IDEM Enough? Evolving Highly Nonlinear Idempotent Boolean Functions

2026-01-31

AI Summary

This work addresses the challenging problem of searching for highly nonlinear Boolean functions under the idempotency constraint, which is inherently difficult due to restrictive algebraic structures. To overcome this, the authors propose a compact genetic encoding strategy based on square orbits, which integrates canonical primitive polynomials with orbit-based encoding in polynomial basis representation. This approach effectively enforces idempotency while preserving solution structure during evolutionary operations, thereby avoiding the structural degradation commonly caused by conventional genetic operators. Using this method, the study successfully evolves high-nonlinearity idempotent Boolean functions efficiently in dimensions ranging from 5 to 12, achieving significant improvements in both search efficiency and solution quality. The results demonstrate the effectiveness and novelty of the proposed encoding mechanism for constrained Boolean function optimization.

Technology Category

• Search and Optimization: Evolutionary Computation
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Evolutionary Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• idempotent Boolean functions
• high nonlinearity
• cryptographic design
• algebraic constraints
• Boolean functions

Innovation

• idempotent Boolean functions
• evolutionary algorithms
• nonlinearity
• orbit encoding
• polynomial basis