A New Hyperbola based Approach to factoring Integers

Abstract

From the results in the literature, the algebraic set of the hyperbola with parameter $n$ defined by $\mathcal{B}_{n}(X, Y, Z)_{\mid_{x\geq 4n}}= \displaystyle \lbrace \left(X: Y: Z\right)\in \mathbb{P}^{2}(\mathbb{Q}) \ \vert \ \displaystyle Y^{2}=X^{2}-4nXZ \rbrace$ where $n$ is a semiprime is proved to be in relation with prime factors of $n$. In the affine space over $\mathbb{Z}_{\geqslant 4n}\times \mathbb{Z}_{\geqslant 0}$, this set has exactly 5 points $\displaystyle\lbrace P_{0}, P_{1}, P_{2}, P_{3}, P_{4} \rbrace$ with $P_{2}+P_{3}=P_{1}+2P_{2}=P_{4}$ for which knowledge of $P_{2}$ or $P_{3}$ yields the factorization of $n$. However, The non cyclicity of this group structure over rationals and integers and moreover its non good reduction over finite fields constitute the main difficulty in finding its solutions. In this paper we describe an approach to finding $P_{2}$ and $P_{3}$. We introduce the concept of Hyperbola X-root and Y-root that the solution's greatest common divisors with $n$ reveal prime factors of $n$. We prove that $P_{2}$ and $P_{3}$ can be found on a singular Weierstrass curve isomorphic to a Jacobi quartic using the Hyperbola X-root and Y-root. We present the mathematical framework for this approach.