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Elliptic Curve Digital Signature Algorithm (view source)

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{{task|Cryptography}}
__NOTOC__
==The Elliptic Curve Digital Signature Algorithm ([https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm ECDSA]).==
 
===;Elliptic curves.===
 
An [[wp:Elliptic_curve|elliptic curve]] E over ℤp (p ≥ 5) is defined by an equation of the form '''y^2 = x^3 + ax + b''',
'''y²= x³ + ax + b''', where a, b ∈ ℤp and the discriminant ≢ 0 (mod p), together with a special point 𝒪
together with a special point 𝒪 called the point at infinity.
called the point at infinity. The set '''E(ℤp)''' consists of all points (x, y), with x, y ∈ ℤp,
The set '''E(ℤp)''' consists of all points (x, y), with x, y ∈ ℤp,
which satisfy the above defining equation, together with 𝒪.
 
Line 14:
 
The addition rule — which can be explained geometrically — is summarized as follows:
<pre>
1. P + &#119978; = &#119978; + P = P for all P &#8712; E(&#8484;p).
 
12. If P += (x, y) &#1199788712; = E(&#1199788484;p), +then inverse -P = P for(x,-y), alland P &#8712;+ E(-P) = &#8484119978;p).
2. If P = (x, y) &#8712; E(&#8484;p), then inverse -P = (x,-y), and P + (-P) = &#119978;.
3. Let P = (xP, yP) and Q = (xQ, yQ), both &#8712; E(&#8484;p), where P &#8800; -Q.
Then R = P + Q = (xR, yR), where
xR = &lambda;^2 - xP - xQ
yR = &lambda;&middot;(xP - xR) - yP,
with
&lambda; = (yP - yQ) / (xP - xQ) if P &#8800; Q,
(3&middot;xP&middot;xP + a) / 2&middot;yP if P = Q (point doubling).
 
3. Let P = (xP, yP) and Q = (xQ, yQ), both &#8712; E(&#8484;p), where P &#8800; -Q.
Then R = P + Q = (xR, yR), where
 
xR = &lambda;^2 - xP - xQ
===Elliptic curve digital signature algorithm.===
yR = &lambda;&middot;(xP - xR) - yP,
 
with
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.
&lambda; = (yP - yQ) / (xP - xQ) if P &#8800; Q,
(3&middot;xP&middot;xP + a) / 2&middot;yP if P = Q (point doubling).
</pre>
Remark: there already is a task page requesting &#8220;a simplified (without modular arithmetic)
version of the [[Elliptic_curve_arithmetic|elliptic curve arithmetic]]&#8221;.
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 [[wp:Elliptic_Curve_Digital_Signature_Algorithm|ECDSA]]:
 
 
;Elliptic curve digital signature algorithm.
 
A [[wp:Digital_signature|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.
Line 47 ⟶ 53:
 
'''ECDSA signature computation.''' To sign a message m, A does the following:<br />
1. Compute message representative f = H(m), using a cryptographic hash function.<br />
[[wp:Cryptographic_hash_function|cryptographic hash function]].<br />
&nbsp;Note that f can be greater than r but not longer (measuring bits).<br />
2. Select a random integer u in the interval [1, r - 1].<br />
Line 62 ⟶ 69:
&nbsp;Accept the signature if and only if c1 = c.
 
To be cryptographically useful, the parameter r should have at least have 250 bits.
The basis for the security of [[wp:Elliptic-curve_cryptography|elliptic curve cryptosystems is the intractability]]
is the intractability of the elliptic curve discrete logarithm problem (ECDLP) in a group of this size:
given two points G, W &#8712; E(&#8484;p), where W lies in the subgroup of order r generated by G,
determine an integer k such that W = kG and 0 &#8804; k &lt; 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 methodway 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 [[wp:Lenstra_elliptic-curve_factorization|elliptic curve factorization]]), but strict where required
&mdash; but strict where required (a point G that is not on E will always cause failure).<br />
Toy ECDSA is of course completely useless for its cryptographic purpose.
If this bothers you, feel free toplease add a multiple-precision version.
 
