Elliptic Curve Digital Signature Algorithm

From Rosetta Code
Elliptic Curve Digital Signature Algorithm is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Elliptic curves.

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

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

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

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

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

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

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

   with

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

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


Elliptic curve digital signature algorithm.

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

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

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

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

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


Task.

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


Reference.

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

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

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

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

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



C

Parallel to: FreeBASIC <lang c> /* subject: Elliptic curve digital signature algorithm, 

        toy version for small modulus N.

tested : gcc 4.6.3, tcc 0.9.27

  • /
  1. include <stdio.h>
  2. include <stdlib.h>
  3. 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;
  }

} </lang>

Output:

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

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

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

aligned hash 789abcde

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

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


FreeBASIC

Parallel to: C <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 </lang>

Output:

(randomize 1, first set only) 

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

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

aligned hash 789ABCDE

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

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

Go

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

}</lang>

Output:

Sample run: 

Private key:
D: 25700608762903774973512323993645267346590725880891580901973011512673451968935

Public key:
X: 37298454876588653961191059192981094503652951300904260069480867699946371240473
Y: 69073688506493709421315518164229531832022167466292360349457318041854718641652

Message: Rosetta Code
Hash   : 0xe6f9ed0d

Signature:
R: 91827099055706804696234859308003894767808769875556550819128270941615405955877
S: 20295707309473352071389945163735458699476300346398176659149368970668313772860

Signature verified: true

Julia

<lang julia>module ToyECDSA

using SHA

import Base.in, Base.==, Base.+, Base.*

export ECDSA_Key, ECDSA_Public_Key, genkey, ECDSA_sign, isverifiedECDSA

  1. 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) * inverse_mod(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

function inverse_mod(a::Number, m::Number) 

   if a < 0 || m <= a
       a = mod1(a, m)
   end
   c, d, r, s = a, m, 0, 1
   while c != 0
       q = d ÷ c
       d, c = c, d - q * c
       temp = 
       r, s = s, r - q * s
   end
   return d > 0 ? r : r + m

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) 

 p = point_a.curve.p
 a = point_a.curve.a
 l = mod((3 * point_a.x^2 + a) * inverse_mod(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)

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((inverse_mod(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 = inverse_mod(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))

</lang>

Output:
ECDSA of message <Bill says this is an elliptic curve digital signature algorithm.> verified: true
ECDSA of message <Bill says this isn't an elliptic curve digital signature algorithm.> verified: false

Phix

Translation of: FreeBASIC

<lang Phix>enum X, Y -- rational ec point enum A, B, N, G, R -- elliptic curve parameters 

                   -- also signature pair(A,B)

constant mxN = 1073741789 -- maximum modulus constant mxr = 1073807325 -- max order G = mxN + 65536 constant inf = -2147483647 -- symbolic infinity

sequence e = {0,0,0,{0,0},0} -- single global curve constant zerO = {inf,0} -- point at infinity zerO

bool inverr -- impossible inverse mod N

function exgcd(atom v, u) -- return mod(v^-1, u) atom q, t, r = 0, s = 1 

   if v<0 then v += u end if

   while v do
       q = floor(u/v)
       t = u-q*v
       u = v
       v = t
       t = r-q*s
       r = s
       s = t
   end while

   if u!=1 then
       printf(1," impossible inverse mod N, gcd = %d\n",{u})
       inverr = true
   end if

   return r

end function

function modn(atom a) -- return mod(a, N) 

   a = mod(a,e[N])
   if a<0 then a += e[N] end if
   return a

end function

function modr(atom a) -- return mod(a, r) 

   a = mod(a,e[R])
   if a<0 then a += e[R] end if
   return a

end function

function disc() -- return the discriminant of E 

   atom a = e[A], b = e[B],
        c = 4*modn(a*modn(a*a))
   return modn(-16*(c+27*modn(b*b)))

end function

function isO(sequence p) -- return true if P = zerO 

   return (p[X]=inf and p[Y]=0)

end function

function ison(sequence p) -- return true if P is on curve E 

   atom r = 0, s = 0
   if not isO(p) then
       r = modn(e[B]+p[X]*modn(e[A]+p[X]*p[X]))
       s = modn(p[Y]*p[Y])
   end if
   return (r=s)

end function

procedure pprint(string f, sequence p) -- print point P with prefix f 

   if isO(p) then
       printf(1,"%s (0)\n",{f})
   else
       atom y = p[Y]
       if y>e[N]-y then y -= e[N] end if
       printf(1,"%s (%d,%d)\n",{f,p[X],y})
   end if

end procedure

function padd(sequence p, q) -- full ec point addition atom la, t 

   if isO(p) then return q end if
   if isO(q) then return p end if

   if p[X]!=q[X] then --                   R := P + Q
       t = p[Y]-q[Y]
       la = modn(t*exgcd(p[X]-q[X], e[N]))

   else --                                 P = Q, R := 2P
       if (p[Y]=q[Y]) and (p[Y]!=0) then
           t = modn(3*modn(p[X]*p[X])+e[A])
           la = modn(t*exgcd(2*p[Y], e[N]))
       else
           return zerO --                  P = -Q, R := O
       end if
   end if

   t = modn(la*la-p[X]-q[X])
   sequence r = zerO
   r[Y] = modn(la*(p[X]-t)-p[Y])
   r[X] = t
   if inverr then r = zerO end if
   return r

end function

function pmul(sequence p, atom k) -- R:= multiple kP 

   sequence s = zerO, q = p

   while k do
       if and_bits(k,1) then
           s = padd(s, q)
       end if
       if inverr then s = zerO; exit end if
       k = floor(k/2)
       q = padd(q, q)
   end while
   return s

