Finding small roots for bivariate polynomials over the ring of integers

Abstract

<p style='text-indent:20px;'>In this paper, we propose the first heuristic algorithm for finding small roots for a bivariate equation modulo an ideal <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{I} $\end{document}</tex-math></inline-formula> over the ring of integers <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R} $\end{document}</tex-math></inline-formula>. Existing algorithms for solving polynomial equations with size constraints only work for bivariate modular equations over integers, and univariate modular equation over number fields.</p><p style='text-indent:20px;'>Both previous algorithms use a relation between the short vector in a skillfully structured lattice and a size constrained solution. Our algorithm also follows this framework, but we additionally use a polynomial factoring algorithm over number fields to recover a 'ring' root of a bivariate polynomial equation.</p><p style='text-indent:20px;'>As a result, when an LLL algorithm is employed to find a short vector, we can recover all small roots of a bivariate polynomial modulo <inline-formula><tex-math id="M5">\begin{document}$ \mathcal{I} $\end{document}</tex-math></inline-formula> in polynomial time under some constraint.</p>