Prime Numbers

6.1 The quadratic sieve factorization method 271
Be that as it may, the large-prime variation does give us something that
we did not have before. By allowing sieve reports of numbers that are close to
the threshold for B-smoothness, but not quite there, we can discover numbers
that have one slightly larger prime. In fact, if a number has all the primes
up to B removed from its prime factorization, and the resulting number is
smaller than B 2 , but larger than 1, then the resulting number must be a
prime. It is this idea that is at work in the large-prime variation. Our sieve
is not perfect, since we are using approximate logarithms and perhaps not
sieving with small primes (see Section 3.2.5), but the added grayness does
not matter much in the mass of numbers being considered. Some numbers
with a large prime factor that might have been reported are possibly passed
over, and some numbers are reported that should not have been, but neither
problem is of great consequence.
So if we can obtain these numbers with a large prime factor for free, how
then can we process them in the linear algebra stage of the algorithm? In
fact, we should not view the numbers with a large prime as having longer
exponent vectors, since this could cause our matrix to be too large. There is
a very cheap way to process these large prime reports. Simply sort them on
the value of the large prime factor. If any large prime appears just once in
the sorted list, then this number cannot possibly be used to make a square
for us, so it is discarded. Say we have k reports with the same large prime:
x 2 i − n = yiP ,fori =1, 2,...,k.Then
(x1xi) 2 ≡ y1yiP 2 (mod n), for i =2,...,k.
So when k ≥ 2 we can use the exponent vectors for the k − 1numbersy1yi,
since the contribution of P 2 to the exponent vector, once it is reduced mod
2, is 0. That is, duplicate large primes lead to exponent vectors on the primes
up to B. Since it is very fast to sort a list, the creation of these new exponent
vectors is like a gift from heaven.
There is one penalty to using these new exponent vectors, though it has
not proved to be a big one. The exponent vector for a y1yi as above is usually
not as sparse as an exponent vector for a fully smooth report. Thus, the
matrix techniques that take advantage of sparseness are somewhat hobbled.
Again, this penalty is not severe, and every important implementation of the
QS method uses the large-prime variation.
One might wonder how likely it is to have a pair of large primes matching.
That is, when we sort our list, could it be that there are very few matches,
and that almost everything is discarded because it appears just once? The
birthday paradox from probability theory suggests that matches will not be
uncommon, once one has plenty of large prime reports. In fact the experience
that factorers have is that the importance of the large prime reports is nil near
the beginning of the run, because there are very few matches, but as the data
set gets larger, the effect of the birthday paradox begins, and the matches for
the large primes blossom and become a significant source of rows for the final
matrix.