/************************************************************************************
* TOP LEVEL FUNCTIONS *
************************************************************************************/
bool bpsw_prime_test(const unsigned long long int n)
{
if (not_base_2_strong_probable_prime(n))
return false;
unsigned long long int big_d = 5;
int neg_jacobi = ((n & 3) == 3) ? -1 : 1;
while (1) {
if (jacobi_symbol(big_d, n, 1) == -1)
return is_strong_lucas_pseudoprime(big_d, n);
big_d += 2;
if (jacobi_symbol(big_d, n, neg_jacobi) == -1)
return is_strong_lucas_pseudoprime(-big_d, n);
big_d += 2;
}
}
static int sqrtmod(giant x, curveParams *cp)
/* If Sqrt[x] (mod p) exists, function returns 1, else 0.
In either case x is modified, but if 1 is returned,
x:= Sqrt[x] (mod p).
*/
{
int rtn;
giant t0 = borrowGiant(cp->maxDigits);
giant t1 = borrowGiant(cp->maxDigits);
giant t2 = borrowGiant(cp->maxDigits);
giant t3 = borrowGiant(cp->maxDigits);
giant t4 = borrowGiant(cp->maxDigits);
giant p = cp->basePrime;
feemod(cp, x); /* Justify the argument. */
gtog(x, t0); /* Store x for eventual validity check on square root. */
if((p->n[0] & 3) == 3) { /* The case p = 3 (mod 4). */
gtog(p, t1);
iaddg(1, t1); gshiftright(2, t1);
powermodg(x, t1, cp);
goto resolve;
}
/* Next, handle case p = 5 (mod 8). */
if((p->n[0] & 7) == 5) {
gtog(p, t1); int_to_giant(1, t2);
subg(t2, t1); gshiftright(2, t1);
gtog(x, t2);
powermodg(t2, t1, cp); /* t2 := x^((p-1)/4) % p. */
iaddg(1, t1);
gshiftright(1, t1); /* t1 := (p+3)/8. */
if(isone(t2)) {
powermodg(x, t1, cp); /* x^((p+3)/8) is root. */
goto resolve;
} else {
int_to_giant(1, t2); subg(t2, t1);
/* t1 := (p-5)/8. */
gshiftleft(2,x);
powermodg(x, t1, cp);
mulg(t0, x); addg(x, x); feemod(cp, x);
/* 2x (4x)^((p-5)/8. */
goto resolve;
}
}
/* Next, handle tougher case: p = 1 (mod 8). */
int_to_giant(2, t1);
while(1) { /* Find appropriate nonresidue. */
gtog(t1, t2);
gsquare(t2); subg(x, t2); feemod(cp, t2);
if(jacobi_symbol(t2, cp) == -1) break;
iaddg(1, t1);
} /* t2 is now w^2 in F_p^2. */
int_to_giant(1, t3);
gtog(p, t4); iaddg(1, t4); gshiftright(1, t4);
powFp2(t1, t3, t2, t4, cp);
gtog(t1, x);
resolve:
gtog(x,t1); gsquare(t1); feemod(cp, t1);
if(gcompg(t0, t1) == 0) {
rtn = 1; /* Success. */
}
else {
rtn = 0; /* no square root */
}
returnGiant(t0);
returnGiant(t1);
returnGiant(t2);
returnGiant(t3);
returnGiant(t4);
return rtn;
}
