Elliptic Curve Digital Signature Algorithm

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 y²= x³ + 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

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
_____

C++

#include <cstdint>
#include <iomanip>
#include <iostream>
#include <random>
#include <stdexcept>
#include <string>
#include <vector>

const int32_t MAX_MODULUS = 1073741789;
const int32_t MAX_ORDER_G = MAX_MODULUS + 65536;

std::random_device random;
std::mt19937 generator(random());
std::uniform_real_distribution<double> distribution(0.0F, 1.0F);

class Point {
public:
	Point(const int64_t& x, const int64_t& y) : x(x), y(y) {}

	Point() : x(0),y(0) {}

	bool isZero() {
		return x == INT64_MAX && y == 0;
	}

	int64_t x, y;
};

const Point ZERO_POINT(INT64_MAX, 0);

class Pair {
public:
	Pair(const int64_t& a, const int64_t& b) : a(a), b(b) {}

	const int64_t a, b;
};

class Parameter {
public:
	Parameter(const int64_t& a, const int64_t& b, const int64_t& n, const Point& g, const int64_t& r)
	: a(a), b(b), n(n), g(g), r(r) {}

	const int64_t a, b, n;
	const Point g;
	const int64_t r;
};

int64_t signum(const int64_t& x) {
	return ( x < 0 ) ? -1 : ( x > 0 ) ? 1 : 0;
}

int64_t floor_mod(const int64_t& num, const int64_t& mod) {
	const int64_t signs = ( signum(num % mod) == -signum(mod) ) ? 1 : 0;
	return ( num % mod ) + signs * mod;
}

int64_t floor_div(const int64_t& number, const int64_t& modulus) {
	const int32_t signs = ( signum(number % modulus) == -signum(modulus) ) ? 1 : 0;
	return ( number / modulus ) - signs;
}

// Return 1 / aV modulus aU
int64_t extended_GCD(int64_t v, int64_t u) {
	if ( v < 0 ) {
		v += u;
	}

	int64_t result = 0;
	int64_t s = 1;
	while ( v != 0 ) {
		const int64_t quotient = floor_div(u, v);
		u = floor_mod(u, v);
		std::swap(u, v);
		result -= quotient * s;
		std::swap(result, s);
	}

	if ( u != 1 ) {
		throw std::runtime_error("Cannot inverse modulo N, gcd = " + u);
	}
	return result;
}

class Elliptic_Curve {
public:
	Elliptic_Curve(const Parameter& parameter) {
		n = parameter.n;
		if ( n < 5 || n > MAX_MODULUS ) {
			throw std::invalid_argument("Invalid value for modulus: " + n);
		}

		a = floor_mod(parameter.a, n);
		b = floor_mod(parameter.b, n);
		g = parameter.g;
		r = parameter.r;

		if ( r < 5 || r > MAX_ORDER_G ) {
			throw std::invalid_argument("Invalid value for the order of g: " + r);
		}

		std::cout << std::endl;
		std::cout << "Elliptic curve: y^2 = x^3 + " << a << "x + " << b << " (mod " << n << ")" << std::endl;
		print_point_with_prefix(g, "base point G");
		std::cout << "order(G, E) = " << r << std::endl;
	}

	Point add(Point p, Point q) {
		if ( p.isZero() ) {
			return q;
		}
		if ( q.isZero() ) {
			return p;
		}

		int64_t la;
		if ( p.x != q.x ) {
			la = floor_mod(( p.y - q.y ) * extended_GCD(p.x - q.x, n), n);
		} else if ( p.y == q.y && p.y != 0 ) {
			la = floor_mod(floor_mod(floor_mod(p.x * p.x, n) * 3 + a, n) * extended_GCD(2 * p.y, n), n);
		} else {
			return ZERO_POINT;
		}

		const int64_t x_coord = floor_mod(la * la - p.x - q.x, n);
		const int64_t y_coord = floor_mod(la * ( p.x - x_coord ) - p.y, n);
		return Point(x_coord, y_coord);
	}

