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

Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

The elliptic curve primality proving (ECPP) algorithm is one of the current fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time O ((log N)^5) to prove the primality of N. An asymptotically fast version of it, attributed to J. O. Shallit, runs in time O ((log N)^4). The aim of this article is to describe this version in more details, leading to actual implementations able to handle numbers with several thousands of decimal digits