x + 671 (mod 1073741789)
base point G (13693, 10088)
order(G, E) = 1073807281

key generation
private key s = 136992620
public key W = sG (1015940633, 345577760)

aligned hash 789abcde
Signature computation
one-time u = 183069978
V = uG (565985545, 303688609)
signature c, d = 565985545, 1060545485

signature verification
h1, h2 = 668000030, 564517193
h1G (801378704, -375265150)
h2W (77432102, 301215724)
+ = (565985545, 303688609)
c’ = 565985545
Valid
——————————————

</pre>

Perl[edit]

Translation of: Go
# 20200828 added Perl programming solution
 
use strict;
use warnings;
 
use Crypt::EC_DSA;
 
my $ecdsa = new Crypt::EC_DSA;
 
my ($pubkey, $prikey) = $ecdsa->keygen;
 
print "Message: ", my $msg = 'Rosetta Code', "\n";
 
print "Private Key :\n$prikey \n";
print "Public key  :\n", $pubkey->x, "\n", $pubkey->y, "\n";
 
my $signature = $ecdsa->sign( Message => $msg, Key => $prikey );
print "Signature  :\n";
for (sort keys %$signature) { print "$_ => $signature->{$_}\n"; }
 
$ecdsa->verify( Message => $msg, Key => $pubkey, Signature => $signature ) and
print "Signature verified.\n"
Output:
Message: Rosetta Code
Private Key :
50896950174101144529764022934807089163030534967278433074982207331912857967110
Public key  :
98639220601877298644829563208621497413822596683110596662237522364057856411416
69976521993624103693270429074404825634551369215777879408776019358694823367135
Signature   :
r => 113328268998856048369024784426817827689451364968174533291969247274701793929451
s => 102496515866716695113707075780391660331998173218829535655149484671019624453603
Signature verified.

Phix[edit]

Translation of: FreeBASIC
requires(64)
enum X, Y            -- rational ec point
enum A, B, N, G, R  -- elliptic curve parameters
                    -- also signature pair(A,B)
 
constant mxN = 1073741789   -- maximum modulus
constant mxr = 1073807325   -- max order G = mxN + 65536
constant inf = -2147483647  -- symbolic infinity
 
sequence e = {0,0,0,{0,0},0}    -- single global curve
constant zerO = {inf,0}         -- point at infinity zerO
 
bool inverr -- impossible inverse mod N
 
function exgcd(atom v, u)
-- return mod(v^-1, u)
atom q, t, r = 0, s = 1
    if v<0 then v += u end if
 
    while v do
        q = floor(u/v)
        t = u-q*v
        u = v
        v = t
        t = r-q*s
        r = s
        s = t
    end while
 
    if u!=1 then
        printf(1," impossible inverse mod N, gcd = %d\n",{u})
        inverr = true
    end if
 
    return r
end function
 
function modn(atom a)
-- return mod(a, N)
    a = mod(a,e[N])
    if a<0 then a += e[N] end if
    return a
end function
 
function modr(atom a)
-- return mod(a, r)
    a = mod(a,e[R])
    if a<0 then a += e[R] end if
    return a
end function
 
function disc()
-- return the discriminant of E
    atom a = e[A], b = e[B],
         c = 4*modn(a*modn(a*a))
    return modn(-16*(c+27*modn(b*b)))
end function
 
function isO(sequence p)
-- return true if P = zerO
    return (p[X]=inf and p[Y]=0)
end function
 
function ison(sequence p)
-- return true if P is on curve E
    atom r = 0, s = 0
    if not isO(p) then
        r = modn(e[B]+p[X]*modn(e[A]+p[X]*p[X]))
        s = modn(p[Y]*p[Y])
    end if
    return (r=s)
end function
 
procedure pprint(string f, sequence p)
-- print point P with prefix f
    if isO(p) then
        printf(1,"%s (0)\n",{f})
    else
        atom y = p[Y]
        if y>e[N]-y then y -= e[N] end if
        printf(1,"%s (%d,%d)\n",{f,p[X],y})
    end if
end procedure
 
function padd(sequence p, q)
-- full ec point addition
atom la, t
 
    if isO(p) then return q end if
    if isO(q) then return p end if
 
    if p[X]!=q[X] then --                   R := P + Q
        t = p[Y]-q[Y]
        la = modn(t*exgcd(p[X]-q[X], e[N]))
 
