Enhanced Algorithms for the Representation of integers by Binary Quadratic forms: Reduction to Subset Sum

TL;DR

This paper introduces efficient algorithms for representing integers with binary quadratic forms by reducing the problem to subset sum, enabling polynomial-time solutions in certain cases and improving existing methods.

Contribution

It presents a novel reduction of the quadratic form representation problem to subset sum, leading to more efficient algorithms, especially when the discriminant is polylogarithmic in m.

Findings

Polynomial-time solution when |disc(f)| = polylog(m)

Quadratic speedup over Cornacchia's method

Applicable to cryptography and elliptic curve problems

Abstract

In this paper, we present efficient algorithms for solving the Diophantine equation $f(x, y) = m$ for an arbitrary definite binary quadratic form $f$, given the factorization of $m$. While Cornacchia's algorithm to solve $x^2 + dy^2 = m$ is efficient in many cases, its runtime becomes exponentially large when $m$ is highly composite and encounters subtleties when generalized to arbitrary forms $f$. To address these issues, we give a reduction from our problem to an instance of the Subset sum, a weakly NP complete problem, allowing for more efficient solutions. Leveraging this approach, we develop deterministic algorithms that adapt to different cases based on $\mathrm{disc}(f)$ and $ m $. In particular, when $|\mathrm{disc}(f)| = \mathrm{polylog}(m) $, we provide a polynomial time solution that remains efficient regardless of the structure of $ m $. For more general cases, we present an algorithm that improves upon Cornacchia's method, achieving a quadratic speedup. Recently, the problem of representing integers by a form $ f $ found important applications in elliptic curves and isogeny based cryptography, where these algorithms are central to solving norm form equations.