11,899 research outputs found
A Generalization of Lehman's Method
A new deterministic algorithm for finding square divisors, and finding
-power divisors in general, is presented. This algorithm is based on
Lehman's method for integer factorization and is straightforward to implement.
While the theoretical complexity of the new algorithm is far from best known,
the algorithm becomes especially effective if even a loose bound on a square
divisor is known. Additionally, we answer a question by D. Harvey and M.
Hittmeir on whether their recent deterministic algorithm for integer
factorization can be adapted to finding -power divisors.Comment: 12 pages, 1 figure, 2 table
Integer factorization as subset-sum problem
This paper elaborates on a sieving technique that has first been applied in
2018 for improving bounds on deterministic integer factorization. We will
generalize the sieve in order to obtain a polynomial-time reduction from
integer factorization to a specific instance of the multiple-choice subset-sum
problem. As an application, we will improve upon special purpose factorization
algorithms for integers composed of divisors with small difference. In
particular, we will refine the runtime complexity of Fermat's factorization
algorithm by a large subexponential factor. Our first procedure is
deterministic, rigorous, easy to implement and has negligible space complexity.
Our second procedure is heuristically faster than the first, but has
non-negligible space complexity.Comment: 22 pages (including appendix
Note on Integer Factoring Methods IV
This note continues the theoretical development of deterministic integer
factorization algorithms based on systems of polynomials equations. The main
result establishes a new deterministic time complexity bench mark in integer
factorization.Comment: 20 Pages, New Versio
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
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