    else --                                 P = Q, R := 2P
        if (p[Y]=q[Y]) and (p[Y]!=0) then
            t = modn(3*modn(p[X]*p[X])+e[A])
            la = modn(t*exgcd(2*p[Y], e[N]))
        else
            return zerO --                  P = -Q, R := O
        end if
    end if
 
    t = modn(la*la-p[X]-q[X])
    sequence r = deep_copy(zerO)
    r[Y] = modn(la*(p[X]-t)-p[Y])
    r[X] = t
    if inverr then r = zerO end if
    return r
end function
 
function pmul(sequence p, atom k)
-- R:= multiple kP
    sequence s = zerO, q = p
 
    while k do
        if and_bits(k,1) then
            s = padd(s, q)
        end if
        if inverr then s = zerO; exit end if
        k = floor(k/2)
        q = padd(q, q)
    end while
    return s
end function
 
function ellinit(sequence i)
-- initialize elliptic curve
atom a = i[1], b = i[2]
    inverr = false
    e[N] = i[3]
 
    if (e[N]<5) or (e[N]>mxN) then return 0 end if
 
    e[A] = modn(a)
    e[B] = modn(b)
    e[G][X] = modn(i[4])
    e[G][Y] = modn(i[5])
    e[R] = i[6]
 
    if (e[R]<5) or (e[R]>mxr) then return 0 end if
 
    printf(1,"E: y^2 = x^3 + %dx + %d (mod %d)\n",{a,b,e[N]})
    pprint("base point G", e[G])
    printf(1,"order(G, E) = %d\n",{e[R]})
 
    return -1
end function
 
function signature(atom s, f)
-- signature primitive
atom c, d, u, u1
sequence V
 
    printf(1,"signature computation\n")
    while true do
        while true do
--          u = rand(e[R]-1)
            u = 571533488       -- match FreeBASIC output
--          u = 605163545       -- match C output
            V = pmul(e[G], u)
            c = modr(V[X])
            if c!=0 then exit end if
        end while
 
        u1 = exgcd(u, e[R])
        d = modr(u1*(f+modr(s*c)))
        if d!=0 then exit end if
    end while
    printf(1,"one-time u = %d\n",u)
    pprint("V = uG", V)
    return {c,d}
end function
 
function verify(sequence W, atom f, sequence sg)
-- verification primitive
atom c = sg[A], d = sg[B],
     t, c1, h1, h2, h
sequence V, V2
 
   --domain check
    t = (c>0) and (c<e[R])
    t = t and (d>0) and (d<e[R])
    if not t then return 0 end if
 
    printf(1,"\nsignature verification\n")
    h = exgcd(d, e[R])
    h1 = modr(f*h)
    h2 = modr(c*h)
    printf(1,"h1,h2 = %d,%d\n",{h1,h2})
    V = pmul(e[G], h1)
    V2 = pmul(W, h2)
    pprint("h1G", V)
    pprint("h2W", V2)
    V = padd(V, V2)
    pprint("+ =", V)
    if isO(V) then return 0 end if
    c1 = modr(V[X])
    printf(1,"c' = %d\n",c1)
 
    return (c1=c)
end function
 
procedure errmsg()
    printf(1,"invalid parameter set\n")
    printf(1,"_____________________\n")
end procedure
 
procedure ec_dsa(atom f, d)
-- digital signature on message hash f, error bit d
atom i, s, t
sequence sg, W
 
   --parameter check
    t = (disc()=0)
    t = t or isO(e[G])
    W = pmul(e[G], e[R])
    t = t or not isO(W)
    t = t or not ison(e[G])
    if t then errmsg() return end if
 
    puts(1,"\nkey generation\n")
--  s = rand(e[R] - 1)
    s = 509100772       -- match FreeBASIC output
--  s = 1343570         -- match C output
    W = pmul(e[G], s)
    printf(1,"private key s = %d\n",{s})
    pprint("public key W = sG", W)
 
   --next highest power of 2 - 1
    t = e[R]
    i = 1
    while i<32 do
        t = or_bits(t,floor(t/power(2,i)))
        i *= 2
    end while
    while f>t do
        f = floor(f/2)
    end while
    printf(1,"\naligned hash %x\n\n",{f})
 
    sg = signature(s, f)
    if inverr then errmsg() return end if
    printf(1,"signature c,d = %d,%d\n",sg)
 
    if d>0 then
        while d>t do
            d = floor(d/2)
        end while
        f = xor_bits(f,d)
        printf(1,"corrupted hash %x\n",{f})
    end if
 
    t = verify(W, f, sg)
    if inverr then errmsg() return end if
    if t then
        printf(1,"Valid\n_____\n")
    else
        printf(1,"invalid\n_______\n")
    end if
end procedure
 
