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Ingy döt Net 2025-08-11 18:05:26 -07:00
parent 4d5544505c
commit 4924dd0264
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Task/Tonelli-Shanks-algorithm/PARI-GP/tonelli-shanks-algorithm.parigp Normal file
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/* Legendre symbol */
legendresymbol(a, p) =
{
lift(Mod(a, p)^((p-1)/2));
}
/* Tonelli-Shanks Algorithm */
tonelli(n, p) =
{
local(q, s, z, c, r, t, m, t2, i, b);
/* Check that n is a quadratic residue mod p */
if(legendresymbol(n, p) != 1,
error("n is not a quadratic residue mod p"));
/* Step 1: Factor out powers of 2 from p - 1 */
q = p - 1;
s = 0;
while(q % 2 == 0,
q = q/2;
s++
);
/* Special case when s == 1 */
if(s == 1,
return(lift(Mod(n, p)^((p+1)/4)))
);
/* Find a non-residue z */
z = 2;
while(z < p,
if(legendresymbol(z, p) == p - 1, break());
z++
);
/* Initialize variables */
c = lift(Mod(z, p)^q);
r = lift(Mod(n, p)^((q+1)/2));
t = lift(Mod(n, p)^q);
m = s;
/* Main loop */
while((t - 1) % p != 0,
t2 = t^2 % p;
i = 1;
while(i < m,
if((t2 - 1) % p == 0, break());
t2 = t2^2 % p;
i++
);
b = lift(Mod(c, p)^(2^(m - i - 1)));
r = (r * b) % p;
c = b^2 % p;
t = (t * c) % p;
m = i
);
return(r);
}
/* Test the functions */
print("Testing Tonelli-Shanks algorithm:");
print();
/* Test case 1 */
n = 10; p = 13;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 2 */
n = 56; p = 101;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 3 */
n = 1030; p = 10009;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 4 */
n = 44402; p = 100049;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 5 */
n = 665820697; p = 1000000009;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 6 */
n = 881398088036; p = 1000000000039;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();
/* Test case 7 */
n = 41660815127637347468140745042827704103445750172002; p = 100000000000000000000000000000000000000000000000577;
r = tonelli(n, p);
print("n = ", n, ", p = ", p);
print("roots: ", r, ", ", p - r);
print("verification: ", r, "^2 mod ", p, " = ", r^2 % p);
print();