Prime Numbers

270 Chapter 6 SUBEXPONENTIAL FACTORING ALGORITHMS
For a discussion of the conjugate gradient method and the Lanczos
method, see [Odlyzko 1985]. For a study of the Lanczos method in a theoretical
setting see [Teitelbaum 1998]. For some practical improvements to the Lanczos
method see [Montgomery 1995].
6.1.4 Large prime variations
As discussed above and in Section 3.2.5, sieving is a very cheap operation.
Unlike trial division, which takes time proportional to the number of trial
divisors, that is, one “unit” of time per prime used as a trial, sieving takes less
and less time per prime sieved as the prime modulus grows. In fact the time
spent per sieve location, on average, for each prime modulus p is proportional
to 1/p. However, there are hidden costs for increasing the list of primes p with
which we sieve. One is that it is unlikely we can fit the entire sieve array into
memory on a computer, so we segment it. If a prime p exceeds the length of
thispartofthesieve,wehavetospendaunitoftimepersegmenttosee
whether this prime will “hit” something or not. Thus, once the prime exceeds
this threshold, the 1/p “philosophy” of the sieve is left behind, and we spend
essentially the same time for each of these larger primes: Sieving begins to
resemble trial division. Another hidden cost is perhaps not so hidden at all.
When we turn to the linear-algebra stage of the algorithm, the matrix will be
that much bigger if more primes are used. Suppose we are using 10 6 primes, a
number that is not inconceivable for the sieving stage. The matrix, if encoded
as a binary (0,1) matrix, would have 10 12 bits. Indeed, this would be a large
object on which to carry out linear algebra! In fact, some of the linear algebra
routines that will be used, see Section 6.1.3, involve a sparse encoding of the
matrix, namely, a listing of where the 1’s appear, since almost all of the entries
are 0’s. Nevertheless, space for the matrix is a worrisome concern, and it puts
a limit on the size of the smoothness bound we take.
The analysis in Section 6.1.1 indicates a third reason for not taking
the smoothness bound too large; namely, it would increase the number of
reports necessary to find a linear dependency. Somehow, though, this reason
is specious. If there is already a dependency around with a subset of our data,
having more data should not destroy this, but just make it a bit harder to
find, perhaps. So we should not take an overshooting of the smoothness bound
as a serious handicap if we can handle the two difficulties mentioned in the
above paragraph.
In its simplest form, the large-prime variation allows us a cheap way to
somewhat increase our smoothness bound, by giving us for free many numbers
that are almost B-smooth, but fail because they have one larger prime factor.
This larger prime could be taken in the interval (B,B 2 ]. It should be noted
from the very start that allowing for numbers that are B-smooth except for
having one prime factor in the interval (B,B 2 ]isnot the same as taking
B 2 -smooth numbers. With B about L(n) 1/2 , as suggested in Section 6.1.1, a
typical B 2 -smooth number near n 1/2+ɛ in fact has many prime factors in the
interval (B,B 2 ], not just one.