Prime Numbers

5.8 Research problems 257
modulo the N to be factored, then the full product has about n 2 M 2 algebraic
factors, implying, in turn, about p 1/2 /M parallel iterations for discovering a
hidden factor p. So the question comes down to this: Can one parallelize the
indicated product, using some sort of fast polynomial evaluation scheme? The
answer is yes, subject to some heuristic controversies, with details in [Crandall
1999d], where it is argued that with M machines one should be able to find a
hidden factor p in
parallel operations.
O
√p ln 2
M
M
5.26. Recall that the Pollard-rho approach to DL solving has the feature
that very little memory is required. What is more, variants of the basic rho
approach are pleasantly varied. The present exercise is to work through a very
simple such variant (that is not computationally optimized), with a view to
solving the specific DL relation
g l ≡ t (mod p),
where t and primitive root g are given as usual. First define a pseudorandom
function on residues z mod p, for example,
f(z) =2+3θ(z − p/2),
that is, f(z) =2forz