--Test vectors: elliptic curve domain parameters,
--short Weierstrass model y^2 = x^3 + ax + b (mod N)
 
constant tests = {
--                  a,   b,  modulus N, base point G, order(G, E), cofactor
                  {355, 671, 1073741789, 13693, 10088, 1073807281},
                  {  0,   7,   67096021,  6580,   779,   16769911}, --   4
                  { -3,   1,     877073,     0,     1,     878159},
                  {  0,  14,      22651,    63,    30,        151}, -- 151
                  {  3,   2,          5,     2,     1,          5},
 
                    --ecdsa may fail if...
                    --the base point is of composite order
                  {  0,   7,   67096021,  2402,  6067,   33539822}, --   2
                    --the given order is a multiple of the true order
                  {  0,   7,   67096021,  6580,   779,   67079644}, --   1
                    --the modulus is not prime (deceptive example)
                  {  0,   7,     877069,     3, 97123,     877069},
                    --fails if the modulus divides the discriminant
                  { 39, 387,      22651,    95,    27,      22651}}
 
--Digital signature on message hash f,
--set d > 0 to simulate corrupted data
atom f = #789ABCDE,
     d = 0
 
--for i=1 to length(tests) do
for i=1 to 1 do
    if not ellinit(tests[i]) then exit end if
    ec_dsa(f, d)
end for
Output:

Note the above only performs tests[1], and assigns literal values in place of rand(), in order to exactly match the FreeBASIC/C output.

E: y^2 = x^3 + 355x + 671 (mod 1073741789)
base point G (13693,10088)
order(G, E) = 1073807281

key generation
private key s = 509100772
public key W = sG (992563138,238074938)

aligned hash 789ABCDE

signature computation
one-time u = 571533488
V = uG (896670665,183547995)
signature c,d = 896670665,728505276

signature verification
h1,h2 = 667118700,709185150
h1G (315367421,343743703)
h2W (1040319975,-262613483)
+ = (896670665,183547995)
c' = 896670665
Valid
_____

Python[edit]

 
from collections import namedtuple
from hashlib import sha256
from math import ceil, log
from random import randint
from typing import NamedTuple
 
# Bitcoin ECDSA curve
secp256k1_data = dict(
p=0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F, # Field characteristic
a=0x0, # Curve param a
b=0x7, # Curve param b
r=0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141, # Order n of basepoint G. Cofactor is 1 so it's ommited.
Gx=0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798, # Base point x
Gy=0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8, # Base point y
)
secp256k1 = namedtuple("secp256k1", secp256k1_data)(**secp256k1_data)
assert (secp256k1.Gy ** 2 - secp256k1.Gx ** 3 - 7) % secp256k1.p == 0
 
 
class CurveFP(NamedTuple):
p: int # Field characteristic
a: int # Curve param a
b: int # Curve param b
 
 
def extended_gcd(aa, bb):
# https://rosettacode.org/wiki/Modular_inverse#Iteration_and_error-handling
lastremainder, remainder = abs(aa), abs(bb)
x, lastx, y, lasty = 0, 1, 1, 0
while remainder:
lastremainder, (quotient, remainder) = remainder, divmod(
lastremainder, remainder
)
x, lastx = lastx - quotient * x, x
y, lasty = lasty - quotient * y, y
return lastremainder, lastx * (-1 if aa < 0 else 1), lasty * (-1 if bb < 0 else 1)
 
 
def modinv(a, m):
# https://rosettacode.org/wiki/Modular_inverse#Iteration_and_error-handling
g, x, _ = extended_gcd(a, m)
if g != 1:
raise ValueError
return x % m
 
 
class PointEC(NamedTuple):
curve: CurveFP
x: int
y: int
 
@classmethod
def build(cls, curve, x, y):
x = x % curve.p
y = y % curve.p
rv = cls(curve, x, y)
if not rv.is_identity():
assert rv.in_curve()
return rv
 
def get_identity(self):
return PointEC.build(self.curve, 0, 0)
 
def copy(self):
return PointEC.build(self.curve, self.x, self.y)
 
def __neg__(self):
return PointEC.build(self.curve, self.x, -self.y)
 
def __sub__(self, Q):
return self + (-Q)
 
