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 $p-1$ algorithm, which finds in random polynomial time the prime divisors $p$ of an integer $n$ such that $p-1$ 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 $k$-th cyclotomic method of factoring ($k\ge 2$) devised by Bach and Shallit. We also investigate reductions of factoring to computing Euler's totient function $\phi$. We point out some explicit sets of integers $n$ that are completely factorable in deterministic polynomial time given $\phi(n)$. These sets consist, roughly speaking, of products of primes $p$ satisfying, with the exception of at most two, certain conditions somewhat weaker than the smoothness of $p-1$. Finally, we prove that $O(\ln n)$ oracle queries for values of $\phi$ are sufficient to completely factor any integer $n$ in less than $\exp\Bigl((1+o(1))(\ln n)^{{1/3}} (\ln\ln n)^{{2/3}}\Bigr)$ deterministic time.Comment: Expanded and heavily revised version, to appear in Mathematics of Computation, 21 page

An Introduction to Quantum Complexity Theory

We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight the interplay between the three scenarios and currently-known quantum algorithms.Comment: 28 pages, LaTeX, 11 figures within the text, to appear in "Collected Papers on Quantum Computation and Quantum Information Theory", edited by C. Macchiavello, G.M. Palma, and A. Zeilinger (World Scientific

Quantum Probabilistic Subroutines and Problems in Number Theory

We present a quantum version of the classical probabilistic algorithms $\grave{a}$ la Rabin. The quantum algorithm is based on the essential use of Grover's operator for the quantum search of a database and of Shor's Fourier transform for extracting the periodicity of a function, and their combined use in the counting algorithm originally introduced by Brassard et al. One of the main features of our quantum probabilistic algorithm is its full unitarity and reversibility, which would make its use possible as part of larger and more complicated networks in quantum computers. As an example of this we describe polynomial time algorithms for studying some important problems in number theory, such as the test of the primality of an integer, the so called 'prime number theorem' and Hardy and Littlewood's conjecture about the asymptotic number of representations of an even integer as a sum of two primes.Comment: 9 pages, RevTex, revised version, accepted for publication on PRA: improvement in use of memory space for quantum primality test algorithm further clarified and typos in the notation correcte