end function

function ellinit(sequence i) -- initialize elliptic curve atom a = i[1], b = i[2] 

   inverr = false
   e[N] = i[3]

   if (e[N]<5) or (e[N]>mxN) then return 0 end if

   e[A] = modn(a)
   e[B] = modn(b)
   e[G][X] = modn(i[4])
   e[G][Y] = modn(i[5])
   e[R] = i[6]

   if (e[R]<5) or (e[R]>mxr) then return 0 end if

   printf(1,"E: y^2 = x^3 + %dx + %d (mod %d)\n",{a,b,e[N]})
   pprint("base point G", e[G])
   printf(1,"order(G, E) = %d\n",{e[R]})

   return -1

end function

function signature(atom s, f) -- signature primitive atom c, d, u, u1 sequence V 

   printf(1,"signature computation\n")
   while true do
       while true do

-- u = rand(e[R]-1) 

           u = 571533488       -- match FreeBASIC output

-- u = 605163545 -- match C output 

           V = pmul(e[G], u)
           c = modr(V[X])
           if c!=0 then exit end if
       end while

       u1 = exgcd(u, e[R])
       d = modr(u1*(f+modr(s*c)))
       if d!=0 then exit end if
   end while
   printf(1,"one-time u = %d\n",u)
   pprint("V = uG", V)
   return {c,d}

end function

function verify(sequence W, atom f, sequence sg) -- verification primitive atom c = sg[A], d = sg[B], 

    t, c1, h1, h2, h

sequence V, V2 

  --domain check
   t = (c>0) and (c<e[R])
   t = t and (d>0) and (d<e[R])
   if not t then return 0 end if

   printf(1,"\nsignature verification\n")
   h = exgcd(d, e[R])
   h1 = modr(f*h)
   h2 = modr(c*h)
   printf(1,"h1,h2 = %d,%d\n",{h1,h2})
   V = pmul(e[G], h1)
   V2 = pmul(W, h2)
   pprint("h1G", V)
   pprint("h2W", V2)
   V = padd(V, V2)
   pprint("+ =", V)
   if isO(V) then return 0 end if
   c1 = modr(V[X])
   printf(1,"c' = %d\n",c1)

   return (c1=c)

end function

procedure errmsg() 

   printf(1,"invalid parameter set\n")
   printf(1,"_____________________\n")

end procedure

procedure ec_dsa(atom f, d) -- digital signature on message hash f, error bit d atom i, s, t sequence sg, W 

  --parameter check
   t = (disc()=0)
   t = t or isO(e[G])
   W = pmul(e[G], e[R])
   t = t or not isO(W)
   t = t or not ison(e[G])
   if t then errmsg() return end if

   puts(1,"\nkey generation\n")

-- s = rand(e[R] - 1) 

   s = 509100772       -- match FreeBASIC output

-- s = 1343570 -- match C output 

   W = pmul(e[G], s)
   printf(1,"private key s = %d\n",{s})
   pprint("public key W = sG", W)

  --next highest power of 2 - 1
   t = e[R]
   i = 1
   while i<32 do
       t = or_bits(t,floor(t/power(2,i)))
       i *= 2
   end while
   while f>t do
       f = floor(f/2)
   end while
   printf(1,"\naligned hash %x\n\n",{f})

   sg = signature(s, f)
   if inverr then errmsg() return end if
   printf(1,"signature c,d = %d,%d\n",sg)

   if d>0 then
       while d>t do
           d = floor(d/2)
       end while
       f = xor_bits(f,d)
       printf(1,"corrupted hash %x\n",{f})
   end if

   t = verify(W, f, sg)
   if inverr then errmsg() return end if
   if t then
       printf(1,"Valid\n_____\n")
   else
       printf(1,"invalid\n_______\n")
   end if

end procedure

--Test vectors: elliptic curve domain parameters, --short Weierstrass model y^2 = x^3 + ax + b (mod N)

constant tests = { -- a, b, modulus N, base point G, order(G, E), cofactor 

                 {355, 671, 1073741789, 13693, 10088, 1073807281},
                 {  0,   7,   67096021,  6580,   779,   16769911}, --   4
                 { -3,   1,     877073,     0,     1,     878159},
                 {  0,  14,      22651,    63,    30,        151}, -- 151
                 {  3,   2,          5,     2,     1,          5},

                   --ecdsa may fail if...
                   --the base point is of composite order
                 {  0,   7,   67096021,  2402,  6067,   33539822}, --   2
                   --the given order is a multiple of the true order
                 {  0,   7,   67096021,  6580,   779,   67079644}, --   1
                   --the modulus is not prime (deceptive example)
                 {  0,   7,     877069,     3, 97123,     877069},
                   --fails if the modulus divides the discriminant
                 { 39, 387,      22651,    95,    27,      22651}}

--Digital signature on message hash f, --set d > 0 to simulate corrupted data atom f = #789ABCDE, 

    d = 0

if machine_bits()!=64 then crash("needs 64 bit") end if --for i=1 to length(tests) do for i=1 to 1 do 

   if not ellinit(tests[i]) then exit end if
   ec_dsa(f, d)

end for</lang>

Output:

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

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

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

aligned hash 789ABCDE

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

signature verification
h1,h2 = 667118700,709185150
h1G (315367421,343743703)
h2W (1040319975,-262613483)
+ = (896670665,183547995)
c' = 896670665
Valid
_____
Retrieved from "https://rosettacode.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm?oldid=285924"