int
egverify(EGpub *pub, EGsig *sig, mpint *m)
{
mpint *p = pub->p, *alpha = pub->alpha;
mpint *r = sig->r, *s = sig->s;
mpint *v1, *v2, *rs;
int rv = -1;
if(mpcmp(r, mpone) < 0 || mpcmp(r, p) >= 0)
return rv;
v1 = mpnew(0);
rs = mpnew(0);
v2 = mpnew(0);
mpexp(pub->key, r, p, v1);
mpexp(r, s, p, rs);
mpmul(v1, rs, v1);
mpmod(v1, p, v1);
mpexp(alpha, m, p, v2);
if(mpcmp(v1, v2) == 0)
rv = 0;
mpfree(v1);
mpfree(rs);
mpfree(v2);
return rv;
}
DSApriv*
dsagen(DSApub *opub)
{
DSApub *pub;
DSApriv *priv;
mpint *exp;
mpint *g;
mpint *r;
int bits;
priv = dsaprivalloc();
pub = &priv->pub;
if(opub != nil){
pub->p = mpcopy(opub->p);
pub->q = mpcopy(opub->q);
} else {
pub->p = mpnew(0);
pub->q = mpnew(0);
DSAprimes(pub->q, pub->p, nil);
}
bits = Dbits*pub->p->top;
pub->alpha = mpnew(0);
pub->key = mpnew(0);
priv->secret = mpnew(0);
// find a generator alpha of the multiplicative
// group Z*p, i.e., of order n = p-1. We use the
// fact that q divides p-1 to reduce the exponent.
exp = mpnew(0);
g = mpnew(0);
r = mpnew(0);
mpsub(pub->p, mpone, exp);
mpdiv(exp, pub->q, exp, r);
if(mpcmp(r, mpzero) != 0)
sysfatal("dsagen foul up");
while(1){
mprand(bits, genrandom, g);
mpmod(g, pub->p, g);
mpexp(g, exp, pub->p, pub->alpha);
if(mpcmp(pub->alpha, mpone) != 0)
break;
}
mpfree(g);
mpfree(exp);
// create the secret key
mprand(bits, genrandom, priv->secret);
mpmod(priv->secret, pub->p, priv->secret);
mpexp(pub->alpha, priv->secret, pub->p, pub->key);
return priv;
}
// find a prime p of length n and a generator alpha of Z^*_p
// Alg 4.86 Menezes et al () Handbook, p.164
void
gensafeprime(mpint *p, mpint *alpha, int n, int accuracy)
{
mpint *q, *b;
q = mpnew(n-1);
while(1){
genprime(q, n-1, accuracy);
mpleft(q, 1, p);
mpadd(p, mpone, p); // p = 2*q+1
if(probably_prime(p, accuracy))
break;
}
// now find a generator alpha of the multiplicative
// group Z*_p of order p-1=2q
b = mpnew(0);
while(1){
mprand(n, genrandom, alpha);
mpmod(alpha, p, alpha);
mpmul(alpha, alpha, b);
mpmod(b, p, b);
if(mpcmp(b, mpone) == 0)
continue;
mpexp(alpha, q, p, b);
if(mpcmp(b, mpone) != 0)
break;
}
mpfree(b);
mpfree(q);
}
mpint*
rsaencrypt(RSApub *rsa, mpint *in, mpint *out)
{
if(out == nil)
out = mpnew(0);
mpexp(in, rsa->ek, rsa->n, out);
return out;
}
int
dsaverify(DSApub *pub, DSAsig *sig, mpint *m)
{
int rv = -1;
mpint *u1, *u2, *v, *sinv;
if(sig->r->sign < 0 || mpcmp(sig->r, pub->q) >= 0)
return rv;
if(sig->s->sign < 0 || mpcmp(sig->s, pub->q) >= 0)
return rv;
u1 = mpnew(0);
u2 = mpnew(0);
v = mpnew(0);
sinv = mpnew(0);
// find (s**-1) mod q, make sure it exists
mpextendedgcd(sig->s, pub->q, u1, sinv, v);
if(mpcmp(u1, mpone) != 0)
goto out;
// u1 = (sinv * m) mod q, u2 = (r * sinv) mod q
mpmul(sinv, m, u1);
mpmod(u1, pub->q, u1);
mpmul(sig->r, sinv, u2);
mpmod(u2, pub->q, u2);
// v = (((alpha**u1)*(key**u2)) mod p) mod q
mpexp(pub->alpha, u1, pub->p, sinv);
mpexp(pub->key, u2, pub->p, v);
mpmul(sinv, v, v);
mpmod(v, pub->p, v);
mpmod(v, pub->q, v);
if(mpcmp(v, sig->r) == 0)
rv = 0;
out:
mpfree(v);
mpfree(u1);
mpfree(u2);
mpfree(sinv);
return rv;
}
DSAsig*
dsasign(DSApriv *priv, mpint *m)
{
DSApub *pub = &priv->pub;
DSAsig *sig;
mpint *qm1, *k, *kinv, *r, *s;
mpint *q = pub->q, *p = pub->p, *alpha = pub->alpha;
int qlen = mpsignif(q);
qm1 = mpnew(0);
kinv = mpnew(0);
r = mpnew(0);
s = mpnew(0);
k = mpnew(0);
mpsub(pub->q, mpone, qm1);
// find a k that has an inverse mod q
while(1){
mprand(qlen, genrandom, k);
if((mpcmp(mpone, k) > 0) || (mpcmp(k, qm1) >= 0))
continue;
mpextendedgcd(k, q, r, kinv, s);
if(mpcmp(r, mpone) != 0)
