Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

Abstract

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruence $\text{m}\text{o}\text{d}\hspace{0.17em}p$ to polynomial congruence $\text{m}\text{o}\text{d}\hspace{0.17em}{p}^k$. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the form $a{x}^n\equiv b(\text{m}\text{o}\text{d}\hspace{0.17em}{p}^k)$, $k\ge \mathrm{1,}$ where $a,\mathrm{b,}$ and $n>0$ are integers which are not divisible by an odd prime $p$. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the form $a{x}^{-n}\equiv b(\text{m}\text{o}\text{d}\hspace{0.17em}{p}^k)$ has also been addressed.