# Entropy modulo a prime

## TL;DR

This paper defines and explores a novel concept of entropy modulo a prime p, extending classical entropy ideas into modular arithmetic and connecting it with polylogarithm identities.

## Contribution

It introduces a unique definition of entropy in modular arithmetic, characterizes it via a functional equation, and links it to polylogarithm-related identities.

## Key findings

- Entropy mod p is uniquely characterized by a functional equation.
- Connections established between real entropy residues and entropy mod p.
- Entropy mod p can be expressed as a polynomial satisfying specific identities.

## Abstract

Building on work of Kontsevich, we introduce a definition of the entropy of a finite probability distribution in which the "probabilities" are integers modulo a prime p. The entropy, too, is an integer mod p. Entropy mod p is shown to be uniquely characterized by a functional equation identical to the one that characterizes ordinary Shannon entropy. We also establish a sense in which certain real entropies have residues mod p, connecting the concepts of entropy over R and over Z/pZ. Finally, entropy mod p is expressed as a polynomial which is shown to satisfy several identities, linking into work of Cathelineau, Elbaz-Vincent and Gangl on polylogarithms.