def __equals__(self, Q):
# TODO: Assert same curve or implement logic for that.
return self.x == Q.x and self.y == Q.y
 
def is_identity(self):
return self.x == 0 and self.y == 0
 
def __add__(self, Q):
# TODO: Assert same curve or implement logic for that.
p = self.curve.p
if self.is_identity():
return Q.copy()
if Q.is_identity():
return self.copy()
if Q.x == self.x and (Q.y == (-self.y % p)):
return self.get_identity()
 
if self != Q:
l = ((Q.y - self.y) * modinv(Q.x - self.x, p)) % p
else:
# Point doubling.
l = ((3 * self.x ** 2 + self.curve.a) * modinv(2 * self.y, p)) % p
l = int(l)
 
Rx = (l ** 2 - self.x - Q.x) % p
Ry = (l * (self.x - Rx) - self.y) % p
rv = PointEC.build(self.curve, Rx, Ry)
return rv
 
def in_curve(self):
return ((self.y ** 2) % self.curve.p) == (
(self.x ** 3 + self.curve.a * self.x + self.curve.b) % self.curve.p
)
 
def __mul__(self, s):
# Naive method is exponential (due to invmod right?) so we use an alternative method:
# https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Montgomery_ladder
r0 = self.get_identity()
r1 = self.copy()
# pdbsas
for i in range(ceil(log(s + 1, 2)) - 1, -1, -1):
if ((s & (1 << i)) >> i) == 0:
r1 = r0 + r1
r0 = r0 + r0
else:
r0 = r0 + r1
r1 = r1 + r1
return r0
 
def __rmul__(self, other):
return self.__mul__(other)
 
 
class ECCSetup(NamedTuple):
E: CurveFP
G: PointEC
r: int
 
 
secp256k1_curve = CurveFP(secp256k1.p, secp256k1.a, secp256k1.b)
secp256k1_basepoint = PointEC(secp256k1_curve, secp256k1.Gx, secp256k1.Gy)
 
 
class ECDSAPrivKey(NamedTuple):
ecc_setup: ECCSetup
secret: int
 
def get_pubkey(self):
# Compute W = sG to get the pubkey
W = self.secret * self.ecc_setup.G
pub = ECDSAPubKey(self.ecc_setup, W)
return pub
 
 
class ECDSAPubKey(NamedTuple):
ecc_setup: ECCSetup
W: PointEC
 
 
class ECDSASignature(NamedTuple):
c: int
d: int
 
 
def generate_keypair(ecc_setup, s=None):
# Select a random integer s in the interval [1, r - 1] for the secret.
if s is None:
s = randint(1, ecc_setup.r - 1)
priv = ECDSAPrivKey(ecc_setup, s)
pub = priv.get_pubkey()
return priv, pub
 
 
def get_msg_hash(msg):
return int.from_bytes(sha256(msg).digest(), "big")
 
 
def sign(priv, msg, u=None):
G = priv.ecc_setup.G
r = priv.ecc_setup.r
 
# 1. Compute message representative f = H(m), using a cryptographic hash function.
# Note that f can be greater than r but not longer (measuring bits).
msg_hash = get_msg_hash(msg)
 
while True:
# 2. Select a random integer u in the interval [1, r - 1].
if u is None:
u = randint(1, r - 1)
 
# 3. Compute V = uG = (xV, yV) and c ≡ xV mod r (goto (2) if c = 0).
V = u * G
c = V.x % r
if c == 0:
print(f"c={c}")
continue
d = (modinv(u, r) * (msg_hash + priv.secret * c)) % r
if d == 0:
print(f"d={d}")
continue
break
 
signature = ECDSASignature(c, d)
return signature
 
 
def verify_signature(pub, msg, signature):
r = pub.ecc_setup.r
G = pub.ecc_setup.G
c = signature.c
d = signature.d
 
# Verify that c and d are integers in the interval [1, r - 1].
def num_ok(n):
return 1 < n < (r - 1)
 
if not num_ok(c):
raise ValueError(f"Invalid signature value: c={c}")
if not num_ok(d):
raise ValueError(f"Invalid signature value: d={d}")
 
