Prime Numbers

4.4 Gauss and Jacobi sums 197
1. [Initialize]
I = −2;
2. [Preparation]
I = I +4;
Find the prime factors of I by trial division, but if I is not squarefree, goto
[Preparation];
Set F equal to the product of the primes q with q − 1|I, but if F 2 ≤ n
goto [Preparation]; // Now I,F are squarefree, and F> √ n.
If n is a prime factor of IF, return “n is prime”;
If gcd(n, IF ) > 1, return “n is composite”;
for(prime q|F ) find the least positive primitive root gq for q;
3. [Probable-prime computation]
for(prime p|I) factor n p−1 − 1=p sp up where p does not divide up;
for(primes p, q with p|I, q|F, p|q − 1) {
Find the first positive integer w(p, q) ≤ sp with
G(p, q) pw(p,q) up ≡ ζ j p (mod n) for some integer j,
but if no such number w(p, q) is found, return “n is composite”;
} // Compute symbolically in the ring Z[ζp,ζq] (see text).
4. [Maximal order search]
for(prime p|I) setw(p) equal to the maximum of w(p, q) over all primes
q|F with p|q − 1, andsetq0(p) equal to the least such prime q with
w(p) =w(p, q);
for(primes p, q with p|I, q|F, p|q − 1) find an integer l(p, q) ∈ [0,p− 1]
with G(p, q) pw(p) up l(p,q)
≡ ζp (mod n);
5. [Coprime check]
for(primes p with p|I) {
H = G(p, q0(p)) pw(p)−1up mod n;
for(0 ≤ j ≤ p − 1) {
if(gcd n, c(H − ζj p) > 1) return “n is composite”;
} // Notation as in Definition 4.4.4.
}
6. [Divisor search]
l(2) = 0;
for(odd prime q|F ) use the Chinese remainder theorem (see Theorem 2.1.6)
to construct an integer l(q) with
l(q) ≡ l(p, q) (modp) for each prime p|q − 1;
Use the Chinese remainder theorem to construct an integer l with
l ≡ g l(q)
q
(mod q) for each prime q|F ;
for(1 ≤ j < I) if l j mod F is a nontrivial factor of n, return “n is
composite”;
return “n is prime”;