Additive index and Carlitz rank

TL;DR

The paper compares complexity measures for functions over finite fields, showing that certain measures cannot both be small, and explores their relationships and implications for cryptographic strength.

Contribution

It demonstrates the incompatibility of small Carlitz rank and additive index, and analyzes their relationships with degree and weight, providing new insights into cryptographic function complexity.

Findings

Carlitz rank and additive index cannot both be small simultaneously.

A function related to discrete logarithm has large degree, weight, additive index, and Carlitz rank.

The measures detect different cryptographic weaknesses in functions.

Abstract

We compare several complexity measures for self-mappings of finite fields. In particular, we show that Carlitz rank and additive index cannot be small simultaneously up to trivial exceptions. That is, these two measures detect cryptographic weaknesses of different classes of functions. We also study the relationship between additive index and degree or weight, respectively, complementing earlier results of Aksoy et al. and G\'omez-P\'erez et al. on the relationship between Carlitz rank and degree or weight, respectively. Finally, we show that a function closely related to the discrete logarithm provides an example in which all four complexity measures, degree, weight, additive index and Carlitz rank, are large.