Higher Order Differentiation over Finite Fields with Applications to Generalising the Cube Attack

TL;DR

This paper extends the concept of higher order differentiation from binary fields to larger finite fields, developing new cryptanalytic techniques like a generalized cube attack applicable to $GF(p)$ and $GF(p^m)$, with theoretical foundations and practical implications.

Contribution

It introduces a novel approach to higher order differentiation over $GF(p)$ and $GF(p^m)$, generalizing the cube attack beyond binary fields and establishing reduction techniques between different field sizes.

Findings

Differentiation over $GF(p)$ can be systematically analyzed and applied to cryptanalysis.

The generalized cube attack can differentiate multiple times per variable, unlike classical methods.

Cube attacks over $GF(p^m)$ are equivalent to those over $GF(p)$ through reduction.

Abstract

Higher order differentiation was introduced in a cryptographic context by Lai. Several attacks can be viewed in the context of higher order differentiations, amongst them the cube attack and the AIDA attack. All of the above have been developed for the binary case. We examine differentiation in larger fields, starting with the field $GF(p)$ of integers modulo a prime $p$. We prove a number of results on differentiating polynomials over such fields and then apply these techniques to generalising the cube attack to $GF(p)$. The crucial difference is that now the degree in each variable can be higher than one, and our proposed attack will differentiate several times with respect to each variable (unlike the classical cube attack and its larger field version described by Dinur and Shamir, both of which differentiate at most once with respect to each variable). Finally we describe differentiation over finite fields $GF(p^m)$ with $p^m$ elements and prove that it can be reduced to differentiation over $GF(p)$, so a cube attack over $GF(p^m)$ would be equivalent to cube attacks over $GF(p)$.