Prime Numbers

244 Chapter 5 EXPONENTIAL FACTORING ALGORITHMS
Such a suitable D is found very easily. Take x0 =
√
n − n2/3 ,sothatif
d = n − x 2 0,thenn 2/3 ≤ d 1 that has
no prime factors below 3n 1/3 . This algorithm decides whether n is prime or
composite, the algorithm giving in the composite case the prime factorization
of n. (Note that any nontrivial factorization must be the prime factorization,
since each prime factor of n exceeds the cube root of n.)
1. [Square test]
If n is a square, say p 2 , return the factorization p · p;
// A number may be tested for squareness via Algorithm 9.2.11.
2. [Side factorization]
√ 2 d = n − n − n2/3 ; // Thus, each prime factor of n is > 2 √ d.
if (gcd(n, d) > 1) return the factorization gcd(n, d) · (n/ gcd(n, d));
By trial division, find the complete prime factorization of d;
3. [Congruences]
for(1 ≤ a ≤ 2
d/3 ) {
Using the prime factorization of d and a method from Section 2.3.2 find
the solutions u1,...,ut of the congruence u2 ≡ 4an (mod 4d);
for(1 ≤ i ≤ t) { // If t =0this loop is not executed.
For all integers u with 0 ≤ u ≤ 2 √ an, u ≡ ui (mod 4d), use
Algorithm 9.2.11 to see whether (4an − u2 )/4d is a square;
If such a square is found, say v2 ,andu≡ ±2x0v (mod 2n), goto
[gcd computation];
}
}
return “n is prime”;
4. [gcd computation]
g =gcd(2x0v − u, n);
return the factorization g · (n/g);
// The factorization is nontrivial and the factors are primes.
Theorem 5.6.4. Consider a procedure that on input of an integer n>1
first removes from n any prime factor up to 3n 1/3 (via trial division), and
if this does not completely factor n, the unfactored portion is used as the
input in Algorithm 5.6.3. In this way, the complete prime factorization of n