Paper 2020/359

4-bit Boolean functions in generation and cryptanalysis of secure 4-bit crypto S-boxes.

Sankhanil Dey and Amlan Chakrabarti and Ranjan Ghosh

Abstract

In modern ciphers of commercial computer cryptography 4-bit crypto substitution boxes or 4-bit crypto S-boxes are of utmost importance since the late sixties. Since then the 4 bit Boolean functions (BFs) are proved to be the best tool to generate the said 4-bit crypto S-boxes. In this paper the crypto related properties of the 4-bit BFs such as the algebraic normal form (ANF) of the 4-bit BFs, the balancedness, the linearity, the nonlinearity, the affinity and the non-affinity of the 4-bit BFs and the strict avalanche criterion (SAC) of 4-bit BFs are studied in detail. An exhaustive study of 4-bit BFs with some new observations and algorithms on SAC of 4-bit BFs is also reported in this paper. A bit later in the end of nineties the Galois field polynomials over Galois field GF(28) are in use to generate the 8-bit crypto S-box of the Advance Encryption Standard (AES). A detailed study on generation of the 4-bit crypto S-boxes with such Galois field polynomials over the binary as well as non-binary extended Galois fields is also given in this paper. The generated 4-bit crypto S-boxes are analyzed with four cryptanalysis techniques and the well-defined SAC algorithms of 4-bit crypto S-boxes to search for the best possible 4-bit crypto S-boxes. Some existing 4-bit crypto S-boxes like the 32 4-bit crypto S-boxes of the Data Encryption Standard (DES) and the four 4-bit crypto S-boxes of the two variants of the Lucifer are analyzed to report the weakness of such S-boxes. A comparative study of the ancient as well as the modern 4-bit crypto S-boxes with the generated 4-bit crypto S-boxes proves the said generated 4-bit crypto S-boxes to be the best possible one.

Note: This is a very lucid and explained foundation work in cryptology.