	Point multiply(Point point, int64_t k) {
		Point result = ZERO_POINT;

		while ( k != 0 ) {
			if ( ( k & 1 ) == 1 ) {
				result = add(result, point);
			}
			point = add(point, point);
			k >>= 1;
		}
		return result;
	}

	bool contains(Point point) {
		if ( point.isZero() ) {
			return true;
		}

		int64_t r = floor_mod(floor_mod(a + point.x * point.x, n) * point.x + b, n);
		int64_t s = floor_mod(point.y * point.y, n);
		return r == s;
	}

	uint64_t discriminant() {
		const int64_t constant = 4 * floor_mod(a * a, n) * floor_mod(a, n);
		return floor_mod(-16 * ( floor_mod(b * b, n) * 27 + constant ), n);
	}

	void print_point_with_prefix(Point point, const std::string& prefix) {
		int64_t y = point.y;
		if ( point.isZero() ) {
			std::cout << prefix + " (0)" << std::endl;
		} else {
			if ( y > n - y ) {
				y -= n;
			}
			std::cout << prefix + " (" << point.x << ", " << y << ")" << std::endl;
		}
	}

	int64_t a, b, n, r;
	Point g;
};

double random_number() {
	return distribution(generator);
}

Pair signature(Elliptic_Curve curve, const int64_t& s, const int64_t& f) {
	int64_t c, d, u;
	Point v;

	while ( true ) {
		while ( true ) {
			u = 1 + (int64_t) ( random_number() * (double) ( curve.r - 1 ) );
			v = curve.multiply(curve.g, u);
			c = floor_mod(v.x, curve.r);
			if ( c != 0 ) {
				break;
			}
		}

		d = floor_mod(extended_GCD(u, curve.r) * floor_mod(f + s * c, curve.r), curve.r);
		if ( d != 0 ) {
			break;
		}
	}

	std::cout << "one-time u = " << u << std::endl;
	curve.print_point_with_prefix(v, "V = uG");
	return Pair(c, d);
}

bool verify(Elliptic_Curve curve, Point point, const int64_t& f, const Pair& signature) {
	if ( signature.a < 1 || signature.a >= curve.r || signature.b < 1 || signature.b >= curve.r ) {
		return false;
	}

	std::cout << "\n" << "signature verification" << std::endl;
	const int64_t h = extended_GCD(signature.b, curve.r);
	const int64_t h1 = floor_mod(f * h, curve.r);
	const int64_t h2 = floor_mod(signature.a * h, curve.r);
	std::cout << "h1, h2 = " << h1 << ", " << h2 << std::endl;
	Point v = curve.multiply(curve.g, h1);
	Point v2 = curve.multiply(point, h2);
	curve.print_point_with_prefix(v, "h1G");
	curve.print_point_with_prefix(v2, "h2W");
	v = curve.add(v, v2);
	curve.print_point_with_prefix(v, "+ =");

	if ( v.isZero() ) {
		return false;
	}
	int64_t c1 = floor_mod(v.x, curve.r);
	std::cout << "c' = " << c1 << std::endl;
	return c1 == signature.a;
}

// Build the digital signature for a message using the hash aF with error bit aD
void ecdsa(Elliptic_Curve curve, int64_t f, int32_t d) {
	Point point = curve.multiply(curve.g, curve.r);

	if ( curve.discriminant() == 0 || curve.g.isZero() || ! point.isZero() || ! curve.contains(curve.g) ) {
		throw std::invalid_argument("Invalid parameter in the method ecdsa");
	}

	std::cout << "\n" << "key generation" << std::endl;
	const int64_t s = 1 + (int64_t) ( random_number() * (double) ( curve.r - 1 ) );
	point = curve.multiply(curve.g, s);
	std::cout << "private key s = " << s << std::endl;
	curve.print_point_with_prefix(point, "public key W = sG");

	// Find the next highest power of two minus one.
	int64_t t = curve.r;
	int64_t i = 1;
	while ( i < 64 ) {
		t |= t >> i;
		i <<= 1;
	}
	while ( f > t ) {
		f >>= 1;
	}
	std::cout << "\n" << "aligned hash " << std::hex << std::setfill('0') << std::setw(8) << f
			  << std::dec << std::endl;

	const Pair signature_pair = signature(curve, s, f);
	std::cout << "signature c, d = " << signature_pair.a << ", " << signature_pair.b << std::endl;

	if ( d > 0 ) {
		while ( d > t ) {
			d >>= 1;
		}
		f ^= d;
		std::cout << "\n" << "corrupted hash " << std::hex << std::setfill('0') << std::setw(8) << f 
                  << std::dec << std::endl;
	}

	std::cout << ( verify(curve, point, f, signature_pair) ? "Valid" : "Invalid" ) << std::endl;
	std::cout << "-----------------" << std::endl;
}

int main() {
	// Test parameters for elliptic curve digital signature algorithm,
	// using the short Weierstrass model: y^2 = x^3 + ax + b (mod N).
	//
	// Parameter: a, b, modulus N, base point G, order of G in the elliptic curve.

