Sidon sets and statistics of the ElGamal function

TL;DR

This paper investigates the ElGamal function's randomness properties, providing experimental and theoretical evidence that it behaves like a random permutation and is equidistributed, though full pseudorandomness remains unproven.

Contribution

It offers the first experimental and theoretical evidence suggesting the ElGamal map exhibits randomness and equidistribution properties, using Sidon set theory.

Findings

Experiments indicate the ElGamal map acts like a uniformly random permutation.

Sidon set theory shows the map's graph is equidistributed.

Open question remains on proving pseudorandomness of the map.

Abstract

In the ElGamal signature and encryption schemes, an element $x$ of the underlying group $G = \mathbb{Z}_p^\times = \{1, \ldots, p-1 \}$ for a prime $p$ is also considered as an exponent, for example in $g^x$, where $g$ is a generator of G. This ElGamal map $x \mapsto g^x$ is poorly understood, and one may wonder whether it has some randomness properties. The underlying map from $G$ to $\mathbb{Z}_{p-1}$ with $x \mapsto x$ is trivial from a computer science point of view, but does not seem to have any mathematical structure. This work presents two pieces of evidence for randomness. Firstly, experiments with small primes suggest that the map behaves like a uniformly random permutation with respect to two properties that we consider. Secondly, the theory of Sidon sets shows that the graph of this map is equidistributed in a suitable sense. It remains an open question to prove more randomness properties, for example, that the ElGamal map is pseudorandom.