# Elliptic Curve Digital Signature Algorithm

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.

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

Reference.

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

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

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

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

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

## C

Parallel to: FreeBASIC

` /*subject: Elliptic curve digital signature algorithm,         toy version for small modulus N.tested : gcc 4.6.3, tcc 0.9.27*/#include <stdio.h>#include <stdlib.h>#include <time.h> // 64-bit integer typetypedef long long int dlong;// rational ec pointtypedef struct {   dlong x, y;} epnt;// elliptic curve parameterstypedef struct {   long a, b;   dlong N;   epnt G;   dlong r;} curve;// signature pairtypedef 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 modulusconst long mxN = 1073741789;// max order G = mxN + 65536const long mxr = 1073807325;// symbolic infinityconst long inf = -2147483647; // single global curvecurve e;// point at infinity zerOepnt zerO;// impossible inverse mod Nint 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 Elong 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 = zerOint isO (epnt p){return (p.x == inf) && (p.y == 0);} // return 1 if P is on curve Eint 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 additionvoid 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 kPvoid 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 fvoid 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 curveint 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 primitivepair 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 primitiveint 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 dvoid ec_dsa (long f, long d){long i, s, t;pair sg;epnt W;    // parameter check   t = (disc() == 0);   t |= isO (e.G);   pmul (&W, e.G, e.r);   t |= ! isO (W);   t |= ! ison (e.G);   if (t) goto errmsg; printf ("\nkey generation\n");   s = 1 + (long)(rnd() * (e.r - 1));   pmul (&W, e.G, s);   printf ("private key s = %d\n", s);   pprint ("public key W = sG", W);    // next highest power of 2 - 1   t = e.r;   for (i = 1; i < 32; i <<= 1)      t |= t >> i;   while (f > t) f >>= 1;   printf ("\naligned hash %x\n", f);    sg = signature (s, f);   if (inverr) goto errmsg;   printf ("signature c,d = %d, %d\n", sg.a, sg.b);    if (d > 0) {      while (d > t) d >>= 1;      f ^= d;      printf ("\ncorrupted hash %x\n", f);   }    t = verify (W, f, sg);   if (inverr) goto errmsg;   if (t)      printf ("Valid\n_____\n");   else      printf ("invalid\n_______\n");    return; errmsg:printf ("invalid parameter set\n");printf ("_____________________\n");}  void main (void){typedef long eparm[6];long d, f;zerO.x = inf; zerO.y = 0;srand(time(NULL)); // Test vectors: elliptic curve domain parameters,// short Weierstrass model y^2 = x^3 + ax + b (mod N)eparm *sp, sets[10] = {//    a,   b,  modulus N, base point G, order(G, E), cofactor   {355, 671, 1073741789, 13693, 10088, 1073807281},   {  0,   7,   67096021,  6580,   779,   16769911}, // 4   { -3,   1,     877073,     0,     1,     878159},   {  0,  14,      22651,    63,    30,        151}, // 151   {  3,   2,          5,     2,     1,          5}, // ecdsa may fail if...// the base point is of composite order   {  0,   7,   67096021,  2402,  6067,   33539822}, // 2// the given order is a multiple of the true order   {  0,   7,   67096021,  6580,   779,   67079644}, // 1// the modulus is not prime (deceptive example)   {  0,   7,     877069,     3, 97123,     877069},// fails if the modulus divides the discriminant   { 39, 387,      22651,    95,    27,      22651},};// Digital signature on message hash f,// set d > 0 to simulate corrupted data   f = 0x789abcde; d = 0;    for (sp = sets; ; sp++) {      if (ellinit (*sp))         ec_dsa (f, d);       else         break;   }} `
Output:

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

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

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

aligned hash 789abcde

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

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

## FreeBASIC

Parallel to: C

` 'subject: Elliptic curve digital signature algorithm,'         toy version for small modulus N.'tested : FreeBasic 1.05.0 'rational ec pointtype epnt   as longint x, yend type'elliptic curve parameterstype curve   as long a, b   as longint N   as epnt G   as longint rend type'signature pairtype pair   as long a, bend 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 modulusconst mxN = 1073741789'max order G = mxN + 65536const mxr = 1073807325'symbolic infinityconst inf = -2147483647 'single global curvedim shared as curve e'point at infinity zerOdim shared as epnt zerO'impossible inverse mod Ndim shared as byte inverr  'return mod(v^-1, u)Function exgcd (byval v as long, byval u as long) as longdim as long q, tdim as long r = 0, s = 1if 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 = rEnd Function 'return mod(a, N)Function modn (byval a as longint) as longint   a mod= e.N   if a < 0 then a += e.Nmodn = aEnd Function 'return mod(a, r)Function modr (byval a as longint) as longint   a mod= e.r   if a < 0 then a += e.rmodr = aEnd Function  'return the discriminant of EFunction disc as longdim 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 = zerOFunction isO (byref p as epnt) as byteisO = (p.x = inf and p.y = 0)End Function 'return -1 if P is on curve EFunction ison (byref p as epnt) as bytedim as long r, sif not isO (p) then   r = modn(e.b + p.x * modn(e.a + p.x * p.x))   s = modn(p.y * p.y)end ifison = (r = s)End Function  'full ec point additionSub 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 subif 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 ifend 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 = zerOEnd Sub 'R:= multiple kPSub 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)   wendr = sEnd Sub  'print point P with prefix fSub 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 ifEnd Sub 'initialize elliptic curveFunction ellinit (i() as long) as bytedim 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 = -1End Function  'signature primitiveFunction signature (byval s as longint, byval f as long) as pairdim as long c, d, u, u1dim as pair sgdim 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 = 0print "one-time u ="; upprint ("V = uG", V) sg.a = c: sg.b = dsignature = sgEnd Function 'verification primitiveFunction verify (byref W as epnt, byval f as long, byref sg as pair) as bytedim as long c = sg.a, d = sg.bdim as long t, c1, h1, h2dim as longint hdim as epnt V, V2verify = 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 dSub ec_dsa (byval f as long, byval d as long)dim as long i, s, tdim as pair sgdim 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  'maindim as long d, f, t, eparm(5)zerO.x = inf: zerO.y = 0randomize 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), cofactordata 355, 671, 1073741789, 13693, 10088, 1073807281data   0,   7,   67096021,  6580,   779,   16769911 '   4data  -3,   1,     877073,     0,     1,     878159data   0,  14,      22651,    63,    30,        151 ' 151data   3,   2,          5,     2,     1,          5 'ecdsa may fail if...'the base point is of composite orderdata   0,   7,   67096021,  2402,  6067,   33539822 '   2'the given order is a multiple of the true orderdata   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 discriminantdata  39, 387,      22651,    95,    27,      22651data   0, 0, 0 'Digital signature on message hash f,'set d > 0 to simulate corrupted dataf = &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 ifloop system `
Output:

(randomize 1, first set only)

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

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

aligned hash 789ABCDE

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

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

## Go

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

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

Sample run:

```Private key:
D: 25700608762903774973512323993645267346590725880891580901973011512673451968935

Public key:
X: 37298454876588653961191059192981094503652951300904260069480867699946371240473
Y: 69073688506493709421315518164229531832022167466292360349457318041854718641652

Message: Rosetta Code
Hash   : 0xe6f9ed0d

Signature:
R: 91827099055706804696234859308003894767808769875556550819128270941615405955877
S: 20295707309473352071389945163735458699476300346398176659149368970668313772860

Signature verified: true
```