# Compute f = H(m) and h ≡ d^-1 mod r.
msg_hash = get_msg_hash(msg)
h = modinv(d, r)
 
# Compute h1 ≡ f·h mod r and h2 ≡ c·h mod r.
h1 = (msg_hash * h) % r
h2 = (c * h) % r
 
# Compute h1G + h2W = (x1, y1) and c1 ≡ x1 mod r.
# Accept the signature if and only if c1 = c.
P = h1 * G + h2 * pub.W
c1 = P.x % r
rv = c1 == c
return rv
 
 
def get_ecc_setup(curve=None, basepoint=None, r=None):
if curve is None:
curve = secp256k1_curve
if basepoint is None:
basepoint = secp256k1_basepoint
if r is None:
r = secp256k1.r
 
# 1. Select an elliptic curve E defined over ℤp.
# The number of points in E(ℤp) should be divisible by a large prime r.
E = CurveFP(curve.p, curve.a, curve.b)
 
# 2. Select a base point G ∈ E(ℤp) of order r (which means that rG = 𝒪).
G = PointEC(E, basepoint.x, basepoint.y)
assert (G * r) == G.get_identity()
 
ecc_setup = ECCSetup(E, G, r)
return ecc_setup
 
 
def main():
ecc_setup = get_ecc_setup()
print(f"E: y^2 = x^3 + {ecc_setup.E.a}x + {ecc_setup.E.b} (mod {ecc_setup.E.p})")
print(f"base point G({ecc_setup.G.x}, {ecc_setup.G.y})")
print(f"order(G, E) = {ecc_setup.r}")
 
print("Generating keys")
priv, pub = generate_keypair(ecc_setup)
print(f"private key s = {priv.secret}")
print(f"public key W = sG ({pub.W.x}, {pub.W.y})")
 
msg_orig = b"hello world"
signature = sign(priv, msg_orig)
print(f"signature ({msg_orig}, priv) = (c,d) = {signature.c}, {signature.d}")
 
validation = verify_signature(pub, msg_orig, signature)
print(f"verify_signature(pub, {msg_orig}, signature) = {validation}")
 
msg_bad = b"hello planet"
validation = verify_signature(pub, msg_bad, signature)
print(f"verify_signature(pub, {msg_bad}, signature) = {validation}")
 
 
if __name__ == "__main__":
main()
 
Output:
E: y^2 = x^3 + 0x + 7 (mod 115792089237316195423570985008687907853269984665640564039457584007908834671663)
base point G(55066263022277343669578718895168534326250603453777594175500187360389116729240, 32670510020758816978083085130507043184471273380659243275938904335757337482424)
order(G, E) = 115792089237316195423570985008687907852837564279074904382605163141518161494337
Generating keys
private key s = 82303800204859706726056108314364152573031639016623313275752312395463491677949
public key W = sG (114124711379379930034967744084997669072230999039555829167372300365264253950871, 3360309271473421344413510933284750262871091919289744186713753032606174460281)
signature (b'hello world', priv) = (c,d) = 1863861291106464538398514909960950901792292936913306559193916523058660671107, 62559673485398527884210590428202308573799357197699075411868464552455123994884
verify_signature(pub, b'hello world', signature) = True
verify_signature(pub, b'hello planet', signature) = False

Raku[edit]

(formerly Perl 6)

Reference: Many routines are translated from this Ruby repository, by Stephen Blackstone. The rest are taken here and there from RC.

use Digest::SHA256::Native;
 
# Following data taken from the C entry
our (\A,\B,\P,\O,\Gx,\Gy) = (355, 671, 1073741789, 1073807281, 13693, 10088);
 
#`{ Following data taken from the Julia entry; 256-bit; tested
our (\A,\B,\P,\O,\Gx,\Gy) = (0, 7, # https://en.bitcoin.it/wiki/Secp256k1
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F,
0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141,
0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8); # }
 
role Horizon { method gist { 'EC Point at horizon' } }
 
class Point { # modified from the Elliptic_curve_arithmetic entry
has ($.x, $.y); # handle modular arithmetic only
multi method new( \x, \y ) { self.bless(:x, :y) }
method gist { "EC Point at x=$.x, y=$.y" }
method isOn { modP(B + $.x * modP(A+$.x²)) == modP($.y²) }
sub modP ($a is copy) { ( $a %= P ) < 0 ?? ($a += P) !! $a }
}
 
multi infix:<>(Point \p, Point \q) {
my= $; # slope
if p.x ~~ q.x and p.y ~~ q.y {
return Horizon if p.y == 0 ;
λ = (3*p.x²+ A) * mult_inv(2*p.y, :modulo(P))
} else {
λ = (p.y - q.y) * mult_inv(p.x - q.x, :modulo(P))
}
my \xr = (λ²- p.x - q.x);
my \yr = (λ*(p.x - xr) - p.y);
return Point.bless: x => xr % P, y => yr % P
}
 
