Complete solving the quadratic equation mod $2^n$

TL;DR

This paper provides a comprehensive analysis of quadratic equations modulo 2^n, including conditions for solutions, their count, and an efficient method to find all solutions in linear time, with implications for cryptography.

Contribution

It offers a complete characterization of solutions for quadratic equations mod 2^n, including solution existence, enumeration, and an efficient solution-finding algorithm.

Findings

Solutions exist only under specific conditions.

Number of solutions is always a power of two.

Solutions can be found in O(n) time.

Abstract

Quadratic functions have applications in cryptography. In this paper, we investigate the modular quadratic equation $$ ax^2+bx+c=0 \quad (mod \,\, 2^n), $$ and provide a complete analysis of it. More precisely, we determine when this equation has a solution and in the case that it has a solution, we not only determine the number of solutions, but also give the set of solutions in $O(n)$ time. One of the interesting results of our research is that, when this equation has a solution, then the number of solutions is a power of two.