Elliptic Curve Digital Signature Algorithm: Difference between revisions
(Remark added, cleaner links) |
No edit summary |
||
Line 484: | Line 484: | ||
_____ |
_____ |
||
</pre> |
</pre> |
||
=={{header|FreeBASIC}}== |
=={{header|FreeBASIC}}== |
Revision as of 20:54, 17 October 2018
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”. This new page is added because modulo operations are precisely essential for implementing ECDSA: we need a discrete, finite field to play in.
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
- /
- 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; }
} </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 _____