multi infix:<>(Int \n, Point \p) {
return 0 if n == 0 ;
return p if n == 1 ;
return p ⊞ ((n-1) ⊠ p ) if n % 2 == 1 ;
return ( n div 2 )( p ⊞ p )
}
 
sub mult_inv($n, :$modulo) { # rosettacode.org/wiki/Modular_inverse#Raku
my ($c, $d, $uc, $vd, $vc, $ud, $q) = $n % $modulo, $modulo, 1, 1, 0, 0, 0;
while $c != 0 {
($q, $c, $d) = ($d div $c, $d % $c, $c);
($uc, $vc, $ud, $vd) = ($ud - $q*$uc, $vd - $q*$vc, $uc, $vc);
}
return $ud % $modulo;
}
 
class Signature {
 
has ($.n, Point $.G); # Order and Generator point
 
method generate_signature(Int \private_key, Str \msg) {
my \z = :16(sha256-hex msg) % $.n; # self ref: Blob.list.fmt("%02X",'')
loop ( my $k = my $s = my $r = 0 ; $s == 0 ; ) {
loop ( $r = $s = 0 ; $r == 0 ; ) {
$r = (( $k = (1..^$.n).roll )$.G).x % $.n;
}
$s = ((z + $r*private_key) * mult_inv $k, :modulo($.n)) % $.n;
}
return $r, $s, private_key ⊠ $.G ;
}
 
method verify_signature(\msg, \r, \s, \public_key) {
my \z = :16(sha256-hex msg) % $.n;
my \w = mult_inv s, :modulo($.n);
my (\u1,\u2) = (z*w, r*w).map: { $_ % $.n }
my \p = (u1 ⊠ $.G )(u2 ⊠ public_key);
return (p.x % $.n) == (r % $.n)
}
}
 
print "The Curve E is  : ";
"𝑦² = 𝑥³ + %s 𝑥 + %s (mod %s) \n".printf(A,B,P);
"with Generator G at  : (%s,%s)\n".printf(Gx,Gy);
my $ec = Signature.new: n => O, G => Point.new: x => Gx, y => Gy ;
say "Order(G, E) is  : ", O;
say "Is G ∈ E ?  : ", $ec.G.isOn;
say "Message  : ", my \message = "Show me the monKey";
say "The private key dA is : ", my \dA = (1..^O).roll;
my ($r, $s, \Qa) = $ec.generate_signature(dA, message);
say "The public key Qa is : ", Qa;
say "Is Qa ∈ E ?  : ", Qa.isOn;
say "Is signature valid?  : ", $ec.verify_signature(message, $r, $s, Qa);
say "Message (Tampered)  : ", my \altered = "Show me the money";
say "Is signature valid?  : ", $ec.verify_signature(altered, $r, $s, Qa)
Output:
The Curve E is        : 𝑦² = 𝑥³ + 355 𝑥 + 671 (mod 1073741789)
with Generator G at   : (13693,10088)
Order(G, E) is        : 1073807281
Is G  ∈ E ?           : True
Message               : Show me the monKey
The private key dA is : 384652035
The public  key Qa is : EC Point at x=919494857, y=18030536
Is Qa ∈ E ?           : True
Is signature valid?   : True
Message (Tampered)    : Show me the money
Is signature valid?   : False

Wren[edit]

Translation of: C
Library: Wren-dynamic
Library: Wren-big
Library: Wren-fmt
Library: Wren-math

As we don't have a signed 64 bit integer type, we use BigInt instead where needed.

import "/dynamic" for Struct
import "/big" for BigInt
import "/fmt" for Fmt
import "/math" for Boolean
import "random" for Random
 
var rand = Random.new()
 
// rational ec point: x and y are BigInts
var Epnt = Struct.create("Epnt", ["x", "y"])
 
// elliptic curve parameters: N is a BigInt, G is an Epnt, rest are integral Nums
var Curve = Struct.create("Curve", ["a", "b", "N", "G", "r"])
 
// signature pair: a and b are integral Nums
var Pair = Struct.create("Pair", ["a", "b"])
 
// maximum modulus
var mxN = 1073741789
 
// max order G = mxN + 65536
var mxr = 1073807325
 
// symbolic infinity
var inf = BigInt.new(-2147483647)
 