continue;
break;
}
// make kinv positive
mpmod(kinv, qm1, kinv);
// r = ((alpha**k) mod p) mod q
mpexp(alpha, k, p, r);
mpmod(r, q, r);
// s = (kinv*(m + ar)) mod q
mpmul(r, priv->secret, s);
mpadd(s, m, s);
mpmul(s, kinv, s);
mpmod(s, q, s);
sig = dsasigalloc();
sig->r = r;
sig->s = s;
mpfree(qm1);
mpfree(k);
mpfree(kinv);
return sig;
}
EGpriv*
eggen(int nlen, int rounds)
{
EGpub *pub;
EGpriv *priv;
priv = egprivalloc();
pub = &priv->pub;
pub->p = mpnew(0);
pub->alpha = mpnew(0);
pub->key = mpnew(0);
priv->secret = mpnew(0);
gensafeprime(pub->p, pub->alpha, nlen, rounds);
mprand(nlen-1, genrandom, priv->secret);
mpexp(pub->alpha, priv->secret, pub->p, pub->key);
return priv;
}
/*
* Miller-Rabin probabilistic primality testing
* Knuth (1981) Seminumerical Algorithms, p.379
* Menezes et al () Handbook, p.39
* 0 if composite; 1 if almost surely prime, Pr(err)<1/4**nrep
*/
int
probably_prime(mpint *n, int nrep)
{
int j, k, rep, nbits, isprime;
mpint *nm1, *q, *x, *y, *r;
if(n->sign < 0)
sysfatal("negative prime candidate");
if(nrep <= 0)
nrep = 18;
k = mptoi(n);
if(k == 2) /* 2 is prime */
return 1;
if(k < 2) /* 1 is not prime */
return 0;
if((n->p[0] & 1) == 0) /* even is not prime */
return 0;
/* test against small prime numbers */
if(smallprimetest(n) < 0)
return 0;
/* fermat test, 2^n mod n == 2 if p is prime */
x = uitomp(2, nil);
y = mpnew(0);
mpexp(x, n, n, y);
k = mptoi(y);
if(k != 2){
mpfree(x);
mpfree(y);
return 0;
}
nbits = mpsignif(n);
nm1 = mpnew(nbits);
mpsub(n, mpone, nm1); /* nm1 = n - 1 */
k = mplowbits0(nm1);
q = mpnew(0);
mpright(nm1, k, q); /* q = (n-1)/2**k */
for(rep = 0; rep < nrep; rep++){
for(;;){
/* find x = random in [2, n-2] */
r = mprand(nbits, prng, nil);
mpmod(r, nm1, x);
mpfree(r);
if(mpcmp(x, mpone) > 0)
break;
}
/* y = x**q mod n */
mpexp(x, q, n, y);
if(mpcmp(y, mpone) == 0 || mpcmp(y, nm1) == 0)
continue;
for(j = 1;; j++){
if(j >= k) {
isprime = 0;
goto done;
}
mpmul(y, y, x);
mpmod(x, n, y); /* y = y*y mod n */
if(mpcmp(y, nm1) == 0)
break;
if(mpcmp(y, mpone) == 0){
isprime = 0;
goto done;
}
}
}
isprime = 1;
done:
mpfree(y);
mpfree(x);
mpfree(q);
mpfree(nm1);
return isprime;
}
// Miller-Rabin probabilistic primality testing
// Knuth (1981) Seminumerical Algorithms, p.379
// Menezes et al () Handbook, p.39
// 0 if composite; 1 if almost surely prime, Pr(err)<1/4**nrep
int
probably_prime(mpint *n, int nrep)
{
int j, k, rep, nbits, isprime = 1;
mpint *nm1, *q, *x, *y, *r;
if(n->sign < 0)
sysfatal("negative prime candidate");
if(nrep <= 0)
nrep = 18;
k = mptoi(n);
if(k == 2) // 2 is prime
return 1;
if(k < 2) // 1 is not prime
return 0;
if((n->p[0] & 1) == 0) // even is not prime
return 0;
// test against small prime numbers
if(smallprimetest(n) < 0)
return 0;
// fermat test, 2^n mod n == 2 if p is prime
x = uitomp(2, nil);
y = mpnew(0);
mpexp(x, n, n, y);
k = mptoi(y);
if(k != 2){
mpfree(x);
mpfree(y);
return 0;
}
nbits = mpsignif(n);
nm1 = mpnew(nbits);
mpsub(n, mpone, nm1); // nm1 = n - 1 */
k = mplowbits0(nm1);
q = mpnew(0);
mpright(nm1, k, q); // q = (n-1)/2**k
for(rep = 0; rep < nrep; rep++){
// x = random in [2, n-2]
r = mprand(nbits, prng, nil);
mpmod(r, nm1, x);
mpfree(r);
if(mpcmp(x, mpone) <= 0)
continue;
// y = x**q mod n
mpexp(x, q, n, y);
if(mpcmp(y, mpone) == 0 || mpcmp(y, nm1) == 0)
goto done;
for(j = 1; j < k; j++){
mpmul(y, y, x);
mpmod(x, n, y); // y = y*y mod n
if(mpcmp(y, nm1) == 0)
goto done;
if(mpcmp(y, mpone) == 0){
isprime = 0;
goto done;
}
}
isprime = 0;
}
done:
mpfree(y);
mpfree(x);
mpfree(q);
mpfree(nm1);
return isprime;
}