Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

TLDR

Classical digital computers are believed to efficiently simulate any physical computing device, but this assumption may fail when quantum mechanics is considered. The paper aims to address integer factorization and discrete logarithm problems, which are computationally hard on classical machines and underpin many cryptosystems. It presents efficient randomized algorithms that solve these problems on a quantum computer in polynomial time relative to the input size.

Abstract

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed to be able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems that are generally thought to be hard on classical computers and that have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, for example, the number of digits of the integer to be factored.