// single global curve
var e = Curve.new(0, 0, BigInt.zero, Epnt.new(inf, BigInt.zero), 0)
 
// impossible inverse mod N
var inverr = false
 
// return mod(v^-1, u)
var exgcd = Fn.new { |v, u|
var r = 0
var s = 1
if (v < 0) v = v + u
while (v != 0) {
var q = (u / v).truncate
var t = u - q * v
u = v
v = t
t = r - q * s
r = s
s = t
}
if (u != 1) {
System.print(" impossible inverse mod N, gcd = %(u)")
inverr = true
}
return r
}
 
// returns mod(a, N), a is a BigInt
var modn = Fn.new { |a|
var b = a.copy()
b = b % e.N
if (b < 0) b = b + e.N
return b
}
 
// returns mod(a, r), a is a BigInt
var modr = Fn.new { |a|
var b = a.copy()
b = b % e.r
if (b < 0) b = b + e.r
return b
}
 
// returns the discriminant of E
var disc = Fn.new {
var a = BigInt.new(e.a)
var b = BigInt.new(e.b)
var c = modn.call(a * modn.call(a * a)) * 4
return modn.call((c + modn.call(b * b) * 27) * (-16)).toSmall
}
 
// return true if P is 'zero' point (at inf, 0)
var isZero = Fn.new { |p| p.x == inf && p.y == 0 }
 
// return true if P is on curve E
var isOn = Fn.new { |p|
var r = 0
var s = 0
if (!isZero.call(p)) {
r = modn.call(p.x * modn.call(p.x * p.x + e.a) + e.b).toSmall
s = modn.call(p.y * p.y).toSmall
}
return r == s
}
 
// full ec point addition
var padd = Fn.new { |p, q|
var la = BigInt.zero
var t = BigInt.zero
if (isZero.call(p)) return Epnt.new(q.x, q.y)
if (isZero.call(q)) return Epnt.new(p.x, p.y)
if (p.x != q.x) { // R = P + Q
t = p.y - q.y
la = modn.call(t * exgcd.call((p.x - q.x).toSmall, e.N.toSmall))
} else { // P = Q, R = 2P
if (p.y == q.y && p.y != 0) {
t = modn.call(modn.call(p.x * p.x) * 3 + e.a)
la = modn.call(t * exgcd.call((p.y * 2).toSmall, e.N.toSmall))
} else {
return Epnt.new(inf, BigInt.zero) // P = -Q, R = O
}
}
if (inverr) return Epnt.new(inf, BigInt.zero)
t = modn.call(la * la - p.x - q.x)
return Epnt.new(t, modn.call(la * (p.x - t) - p.y))
}
 
// R = multiple kP
var pmul = Fn.new { |p, k|
var s = Epnt.new(inf, BigInt.zero)
var q = Epnt.new(p.x, p.y)
while (k != 0) {
if (k % 2 == 1) s = padd.call(s, q)
if (inverr) {
s.x = inf
s.y = BigInt.zero
break
}
q = padd.call(q, q)
k = (k/2).floor
}
return s
}
 
// print point P with prefix f
var pprint = Fn.new { |f, p|
var y = p.y
if (isZero.call(p)) {
Fmt.print("$s (0)", f)
} else {
if (y > e.N - y) y = y - e.N
Fmt.print("$s ($i, $i)", f, p.x, y)
}
}
 
// initialize elliptic curve
var ellinit = Fn.new { |i|
var a = BigInt.new(i[0])
var b = BigInt.new(i[1])
e.N = BigInt.new(i[2])
inverr = false
if (e.N < 5 || e.N > mxN) return false
e.a = modn.call(a).toSmall
e.b = modn.call(b).toSmall
e.G.x = modn.call(BigInt.new(i[3]))
e.G.y = modn.call(BigInt.new(i[4]))
e.r = i[5]
if (e.r < 5 || e.r > mxr) return false
Fmt.write("\nE: y^2 = x^3 + $ix + $i", a, b)
Fmt.print(" (mod $i)", e.N)
pprint.call("base point G", e.G)
Fmt.print("order(G, E) = $d", e.r)
return true
}
 
// signature primitive
var signature = Fn.new { |s, f|
var c
var d
var u
var u1
var sg = Pair.new(0, 0)
var V
System.print("\nsignature computation")
while (true) {
while (true) {
u = 1 + (rand.float() * (e.r - 1)).truncate
V = pmul.call(e.G, u)
c = modr.call(V.x).toSmall
if (c != 0) break
}
u1 = exgcd.call(u, e.r)
d = modr.call((modr.call(s * c) + f) * u1).toSmall
if (d != 0) break
}
Fmt.print("one-time u = $d", u)
pprint.call("V = uG", V)
sg.a = c
sg.b = d
return sg
}
 