	const std::vector<Parameter> parameters {
		Parameter( 355, 671, 1073741789, Point(13693, 10088), 1073807281 ),
		Parameter(   0,   7,   67096021, Point( 6580,   779),   16769911 ),
		Parameter(  -3,   1,     877073, Point(    0,     1),     878159 ),
		Parameter(   0,  14,      22651, Point(   63,    30),        151 ),
		Parameter(   3,   2,          5, Point(    2,     1),          5 ) };

		// Parameters which cause failure of the algorithm for the given reasons
		// the base point is of composite order
//		Parameter(   0,   7,   67096021, Point( 2402,  6067),   33539822 ),
		// the given order is of composite order
//		Parameter(   0,   7,   67096021, Point( 6580,   779),   67079644 ),
		// the modulus is not prime (deceptive example)
//		Parameter(   0,   7,     877069, Point(    3, 97123),     877069 ),
		// fails if the modulus divides the discriminant
//		Parameter(  39, 387,      22651, Point(   95,    27),      22651 ) );

	const int64_t f = 0x789abcde; // The message's digital signature hash which is to be verified
	const int32_t d = 0;           // Set d > 0 to simulate corrupted data

	for ( const Parameter& parameter : parameters ) {
		Elliptic_Curve elliptic_curve(parameter);
		ecdsa(elliptic_curve, f, d);
	}
}

Output:

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

key generation
private key s = 1024325479
public key W = sG (717544485, -463545080)

aligned hash 789abcde
one-time u = 76017594
V = uG (231970881, 178344325)
signature c, d = 231970881, 762656688

signature verification
h1, h2 = 205763719, 179248000
h1G (757236523, -510136355)
h2W (118269957, 451962485)
+ = (231970881, 178344325)
c' = 231970881
Valid
-----------------

// continues ...

FreeBASIC

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

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

Java

import java.util.List;
import java.util.concurrent.ThreadLocalRandom;

public final class EllipticCurveDigitalSignatureAlgorithm {

	public static void main(String[] aArgs) { 
		// Test parameters for elliptic curve digital signature algorithm,
		// using the short Weierstrass model: y^2 = x^3 + ax + b (mod N).
		//
		// Parameter: a, b, modulus N, base point G, order of G in the elliptic curve.

		List<Parameter> parameters = List.of(				
		    new Parameter( 355, 671, 1073741789, new Point(13693, 10088), 1073807281 ),
		    new Parameter(   0,   7,   67096021, new Point( 6580,   779),   16769911 ),
		    new Parameter(  -3,   1,     877073, new Point(    0,     1),     878159 ),
		    new Parameter(   0,  14,      22651, new Point(   63,    30),        151 ),
		    new Parameter(   3,   2,          5, new Point(    2,     1),          5 ) );

			// Parameters which cause failure of the algorithm for the given reasons
		    // the base point is of composite order
//		    new Parameter(   0,   7,   67096021, new Point( 2402,  6067),   33539822 ),
		    // the given order is of composite order
//		    new Parameter(   0,   7,   67096021, new Point( 6580,   779),   67079644 ),
		    // the modulus is not prime (deceptive example)
//		    new Parameter(   0,   7,     877069, new Point(    3, 97123),     877069 ),
		    // fails if the modulus divides the discriminant
//		    new Parameter(  39, 387,      22651, new Point(   95,    27),      22651 ) );		
		
		final long f = 0x789abcde; // The message's digital signature hash which is to be verified
		final int d = 0;           // Set d > 0 to simulate corrupted data

		for ( Parameter parameter : parameters ) {
			EllipticCurve ellipticCurve = new EllipticCurve(parameter);			
		    ecdsa(ellipticCurve, f, d);		    
		}
	}
	
	// Build the digital signature for a message using the hash aF with error bit aD
	private static void ecdsa(EllipticCurve aCurve, long aF, int aD) {
		Point point = aCurve.multiply(aCurve.g, aCurve.r);
		
		if ( aCurve.discriminant() == 0 || aCurve.g.isZero() || ! point.isZero() || ! aCurve.contains(aCurve.g) ) {		
		    throw new AssertionError("Invalid parameter in method ecdsa");
		}