 
===;Reference.===
 
Elliptic curves are in the [https://perso.telecom-paristech.fr/guilley/recherche/cryptoprocesseurs/ieee/00891000.pdf IEEE Std 1363-2000] (Standard Specifications for Public-Key Cryptography), see:
Line 99 ⟶ 106:
A.9 Elliptic curves: overview (p. 115)<br />
A.10 Elliptic curves: algorithms (p. 121)
 
 
 
__TOC__
 
=={{header|C}}==
Parallel to: FreeBASIC
<syntaxhighlight lang="c">
/*
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;
}
}
</syntaxhighlight>
{{out}}
(tcc, srand(1); first set only)
<pre>
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
_____
</pre>
 
=={{header|C++}}==
<syntaxhighlight lang="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);
}
}
</syntaxhighlight>
{{ out }}
<pre>
 
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 ...
</pre>
 
=={{header|FreeBASIC}}==
Parallel to: C
<syntaxhighlight lang="freebasic">
'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
</syntaxhighlight>
{{out}}
(randomize 1, first set only)
<pre>
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
_____
</pre>
 
=={{header|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.
<syntaxhighlight lang="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)
}</syntaxhighlight>
 
{{out}}
Sample run:
<pre>
Private key:
D: 25700608762903774973512323993645267346590725880891580901973011512673451968935
 
Public key:
X: 37298454876588653961191059192981094503652951300904260069480867699946371240473
Y: 69073688506493709421315518164229531832022167466292360349457318041854718641652
 
Message: Rosetta Code
Hash : 0xe6f9ed0d
 
Signature:
R: 91827099055706804696234859308003894767808769875556550819128270941615405955877
S: 20295707309473352071389945163735458699476300346398176659149368970668313772860
 
Signature verified: true
</pre>
 
=={{header|Java}}==
<syntaxhighlight lang="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();
 
}
</syntaxhighlight>
{{ out }}
<pre>
 
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 ...
</pre>
 
=={{header|Julia}}==
<syntaxhighlight lang="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))
</syntaxhighlight>{{out}}
<pre>
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
</pre>
 
=={{header|Nim}}==
{{trans|C}}
<syntaxhighlight lang="nim">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 "——————————————"</syntaxhighlight>
 
{{out}}
First set only.
<pre>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>
</pre>
 
=={{header|Perl}}==
{{trans|Go}}
<syntaxhighlight lang="perl">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"</syntaxhighlight>
{{out}}
<pre>Message: Rosetta Code
Private Key :
50896950174101144529764022934807089163030534967278433074982207331912857967110
Public key :
98639220601877298644829563208621497413822596683110596662237522364057856411416
69976521993624103693270429074404825634551369215777879408776019358694823367135
Signature :
r => 113328268998856048369024784426817827689451364968174533291969247274701793929451
s => 102496515866716695113707075780391660331998173218829535655149484671019624453603
Signature verified.</pre>
 
