Efficiently Checking Separating Indeterminates

TL;DR

This paper develops fast algorithms for verifying specific variable elimination conditions in polynomial ideals, enhancing techniques for algebraic elimination and cryptanalysis of AES variants.

Contribution

It introduces efficient algorithms to check for $Z$-separating re-embeddings, improving the process of polynomial ideal elimination and cryptanalysis.

Findings

Algorithms effectively verify $Z$-separating tuples in polynomial ideals.

Extended methods apply to Boolean polynomials in cryptanalysis.

Applied to faster cryptanalysis of reduced AES versions.

Abstract

In this paper we continue the development of a new technique for computing elimination ideals by substitution which has been called $Z$-separating re-embeddings. Given an ideal $I$ in the polynomial ring $K[x_1,\dots,x_n]$ over a field $K$, this method searches for tuples $Z=(z_1,\dots,z_s)$ of indeterminates with the property that $I$ contains polynomials of the form $f_i = z_i - h_i$ for $i=1,\dots,s$ such that no term in $h_i$ is divisible by an indeterminate in $Z$. As there are frequently many candidate tuples $Z$, the task addressed by this paper is to efficiently check whether a given tuple $Z$ has this property. We construct fast algorithms which check whether the vector space spanned by the generators of $I$ or a somewhat enlarged vector space contain the desired polynomials $f_i$. We also extend these algorithms to Boolean polynomials and apply them to cryptoanalyse round reduced versions of the AES cryptosystem faster.