		System.out.println(System.lineSeparator() + "key generation");
		final long s = 1 + (long) ( random() * (double) ( aCurve.r - 1 ) );
		point = aCurve.multiply(aCurve.g, s);
		System.out.println("private key s = " + s);
		aCurve.printPointWithPrefix(point, "public key W = sG");

		// Find the next highest power of two minus one.
		long t = aCurve.r;
		long i = 1;
		while ( i < 64 ) {
		    t |= t >> i;
		    i <<= 1;
		}
		long f = aF;
		while ( f > t ) {
			f >>= 1;
		}
		System.out.println(System.lineSeparator() + "aligned hash " + String.format("%08x", f));

		Pair signature = signature(aCurve, s, f);
		System.out.println("signature c, d = " + signature.a + ", " + signature.b);

		long d = aD;
		if ( d > 0 ) {
		    while ( d > t ) {
		    	d >>= 1;
		    }
		    f ^= d;
		    System.out.println(System.lineSeparator() + "corrupted hash " + String.format("%08x", f));
		}

		System.out.println(verify(aCurve, point, f, signature) ? "Valid" : "Invalid");
		System.out.println("-----------------");
	}
	
	private static boolean verify(EllipticCurve aCurve, Point aPoint, long aF, Pair aSignature) {
		if ( aSignature.a < 1 || aSignature.a >= aCurve.r || aSignature.b < 1 || aSignature.b >= aCurve.r ) {
		    return false;
		}

		System.out.println(System.lineSeparator() + "signature verification");
		final long h = extendedGCD(aSignature.b, aCurve.r);
		final long h1 = Math.floorMod(aF * h, aCurve.r);
		final long h2 = Math.floorMod(aSignature.a * h, aCurve.r);
		System.out.println("h1, h2 = " + h1 + ", " + h2);
		Point v = aCurve.multiply(aCurve.g, h1);
		Point v2 = aCurve.multiply(aPoint, h2);
		aCurve.printPointWithPrefix(v, "h1G");
		aCurve.printPointWithPrefix(v2, "h2W");
		v = aCurve.add(v, v2);
		aCurve.printPointWithPrefix(v, "+ =");
		
		if ( v.isZero() ) {
			return false;
		}
		long c1 = Math.floorMod(v.x, aCurve.r);
		System.out.println("c' = " + c1);
		return c1 == aSignature.a;
	}
	
	private static Pair signature(EllipticCurve aCurve, long aS, long aF) {
		long c = 0;
		long d = 0;
		long u;
		Point v;
		System.out.println("Signature computation");

		while ( true ) {
		    while ( true ) {
		        u = 1 + (long) ( random() * (double) ( aCurve.r - 1 ) );
		        v = aCurve.multiply(aCurve.g, u);
		        c = Math.floorMod(v.x, aCurve.r);
		        if ( c != 0 ) {
		        	break;
		        }
		    }
		    
		    d = Math.floorMod(extendedGCD(u, aCurve.r) * Math.floorMod(aF + aS * c, aCurve.r), aCurve.r);
		    if ( d != 0 ) {
		    	break;
		    }
		}

		System.out.println("one-time u = " + u);
		aCurve.printPointWithPrefix(v, "V = uG");
		return new Pair(c, d);
	}		
	
	// Return 1 / aV modulus aU
	private static long extendedGCD(long aV, long aU) {
		if ( aV < 0 ) {
			aV += aU;
		}

		long result = 0;
		long s = 1;
		while ( aV != 0 ) {
		    final long quotient = Math.floorDiv(aU, aV);
		    aU = Math.floorMod(aU, aV);
		    long temp = aU; aU = aV; aV = temp;
		    result -= quotient * s;
		    temp = result; result = s; s = temp;
		}

		if ( aU != 1 ) {
		    throw new AssertionError("Cannot inverse modulo N, gcd = " + aU);
		}
		return result;
	}
	
	private static double random() {
		return RANDOM.nextDouble();
	}

	private static class EllipticCurve {
		
		public EllipticCurve(Parameter aParameter) {
			n = aParameter.n;
			if ( n < 5 || n > MAX_MODULUS ) {
			    throw new AssertionError("Invalid value for modulus: " + n);
			}
			
			a = Math.floorMod(aParameter.a, n);
			b = Math.floorMod(aParameter.b, n);
			g = aParameter.g;
			r = aParameter.r;

			if ( r < 5 || r > MAX_ORDER_G ) {
			    throw new AssertionError("Invalid value for the order of g: " + r);
			}

