Prime Numbers

6.2 Number field sieve 293
Algorithm 6.2.5 (Number field sieve). We are given an odd composite
number n that is not a power. This algorithm attempts to find a nontrivial
factorization of n.
1. [Setup]
d = (3 ln n/ ln ln n) 1/3 ; // This d has d2d2 Bsuch that R(qj) contains some
element sj with f ′ (sj) ≡ 0(modqj), storing the k pairs (qj,sj);
B ′ =
p≤B #R(p);
V =1+π(B)+B ′ + k;
M = B;
2. [The sieve]
UseasievetofindasetS ′ of coprime integer pairs (a, b) with 0 < |a|,b≤
M, andF (a, b)G(a, b) being B-smooth, until #S ′ >V, or failing this,
increase M and try again, or goto [Setup] and increase B;
3. [The matrix]
// We shall build a V × #S ′ binary matrix, one row per (a, b) pair.
// We shall compute v(a−bα), the binary exponent vector for a−bα
having V bits (coordinates) as follows:
Set the first bit of v to1ifG(a, b) < 0, else set this bit to 0;
// The next π(B) bits depend on the primes p ≤ B: Definepγas the power of p in the prime factorization of |G(a, b)|.
Set the bit for p to1ifγ is odd, else set this bit to 0;
// The next B ′ bits are to correspond to the pairs p, r where p is
a prime not exceeding B and r ∈ R(p). We use the notation
vp,r(a − bα) defined prior to Lemma 6.2.1.
Set the bit for p, r to1ifvp,r(a − bα) is odd, else set it to 0;
// Next, the last k bits correspond to the pairs qj,sj.
Set the bit for qj,sj to1if a−bsj
qj
is −1, else set it to 0;