Prime Numbers

266 Chapter 6 SUBEXPONENTIAL FACTORING ALGORITHMS
6.1.2 Basic QS: A summary
We have described the basic QS algorithm in the above discussion. We now
give a summary description.
Algorithm 6.1.1 (Basic quadratic sieve). We are given an odd composite
number n that is not a power. This algorithm attempts to give a nontrivial
factorization of n.
1. [Initialization]
B = L(n) 1/2 ; // Or tune B to taste.
Set p1 =2and a1 =1;
Find the odd primes p ≤ B for which
n
p =1, and label them p2,...,pK;
for(2 ≤ i ≤ K) find roots ±ai with a2 i ≡ n (mod pi);
// Find such roots via Algorithm 2.3.8 or 2.3.9.
2. [Sieving]
Sieve the sequence (x 2 −n), x = ⌈ √ n⌉ , ⌈ √ n⌉+1,...for B-smooth values,
until K +1such pairs (x, x 2 − n) are collected in a set S;
// See Sections 3.2.5, 3.2.6, and remarks (2), (3), (4).
3. [Linear algebra]
for((x, x2 − n) ∈ S) {
Establish prime factorization x2 − n = K i=1 pei i ;
v(x2 − n) =(e1,e2,...,eK); // Exponent vector.
}
Form the (K +1)×K matrix with rows being the various vectors v(x2 −n)
reduced mod 2;
Use algorithms of linear algebra to find a nontrivial subset of the rows of
the matrix that sum to the 0-vector (mod 2), sayv(x1)+v(x2)+···+
v(xk) =0;
4. [Factorization]
x = x1x2 ···xk mod n;
y = (x2 1 − n)(x22 − n) ...(x2 k − n) modn;
// Infer this root directly from the known prime factorization of the
perfect square (x2 1 − n)(x2 2 − n) ...(x2 k − n), see remark (6).
d =gcd(x−y, n);
return d;
There are several points that should be made about this algorithm:
(1) In practice, people generally use a somewhat smaller value of B than that
given by the formula in Step [Initialization]. Any value of B of order of
magnitude L(n) 1/2 will lead to the same overall complexity, and there
are various practical issues that mitigate toward a smaller value, such as
the size of the matrix that one deals with in Step [Linear algebra], and
the size of the moduli one sieves with in comparison to cache size on the
machine used in Step [Sieving]. The optimal B-value is more of an art
than a science, and is perhaps best left to experimentation.