Prime Numbers

5.8 Research problems 259
In judging the efficacy of such a factoring method, one should address at
least the following questions. How, in this case, do we find an initial point
(x0,y0,w0,z0) in the group? How many field operations are required for point
doubling, and for arbitrary point addition?
Explore any algebraic connections of the circle and hyperspherical groups
(and perhaps further relatives of these) with groups of matrices (mod p).
For example, all n × n matrices having determinant 1 modulo p form a
group that can for better or worse be used to forge some kind of factoring
algorithm. These relations are well known, including yet more relations with
so-called cyclotomic factoring. But an interesting line of research is based on
this question: How do we design efficient factoring algorithms, if any, using
these group/matrix ideas? We already know that complex multiplication, for
example, can be done in three multiplies instead of four, and large-matrix
multiplication can be endowed with its own special speedups, such as Strassen
recursion [Crandall 1994b] and number-theoretical transform acceleration
[Yagle 1995]; see Exercise 9.84.
5.29. Investigate the possibility of modifying the polynomial evaluation
method of Pollard and Strassen for application to the factorization of Fermat
numbers Fn =22n + 1. Since we may restrict factor searches to primes of the
form p = k2n+2 + 1, consider the following approach. Form a product
P =
ki2 n+2 +1
i
(all modulo Fn), where the {ki} constitute some set of cleverly chosen integers,
with a view to eventual taking of gcd(Fn,P). The Pollard–Strassen notion of
evaluating products of consecutive integers is to be altered: Now we wish to
form the product over a special multiplier set. So investigate possible means
for efficient creation of P . There is the interesting consideration that we should
be able somehow to presieve the {ki}, or even to alter the exponents n +2
in some i-dependent manner. Does it make sense to describe the multiplier
set {ki} as a union of disjoint arithmetic progressions (as would result from a
presieving operation)? One practical matter that would be valuable to settle is
this: Does a Pollard–Strassen variant of this type have any hope of exceeding
the performance of direct, conventional sieving (in which one simply checks
22n (mod p) for various p = k2n+2 + 1)? The problem is not without merit,
since beyond F20 or thereabouts, direct sieving has been the only recourse to
date for discovering factors of the mighty Fn.