An Algebraic-Geometry Approach to Prime Factorization

TL;DR

This paper introduces an algebraic-geometry approach to prime factorization, reducing the problem to finding points on parametrizable varieties over finite fields, with potential implications for cryptography and quantum computing.

Contribution

It proposes a novel algebraic-geometry framework for prime factorization, constructing specific varieties with efficient point evaluation methods, advancing classical complexity analysis.

Findings

Constructed varieties with points evaluable in sublinear parameters

Demonstrated varieties with points evaluable in n/2 and n/3 parameters

Provided insights into algebraic structures relevant to factorization complexity

Abstract

New algorithms for prime factorization that outperform the existing ones or take advantage of particular properties of the prime factors can have a practical impact on present implementations of cryptographic algorithms that rely on the complexity of factorization. Currently used keys are chosen on the basis of the present algorithmic knowledge and, thus, can potentially be subject to future breaches. For this reason, it is worth to investigate new approaches which have the potentiality of giving a computational advantage. The problem has also relevance in quantum computation, as an efficient quantum algorithm for prime factorization already exists. Thus, better classical asymptotic complexity can provide a better understanding of the advantages offered by quantum computers. In this paper, we reduce the factorization problem to the search of points of parametrizable varieties, in particular curves, over finite fields. The varieties are required to have an arbitrarily large number of intersection points with some hypersurface over the base field. For a subexponential or poly- nomial factoring complexity, the number of parameters have to scale sublinearly in the space dimension n and the complexity of computing a point given the parameters has to be subexponential or polynomial, respectively. We outline a procedure for building these varieties, which is illustrated with two constructions. In one case, we show that there are varieties whose points can be evaluated efficiently given a number of parameters not greater than n/2. In the other case, the bound is dropped to n/3. Incidentally, the first construction resembles a kind of retro-causal model. Retro-causality is considered one possible explanation of quantum weirdness.