An Empirical Comparison of the Quadratic Sieve Factoring Algorithm and the Pollard Rho Factoring Algorithm

TL;DR

This paper empirically compares the performance of the quadratic sieve and Pollard Rho algorithms for factoring large integers, analyzing their efficiency, theoretical versus actual complexity, and the impact of number size.

Contribution

It provides a practical performance comparison of two key factoring algorithms and examines how number size influences their efficiency.

Findings

Quadratic sieve outperforms Pollard Rho for numbers larger than 80 bits.

Actual time complexities differ from theoretical predictions.

Bit length significantly affects quadratic sieve's performance.

Abstract

One of the most significant challenges on cryptography today is the problem of factoring large integers since there are no algorithms that can factor in polynomial time, and factoring large numbers more than some limits(200 digits) remain difficult. The security of the current cryptosystems depends on the hardness of factoring large public keys. In this work, we want to implement two existing factoring algorithms - pollard-rho and quadratic sieve - and compare their performance. In addition, we want to analyze how close is the theoretical time complexity of both algorithms compared to their actual time complexity and how bit length of numbers can affect quadratic sieve's performance. Finally, we verify whether the quadratic sieve would do better than pollard-rho for factoring numbers smaller than 80 bits.