// verification primitive
var verify = Fn.new { |W, f, sg|
var c = sg.a
var d = sg.b
 
// domain check
var t = (c > 0) && (c < e.r)
t = Boolean.and(t, d > 0 && d < e.r)
if (!t) return false
System.print("\nsignature verification")
var h = BigInt.new(exgcd.call(d, e.r))
var h1 = modr.call(h * f).toSmall
var h2 = modr.call(h * c).toSmall
Fmt.print ("h1, h2 = $d, $d", h1, h2)
var V = pmul.call(e.G, h1)
var V2 = pmul.call(W, h2)
pprint.call("h1G", V)
pprint.call("h2W", V2)
V = padd.call(V, V2)
pprint.call("+ =", V)
if (isZero.call(V)) return false
var c1 = modr.call(V.x).toSmall
Fmt.print("c' = $d", c1)
return c1 == c
}
 
var errmsg = Fn.new {
System.print("invalid parameter set")
System.print("_____________________")
}
 
// digital signature on message hash f, error bit d
var ec_dsa = Fn.new { |f, d|
// parameter check
var t = disc.call() == 0
t = Boolean.or(t, isZero.call(e.G))
var W = pmul.call(e.G, e.r)
t = Boolean.or(t, !isZero.call(W))
t = Boolean.or(t, !isOn.call(e.G))
if (t) {
errmsg.call()
return
}
System.print("\nkey generation")
var s = 1 + (rand.float() * (e.r - 1)).truncate
W = pmul.call(e.G, s)
Fmt.print("private key s = $d\n", s)
pprint.call("public key W = sG", W)
 
// next highest power of 2 - 1
t = e.r
var i = 1
while (i < 32) {
t = t | (t >> i)
i = i << 1
}
while (f > t) f = f >> 1
Fmt.print("\naligned hash $x", f)
var sg = signature.call(BigInt.new(s), f)
if (inverr) {
errmsg.call()
return
}
Fmt.print("signature c, d = $d, $d", sg.a, sg.b)
if (d > 0) {
while (d > t) d = d >> 1
f = f ^ d
Fmt.print("\ncorrupted hash $x", f)
}
t = verify.call(W, f, sg)
if (inverr) {
errmsg.call()
return
}
if (t) {
System.print("Valid\n_____")
} else {
System.print("invalid\n_______")
}
}
 
// Test vectors: elliptic curve domain parameters,
// short Weierstrass model y^2 = x^3 + ax + b (mod N)
var sets = [
// a, b, modulus N, base point G, order(G, E), cofactor
[355, 671, 1073741789, 13693, 10088, 1073807281],
[ 0, 7, 67096021, 6580, 779, 16769911], // 4
[ -3, 1, 877073, 0, 1, 878159],
[ 0, 14, 22651, 63, 30, 151], // 151
[ 3, 2, 5, 2, 1, 5],
 
// ecdsa may fail if...
// the base point is of composite order
[ 0, 7, 67096021, 2402, 6067, 33539822], // 2
// the given order is a multiple of the true order
[ 0, 7, 67096021, 6580, 779, 67079644], // 1
// the modulus is not prime (deceptive example)
[ 0, 7, 877069, 3, 97123, 877069],
// fails if the modulus divides the discriminant
[ 39, 387, 22651, 95, 27, 22651]
]
// Digital signature on message hash f,
// set d > 0 to simulate corrupted data
var f = 0x789abcde
var d = 0
 
for (s in sets) {
if (ellinit.call(s)) {
ec_dsa.call(f, d)
} else {
break
}
}
Output:

Sample output - first set only.

E: y^2 = x^3 + 355x + 671 (mod 1073741789)
base point G (13693, 10088)
order(G, E) = 1073807281

key generation
private key s = 121877962

public key W = sG (320982025, 402911160)

aligned hash 789abcde

signature computation
one-time u = 899439563
V = uG (105563482, 310387297)
signature c, d = 105563482, 270442619

signature verification
h1, h2 = 123954960, 653133035
h1G (623038071, 220475456)
h2W (893517020, -249809739)
+ = (105563482, 310387297)
c' = 105563482
Valid