Probability Analysis and Comparison of Well-Known Integer Factorization Algorithms

TL;DR

This paper analyzes and compares the success probabilities of well-known integer factorization algorithms, specifically the general integer sieve, quadratic sieve, and elliptic curve methods, highlighting the efficiency of elliptic curve factorization.

Contribution

It provides a probability-based analysis of these algorithms and demonstrates the superior asymptotic efficiency of elliptic curve factorization under certain assumptions.

Findings

Elliptic curve method is a probabilistic polynomial time algorithm.

Elliptic curve method is more likely to succeed and faster asymptotically.

Heuristic estimates compare the success probabilities of different algorithms.

Abstract

Two prominent methods for integer factorization are those based on general integer sieve and elliptic curve. The general integer sieve method can be specialized to quadratic integer sieve method. In this paper, a probability analysis for the success of these methods is described, under some reasonable conditions. The estimates presented are specialized for the elliptic curve factorization. These methods are compared through heuristic estimates. It is shown that the elliptic curve method is a probabilistic polynomial time algorithm under the assumption of uniform probability distribution for the arising group orders and clearly more likely to succeed, faster asymptotically.