			System.out.println();
			System.out.println("Elliptic curve: y^2 = x^3 + " + a + "x + " + b + " (mod " + n + ")");
			printPointWithPrefix(g, "base point G");
			System.out.println("order(G, E) = " + r);
		}	
		
		private Point add(Point aP, Point aQ) {
			if ( aP.isZero() ) {
				return aQ;
			}
			if ( aQ.isZero() ) {
				return aP;
			}

			long la;
			if ( aP.x != aQ.x ) {
			    la = Math.floorMod(( aP.y - aQ.y ) * extendedGCD(aP.x - aQ.x, n), n);			    
			} else if ( aP.y == aQ.y && aP.y != 0 ) {
				la = Math.floorMod(Math.floorMod(Math.floorMod(
					aP.x * aP.x, n) * 3 + a, n) * extendedGCD(2 * aP.y, n), n);
			} else {
			    return Point.ZERO;
			}

			final long xCoordinate = Math.floorMod(la * la - aP.x - aQ.x, n);
			final long yCoordinate = Math.floorMod(la * ( aP.x - xCoordinate ) - aP.y, n);
			return new Point(xCoordinate, yCoordinate);
		}
		
		public Point multiply(Point aPoint, long aK) {
			Point result = Point.ZERO;

			while ( aK != 0 ) {
			    if ( ( aK & 1 ) == 1 ) {
			        result = add(result, aPoint);
			    }
			    aPoint = add(aPoint, aPoint);
			    aK >>= 1;
			}
			return result;
		}
		
		public boolean contains(Point aPoint) {
			if ( aPoint.isZero() ) {
				return true;
			}
			
			final long r = Math.floorMod(Math.floorMod(a + aPoint.x * aPoint.x, n) * aPoint.x + b, n);
			final long s = Math.floorMod(aPoint.y * aPoint.y, n);
			return r == s;
		}	
		
		public long discriminant() {			  
			final long constant = 4 * Math.floorMod(a * a, n) * Math.floorMod(a, n);
			return Math.floorMod(-16 * ( Math.floorMod(b * b, n) * 27 + constant ), n);
		}	
		
		public void printPointWithPrefix(Point aPoint, String aPrefix) {		
			 long y = aPoint.y;
			 if ( aPoint.isZero() ) {
				 System.out.println(aPrefix + " (0)");
			 } else {
			    if ( y > n - y ) {
			    	y -= n;
			    }
			    System.out.println(aPrefix + " (" + aPoint.x + ", " + y + ")");
			 }
		}		
		
		private final long a, b, n, r;
		private final Point g;
		
	}
	
	private static class Point {
		
		public Point(long aX, long aY) {
			x = aX;
			y = aY;
		}	
		
		public boolean isZero() {
			return x == INFINITY && y == 0;
		}
					
		private long x, y;
		
		private static final long INFINITY = Long.MAX_VALUE;
		private static final Point ZERO = new Point(INFINITY, 0);
			
	}

	private static record Pair(long a, long b) {}

	private static record Parameter(long a, long b, long n, Point g, long r) {}
	
	private static final int MAX_MODULUS = 1073741789;
	private static final int MAX_ORDER_G = MAX_MODULUS + 65536;
	
	private static final ThreadLocalRandom RANDOM = ThreadLocalRandom.current();

}

Output:

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

key generation
private key s = 631324603
public key W = sG (789657939, 162653307)

aligned hash 789abcde
Signature computation
one-time u = 978377271
V = uG (40748926, -184905705)
signature c, d = 40748926, 938914492

signature verification
h1, h2 = 903203729, 29256237
h1G (286322818, -482840260)
h2W (31139810, 45550525)
+ = (40748926, -184905705)
c' = 40748926
Valid
-----------------

// continues ...