=={{header|Phix}}==
{{trans|FreeBASIC}}
<!--<syntaxhighlight lang="phix">(notonline)-->
<span style="color: #7060A8;">requires</span><span style="color: #0000FF;">(</span><span style="color: #000000;">64</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">enum</span> <span style="color: #000000;">X</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">Y</span> <span style="color: #000080;font-style:italic;">-- rational ec point</span>
<span style="color: #008080;">enum</span> <span style="color: #000000;">A</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">B</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">N</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">G</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">R</span> <span style="color: #000080;font-style:italic;">-- elliptic curve parameters
-- also signature pair(A,B)</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">mxN</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1073741789</span> <span style="color: #000080;font-style:italic;">-- maximum modulus</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">mxr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1073807325</span> <span style="color: #000080;font-style:italic;">-- max order G = mxN + 65536</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">inf</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2147483647</span> <span style="color: #000080;font-style:italic;">-- symbolic infinity</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">e</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">},</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- single global curve</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">zerO</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">inf</span><span style="color: #0000FF;">,</span><span style="color: #000000;">0</span><span style="color: #0000FF;">}</span> <span style="color: #000080;font-style:italic;">-- point at infinity zerO</span>
<span style="color: #004080;">bool</span> <span style="color: #000000;">inverr</span> <span style="color: #000080;font-style:italic;">-- impossible inverse mod N</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">exgcd</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return mod(v^-1, u)</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">v</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000000;">v</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">u</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">v</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">/</span><span style="color: #000000;">v</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">-</span><span style="color: #000000;">q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">v</span>
<span style="color: #000000;">u</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span>
<span style="color: #000000;">v</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">-</span><span style="color: #000000;">q</span><span style="color: #0000FF;">*</span><span style="color: #000000;">s</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" impossible inverse mod N, gcd = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">u</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">inverr</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">true</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">r</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return mod(a, N)</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return mod(a, r)</span>
<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">mod</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">a</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">disc</span><span style="color: #0000FF;">()</span>
<span style="color: #000080;font-style:italic;">-- return the discriminant of E</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">A</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">B</span><span style="color: #0000FF;">],</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">*</span><span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(-</span><span style="color: #000000;">16</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">+</span><span style="color: #000000;">27</span><span style="color: #0000FF;">*</span><span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return true if P = zerO</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">inf</span> <span style="color: #008080;">and</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">ison</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- return true if P is on curve E</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">B</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">A</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]))</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- print point P with prefix f</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s (0)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">else</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">></span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">y</span> <span style="color: #008080;">then</span> <span style="color: #000000;">y</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s (%d,%d)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">],</span><span style="color: #000000;">y</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">padd</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- full ec point addition</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">la</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">q</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">p</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">q</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- R := P + Q</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">q</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">la</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">exgcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">q</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]))</span>
<span style="color: #008080;">else</span> <span style="color: #000080;font-style:italic;">-- P = Q, R := 2P</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">q</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]!=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">])+</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">A</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">la</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">exgcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]))</span>
<span style="color: #008080;">else</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">zerO</span> <span style="color: #000080;font-style:italic;">-- P = -Q, R := O</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">la</span><span style="color: #0000FF;">*</span><span style="color: #000000;">la</span><span style="color: #0000FF;">-</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">q</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">])</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">deep_copy</span><span style="color: #0000FF;">(</span><span style="color: #000000;">zerO</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">la</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)-</span><span style="color: #000000;">p</span><span style="color: #0000FF;">[</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">r</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">inverr</span> <span style="color: #008080;">then</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">zerO</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">r</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- R:= multiple kP</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">zerO</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">k</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #7060A8;">and_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">padd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">inverr</span> <span style="color: #008080;">then</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">zerO</span><span style="color: #0000FF;">;</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">padd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">q</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">s</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">ellinit</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- initialize elliptic curve</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">b</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">inverr</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">3</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]<</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]></span><span style="color: #000000;">mxN</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">A</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">B</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">b</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">][</span><span style="color: #000000;">X</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">4</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">][</span><span style="color: #000000;">Y</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modn</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">5</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">[</span><span style="color: #000000;">6</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">if</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]<</span><span style="color: #000000;">5</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">or</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]></span><span style="color: #000000;">mxr</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"E: y^2 = x^3 + %dx + %d (mod %d)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b</span><span style="color: #0000FF;">,</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">N</span><span style="color: #0000FF;">]})</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"base point G"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"order(G, E) = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]})</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">signature</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- signature primitive</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">u1</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">V</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"signature computation\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
<span style="color: #000080;font-style:italic;">-- u = rand(e[R]-1)</span>
<span style="color: #000000;">u</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">571533488</span> <span style="color: #000080;font-style:italic;">-- match FreeBASIC output
-- u = 605163545 -- match C output</span>
<span style="color: #000000;">V</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">u</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">V</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">c</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">u1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">exgcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">u1</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">+</span><span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)))</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"one-time u = %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">u</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"V = uG"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">c</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">verify</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">W</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">sequence</span> <span style="color: #000000;">sg</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- verification primitive</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">c</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sg</span><span style="color: #0000FF;">[</span><span style="color: #000000;">A</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sg</span><span style="color: #0000FF;">[</span><span style="color: #000000;">B</span><span style="color: #0000FF;">],</span>
<span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">c1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">h1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">h2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">h</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">V</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V2</span>
