IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions

TL;DR

This paper explores evolutionary algorithms for constructing highly nonlinear idempotent Boolean functions, highlighting the challenges due to algebraic constraints and proposing an orbit-based encoding to improve search efficiency.

Contribution

It introduces an orbit-based encoding method for evolving highly nonlinear idempotent Boolean functions, addressing the difficulty caused by their algebraic structure.

Findings

Evolutionary methods face challenges due to disruptive genetic operators.

Orbit-based encoding effectively enforces idempotence in Boolean functions.

Constructed functions exhibit high nonlinearity within the studied dimensions.

Abstract

Idempotent Boolean functions form a highly structured subclass of Boolean functions that is closely related to rotation symmetry under a normal-basis representation and to invariance under a fixed linear map in a polynomial basis. These functions are attractive as candidates for cryptographic design, yet their additional algebraic constraints make the search for high nonlinearity substantially more difficult than in the unconstrained case. In this work, we investigate evolutionary methods for constructing highly nonlinear idempotent Boolean functions for dimensions $n=5$ up to $n=12$ using a polynomial basis representation with canonical primitive polynomials. Our results show that the problem of evolving idempotent functions is difficult due to the disruptive nature of crossover and mutation operators. Next, we show that idempotence can be enforced by encoding the truth table on orbits, yielding a compact genome of size equal to the number of distinct squaring orbits.