Linear Hensel Lifting for Zp[x,y] for n Factors with Cubic Cost

Abstract

We present a new algorithm for performing linear Hensel lifting on bivariate polynomials over the finite field Zp for some prime p. Our algorithm lifts n monic, univariate polynomials to recover the factors of a polynomial A(x,y) in Zp[x,y] which is monic in x, and bounded by degrees dx = deg(A,x) and dy = deg(A,y). Our algorithm improves upon Bernardin's algorithm in [1] and reduces the number of arithmetic operations in Zp from O(n dx^2 dy^2) to O(dx^2 dy + dx dy^2) for p >= dx. Experimental results in C verify that our algorithm compares favorably with Bernardin's for large degree polynomials. Moreover, we've implemented a Quadratic Hensel lifting algorithm in Magma to show that our cubic Linear Hensel lifting algorithm outperforms Magma's Quadratic Hensel lifting for a wide range of input sizes.