Doubly Perfect Nonlinear Boolean Permutations

TL;DR

This paper introduces doubly perfect nonlinear permutations, which are robust against differential attacks based on both finite field multiplication and exponentiation, enhancing cryptographic resistance.

Contribution

It defines and analyzes doubly perfect nonlinear permutations that resist multiple differential cryptanalysis methods, a novel concept in cryptographic function design.

Findings

Identifies functions resistant to multiplication-based differential attacks

Shows existence of permutations resistant to both multiplication and exponentiation attacks

Provides theoretical framework for designing more secure cryptographic permutations

Abstract

Due to implementation constraints the XOR operation is widely used in order to combine plaintext and key bit-strings in secret-key block ciphers. This choice directly induces the classical version of the differential attack by the use of XOR-kind differences. While very natural, there are many alternatives to the XOR. Each of them inducing a new form for its corresponding differential attack (using the appropriate notion of difference) and therefore block-ciphers need to use S-boxes that are resistant against these nonstandard differential cryptanalysis. In this contribution we study the functions that offer the best resistance against a differential attack based on a finite field multiplication. We also show that in some particular cases, there are robust permutations which offers the best resistant against both multiplication and exponentiation base differential attacks. We call them doubly perfect nonlinear permutations.