3 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
Majorana neutrino mass matrix with CP symmetry breaking
From the new existing data with not vanishing theta13 mixing angle we
determine the possible shape of the Majorana neutrino mass matrix. We assume
that CP symmetry is broken and all Dirac and Majorana phases are taken into
account. Two possible approaches "bottom-up" and "top down" are presented. The
problem of unphysical phases is examined in detail.Comment: 6 pages, 2 figures, presented at the XXXV International Conference of
Theoretical Physics "Matter to the Deepest 2011", Ustron, Poland, September
12-18, 201