A lower bound on the average entropy of a function determined up to a diagonal linear map on F_q^n

TL;DR

This paper establishes a lower bound on the average entropy of functions over finite fields when subjected to diagonal linear transformations, highlighting fundamental entropy limits in such transformations.

Contribution

It provides a novel lower bound on the average Shannon and Renyi entropy for functions under diagonal linear maps over finite fields.

Findings

Average entropy is at least about log2(q^n)-n.

Collision probability is at most about 2^n/q^n.

Results apply to any function with uniformly random diagonal linear shifts.

Abstract

In this note, it is shown that if $f\colon\efq^n\to\efq^n$ is any function and $\bA=(A_1,..., A_n)$ is uniformly distributed over $\efq^n$, then the average over $(k_1,...,k_n)\in \efq^n$ of the Renyi (and hence, of the Shannon) entropy of $f(\bA)+(k_1A_1,...,k_nA_n)$ is at least about $\log_2(q^n)-n$. In fact, it is shown that the average collision probability of $f(\bA)+(k_1A_1,...,k_nA_n)$ is at most about $2^n/q^n$.