Julia

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

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
——————————————

Perl

Translation of: Go

use strict;
use warnings;
use Crypt::EC_DSA;

my $ecdsa = Crypt::EC_DSA->new;

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

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

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

(formerly Perl 6)

module EC {
  use FiniteField;

  our class Point {
    has ($.x, $.y);
    submethod TWEAK { fail unless $!y**2 == $!x**3 + $*a*$!x + $*b }
    multi method gist(::?CLASS:U:) { "Point at Horizon" }
    multi method new($x, $y) { samewith :$x, :$y }
  }
  multi infix:<==>(Point:D $A, Point:D $B) is export { $A.x == $B.x and $A.y == $B.y }

  multi prefix:<->(Point $P) returns Point is export { Point.new: $P.x, -$P.y }
  multi infix:<+>(Point $A, Point:U) is export { $A }
  multi infix:<+>(Point:U, Point $B) is export { $B }
  multi infix:<+>(Point:D $A, Point:D $B) returns Point is export {
    my $λ;
    if $A.x == $B.x and $A.y == -$B.y { return Point }
    elsif $A == $B {
      return Point if $A.y == 0;
      $λ = (3*$A.x² + $*a) / (2*$A.y);
    }
    else { $λ = ($A.y - $B.y) / ($A.x - $B.x); }

    given $λ**2 - $A.x - $B.x {
      return Point.new: $_, $λ*($A.x - $_) - $A.y;
    }
  }
  multi infix:<*>(0, Point     ) is export { Point }
  multi infix:<*>(1, Point   $p) is export { $p }
  multi infix:<*>(2, Point:D $p) is export { $p + $p }
  multi infix:<*>(Int $n, Point $p) is export { 2*(($n div 2)*$p) + ($n mod 2)*$p }
}

import EC;

module ECDSA {
  use Digest::SHA256::Native;
  our class Signature {
    has UInt ($.c, $.d);
    multi method new(Str $message, UInt :$private-key) {
      my $z = :16(sha256-hex $message) % $*n;
      loop (my $k = my $s = my $r = 0; $r == 0; ) {
        loop ( $r = $s = 0 ; $r == 0 ; ) {
          $r = (( $k = (1..^$*n).roll ) * $*G).x % $*n;
        }
        {
          use FiniteField; my $*modulus = $*n;
          $s = ($z + $r*$private-key) / $k;
        }
      }
      samewith c => $r, d => $s;
    }
    multi method verify(Str $message, EC::Point :$public-key) {
      my $z = :16(sha256-hex $message) % $*n;
      my ($u1, $u2);
      {
        use FiniteField;
        my $*modulus = $*n;
        my $w = 1/$!d;
        ($u1, $u2) = $z*$w, $!c*$w;
      }
      my $p = ($u1 * $*G) + ($u2 * $public-key);
      die unless ($p.x mod $*n) == ($!c mod $*n);
    }
  }
}

my ($*a, $*b) = 355, 671;
my $*modulus = my $*p = 1073741789;

my $*G = EC::Point.new: 13693, 10088;
my $*n = 1073807281;

die "G is not of order n" if $*n*$*G;

my $private-key = ^$*n .pick;
my $public-key = $private-key*$*G; 

my $message = "Show me the monKey";
my $signature = ECDSA::Signature.new: $message, :$private-key;

printf "The curve E is        : 𝑦² = 𝑥³ + %s 𝑥 + %s (mod %s)\n", $*a, $*b, $*p;
printf "with generator G at   : (%s, %s)\n", $*G.x, $*G.y;
printf "G's order is          : %d\n", $*n;
printf "The private key is    : %d\n", $private-key;
printf "The public key is at  : (%s, %s)\n", $public-key.x, $public-key.y;
printf "The message is        : %s\n", $message;
printf "The signature is      : (%s, %s)\n", $signature.c, $signature.d;

{
  use Test;

  lives-ok {
    $signature.verify: $message, :$public-key;
  }, "good signature for <$message>";

  my $altered = $message.subst(/monKey/, "money");
  dies-ok {
    $signature.verify: $altered, :$public-key
  }, "wrong signature for <$altered>";
}

Output:

The curve E is        : 𝑦² = 𝑥³ + 355 𝑥 + 671 (mod 1073741789)
with generator G at   : (13693, 10088)
G's order is          : 1073807281
The private key is    : 405586338
The public key is at  : (457744420, 236326628)
The message is        : Show me the monKey
The signature is      : (839907635, 23728690)
ok 1 - good signature for <Show me the monKey>
ok 2 - wrong signature for <Show me the money>

Wren

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