<span style="color: #000080;font-style:italic;">--domain check</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;"><</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">and</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;"><</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">t</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nsignature verification\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">h</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">exgcd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">h1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">*</span><span style="color: #000000;">h</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">h2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">c</span><span style="color: #0000FF;">*</span><span style="color: #000000;">h</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"h1,h2 = %d,%d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">h1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">h2</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">V</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">h1</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">V2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">W</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">h2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"h1G"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"h2W"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">V</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">padd</span><span style="color: #0000FF;">(</span><span style="color: #000000;">V</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"+ ="</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">V</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">V</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">c1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">modr</span><span style="color: #0000FF;">(</span><span style="color: #000000;">V</span><span style="color: #0000FF;">[</span><span style="color: #000000;">X</span><span style="color: #0000FF;">])</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"c' = %d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">c1</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">c1</span><span style="color: #0000FF;">=</span><span style="color: #000000;">c</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">errmsg</span><span style="color: #0000FF;">()</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"invalid parameter set\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"_____________________\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">ec_dsa</span><span style="color: #0000FF;">(</span><span style="color: #004080;">atom</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- digital signature on message hash f, error bit d</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">sg</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">W</span>
<span style="color: #000080;font-style:italic;">--parameter check</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">disc</span><span style="color: #0000FF;">()=</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #008080;">or</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">W</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">])</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #008080;">or</span> <span style="color: #008080;">not</span> <span style="color: #000000;">isO</span><span style="color: #0000FF;">(</span><span style="color: #000000;">W</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span> <span style="color: #008080;">or</span> <span style="color: #008080;">not</span> <span style="color: #000000;">ison</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">t</span> <span style="color: #008080;">then</span> <span style="color: #000000;">errmsg</span><span style="color: #0000FF;">()</span> <span style="color: #008080;">return</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">puts</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\nkey generation\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">-- s = rand(e[R] - 1)</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">509100772</span> <span style="color: #000080;font-style:italic;">-- match FreeBASIC output
-- s = 1343570 -- match C output</span>
<span style="color: #000000;">W</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">pmul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">G</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"private key s = %d\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">s</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">pprint</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"public key W = sG"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">W</span><span style="color: #0000FF;">)</span>
<span style="color: #000080;font-style:italic;">--next highest power of 2 - 1</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">e</span><span style="color: #0000FF;">[</span><span style="color: #000000;">R</span><span style="color: #0000FF;">]</span>
<span style="color: #000000;">i</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">i</span><span style="color: #0000FF;"><</span><span style="color: #000000;">32</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">or_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">/</span><span style="color: #7060A8;">power</span><span style="color: #0000FF;">(</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)))</span>
<span style="color: #000000;">i</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">2</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">></span><span style="color: #000000;">t</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\naligned hash %x\n\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">sg</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">signature</span><span style="color: #0000FF;">(</span><span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">inverr</span> <span style="color: #008080;">then</span> <span style="color: #000000;">errmsg</span><span style="color: #0000FF;">()</span> <span style="color: #008080;">return</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"signature c,d = %d,%d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">sg</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">></span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span>
<span style="color: #008080;">while</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">></span><span style="color: #000000;">t</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
<span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">xor_bits</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span><span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"corrupted hash %x\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">f</span><span style="color: #0000FF;">})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">verify</span><span style="color: #0000FF;">(</span><span style="color: #000000;">W</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sg</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">inverr</span> <span style="color: #008080;">then</span> <span style="color: #000000;">errmsg</span><span style="color: #0000FF;">()</span> <span style="color: #008080;">return</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">if</span> <span style="color: #000000;">t</span> <span style="color: #008080;">then</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Valid\n_____\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">else</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"invalid\n_______\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
<span style="color: #000080;font-style:italic;">--Test vectors: elliptic curve domain parameters,
--short Weierstrass model y^2 = x^3 + ax + b (mod N)</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">tests</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span>
<span style="color: #000080;font-style:italic;">-- a, b, modulus N, base point G, order(G, E), cofactor</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">355</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">671</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1073741789</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">13693</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">10088</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1073807281</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">67096021</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">6580</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">779</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">16769911</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- 4</span>
<span style="color: #0000FF;">{</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">877073</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">878159</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">14</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">22651</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">63</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">30</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">151</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- 151</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">},</span>
<span style="color: #000080;font-style:italic;">--ecdsa may fail if...
--the base point is of composite order</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">67096021</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2402</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">6067</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">33539822</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- 2
--the given order is a multiple of the true order</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">67096021</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">6580</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">779</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">67079644</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- 1
--the modulus is not prime (deceptive example)</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">877069</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">97123</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">877069</span><span style="color: #0000FF;">},</span>
<span style="color: #000080;font-style:italic;">--fails if the modulus divides the discriminant</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">39</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">387</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">22651</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">95</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">27</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">22651</span><span style="color: #0000FF;">}}</span>
<span style="color: #000080;font-style:italic;">--Digital signature on message hash f,
--set d &gt; 0 to simulate corrupted data</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">f</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">#789ABCDE</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">d</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
<span style="color: #000080;font-style:italic;">--for i=1 to length(tests) do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">ellinit</span><span style="color: #0000FF;">(</span><span style="color: #000000;">tests</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">])</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
<span style="color: #000000;">ec_dsa</span><span style="color: #0000FF;">(</span><span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</syntaxhighlight>-->
{{out}}
Note the above only performs tests[1], and assigns literal values in place of rand(), in order to exactly match the FreeBASIC/C output.
<pre>
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
_____
</pre>
 
=={{header|Python}}==
<syntaxhighlight lang="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()
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
 
=={{header|Raku}}==
(formerly Perl 6)
<syntaxhighlight lang="raku" line>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>";
}</syntaxhighlight>
{{out}}
<pre>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></pre>
 
=={{header|Wren}}==
{{trans|C}}
{{libheader|Wren-dynamic}}
{{libheader|Wren-big}}
{{libheader|Wren-fmt}}
{{libheader|Wren-math}}
As we don't have a signed 64 bit integer type, we use BigInt instead where needed.
<syntaxhighlight lang="wren">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
}
}</syntaxhighlight>
 
{{out}}
Sample output - first set only.
<pre>
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
</pre>
PureFox
9,476

edits

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