1,028 research outputs found
A deterministic version of Pollard's p-1 algorithm
In this article we present applications of smooth numbers to the
unconditional derandomization of some well-known integer factoring algorithms.
We begin with Pollard's algorithm, which finds in random polynomial
time the prime divisors of an integer such that is smooth. We
show that these prime factors can be recovered in deterministic polynomial
time. We further generalize this result to give a partial derandomization of
the -th cyclotomic method of factoring () devised by Bach and
Shallit.
We also investigate reductions of factoring to computing Euler's totient
function . We point out some explicit sets of integers that are
completely factorable in deterministic polynomial time given . These
sets consist, roughly speaking, of products of primes satisfying, with the
exception of at most two, certain conditions somewhat weaker than the
smoothness of . Finally, we prove that oracle queries for
values of are sufficient to completely factor any integer in less
than deterministic
time.Comment: Expanded and heavily revised version, to appear in Mathematics of
Computation, 21 page
Linearizing torsion classes in the Picard group of algebraic curves over finite fields
We address the problem of computing in the group of -torsion rational
points of the jacobian variety of algebraic curves over finite fields, with a
view toward computing modular representations.Comment: To appear in Journal of Algebr
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