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| void tonelliShanks(mpz_t res, mpz_t n, mpz_t p) {
// check if p is prime
if (mpz_probab_prime_p(p, PRIME_TRIES) == 0) {
gmp_printf("p = %Zd is definately not prime!", p);
return;
}
// check if n is quadratic residue
if (mpz_legendre(n, p) == -1) {
gmp_printf("n = %Zd is not a quadratic residue\n", n);
return;
}
// if p = 3 mod 4 use fermat calculation
if (isXmodN(p, 3, 4) == 1) {
sqrtmodFermat(res, n, p);
return;
}
// initialize variables
mpz_t q, s, z, t, r, c, m, i, b, exp_temp;
mpz_inits(q, s, z, t, r, c, m, i, b, exp_temp, NULL);
// find Q and S such that p-1 = Q*2^S con Q dispari
factorOddPart(q, s, p);
// find a Z that is not a quadratic residue
findQuadraticNonResidue(z, p);
// t = n ** Q mod p
mpz_powm(t, n, q, p);
// r = n ** (q+1)/2 mod p
mpz_set_ui(r, 1);
mpz_add(r, q, r);
mpz_tdiv_q_ui(r, r, 2);
mpz_powm(r, n, r, p);
// c = z**q mod p
mpz_powm(c, z, q, p);
// m = s
mpz_set(m, s);
// iterate
while (mpz_cmp_ui(t, 1) != 0) {
// if t == 0 return r = 0
if (mpz_cmp_ui(t, 0) == 0) {
mpz_set_ui(res, 0);
mpz_clears(q, s, z, t, r, c, m, i, b, exp_temp, NULL);
return;
}
// repeated squaring to find the least i < 0 < M
repeatedSquaring(i, t, m, p);
// b = c ** (2**(m-i-1)) mod p
mpz_ui_pow_ui(exp_temp, 2, mpz_get_ui(m) - mpz_get_ui(i) - 1);
mpz_powm(b, c, exp_temp, p);
// r = r*b mod p
mpz_mul(r, r, b);
mpz_mod(r, r, p);
// c = b**2
mpz_powm_ui(c, b, 2, p);
// t = t * b**2
mpz_mul(t, t, c);
mpz_mod(t, t, p);
// set m = i
mpz_set(m, i);
}
// result is r (and -r mod p)
mpz_set(res, r);
mpz_clears(q, s, z, t, r, c, m, i, b, exp_temp, NULL);
}
void repeatedSquaring(mpz_t i, mpz_t t, mpz_t m, mpz_t p) {
// find the lowest i for that t ** (2**i) mod p = 1
mpz_t res, exp;
mpz_inits(exp, res, NULL);
// il più piccolo i è il primo i
for (int iter = 1; iter < mpz_get_ui(m); iter++) {
// exp = 2 ** i
mpz_ui_pow_ui(exp, 2, iter);
// res = t ** (2 ** i) mod p
mpz_powm(res, t, exp, p);
// if res == 1 return iter
if (mpz_cmp_ui(res, 1) == 0) {
mpz_set_ui(i, iter);
break;
}
}
mpz_clears(exp, res, NULL);
}
void findQuadraticNonResidue(mpz_t z, mpz_t p) {
// use random approach (efficient as 50% is !qr)
gmp_randstate_t state;
gmp_randinit_default(state);
gmp_randseed_ui(state, time(0));
do {
mpz_urandomm(z, state, p);
} while (mpz_legendre(z, p) != -1);
gmp_randclear(state);
}
void factorOddPart(mpz_t q, mpz_t s, mpz_t p) {
mpz_t p_meno_uno, due;
mpz_inits(p_meno_uno, NULL);
mpz_init_set_ui(due, 2);
// p_meno_uno = p-1
mpz_sub_ui(p_meno_uno, p, 1);
// find Q and S such that p-1 = Q*2^S con Q dispa
unsigned long int temp = mpz_remove(q, p_meno_uno, due);
mpz_set_ui(s, temp);
mpz_clears(p_meno_uno, due, NULL);
}
int isXmodN(mpz_t val, int x, int n) {
mpz_t temp;
mpz_init(temp);
// temp = p mod 4
mpz_mod_ui(temp, val, n);
int result = (mpz_cmp_ui(temp, x) == 0) ? 1 : 0;
mpz_clear(temp);
return result;
}
void sqrtmodFermat(mpz_t res, mpz_t qr, mpz_t p) {
// se p = 3 mod 4 -> p+1//4 + p+1//4 = p+1//2 = (p-1)+2//2 = (p-1)/2 + 1
// quindi radice quadrata modulare = x ^ p+1//4
mpz_t temp, exp;
mpz_inits(temp, exp, NULL);
// check p=3 mod4
mpz_mod_ui(temp, p, 4);
if (mpz_cmp_ui(temp, 3) == 0) {
// res = qr ** (p+1)//4 mod p
mpz_set(temp, p);
mpz_add_ui(temp, p, 1);
mpz_tdiv_q_ui(exp, temp, 4);
mpz_powm(res, qr, exp, p);
} else
gmp_printf("p non è = 3 mod 4 (p mod 4 = %Zd)\n", temp);
mpz_clears(temp, exp, NULL);
}
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