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)


Contents


C[edit]

Parallel to: FreeBASIC

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

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

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

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

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

aligned hash 789abcde

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

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


FreeBASIC[edit]

Parallel to: C

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

(randomize 1, first set only)

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

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

aligned hash 789ABCDE

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

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

Go[edit]

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

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

Sample run:

Private key:
D: 25700608762903774973512323993645267346590725880891580901973011512673451968935

Public key:
X: 37298454876588653961191059192981094503652951300904260069480867699946371240473
Y: 69073688506493709421315518164229531832022167466292360349457318041854718641652

Message: Rosetta Code
Hash   : 0xe6f9ed0d

Signature:
R: 91827099055706804696234859308003894767808769875556550819128270941615405955877
S: 20295707309473352071389945163735458699476300346398176659149368970668313772860

Signature verified: true