Prime Numbers

6.1 The quadratic sieve factorization method 275
For numbers in the range of 50 to 150 digits, typical choices for B are in
the range 10 4 to 10 7 , approximately. It turns out that sieving is so fast an
operation, that if we changed polynomials every time we sieved B numbers,
the overhead in making the change would be so time-consuming that overall
efficiency would suffer. This overhead is principally to solve the initialization
problem. That is, given a, b, c as in Section 6.1.5, for each odd prime p ≤ B
with
n
p = 1, we have to solve the congruence
ax 2 +2bx + c ≡ 0(modp)
for the two roots r(p) modp and s(p) modp (we assume here that p does not
divide an). Thus, we have
where
r(p) =(−b + t(p))a −1 mod p, s(p) =(−b − t(p))a −1 mod p, (6.3)
t(p) 2 ≡ n (mod p).
For each polynomial, we can use the exact same residue t(p) each time when
we come to finding r(p),s(p). So the principal work in using (6.3) is in
computing a −1 mod p for each p (say by Algorithm 2.1.4) and the two mod p
multiplications. If there are many primes p for which this needs to be done,
it is enough work that we do not want to do it too frequently.
The idea of self initialization is to amortize the work in (6.3) over several
polynomials with the same value of a. For each value of a, we choose b such
that b 2 ≡ n (mod a)and0