Paper 2020/878

Bent Boolean functions A better procedure to generate non-crypto 4-bit S-boxes

Sankhanil De and Ranjan Ghosh

Abstract

crypto 4-bit substitution boxes or crypto 4-bit S-boxes are used in block ciphers for nonlinear substitution very frequently. If the 16 elements of a 4-bit S-box are unique, distinct and vary between 0 and f in hex then the said 4-bit S-box is called as a crypto 4-bit S-box. There are 16! crypto 4-bit S-boxes available in crypto literature. Other than crypto 4-bit S-boxes there are another type of 4-bit S-boxes exist. In such 4-bit S-boxes 16 elements of the 4-bit S-box are not unique and distinct i.e. at least one element must repeat more than one time. They are called as non-crypto 4-bit S-boxes. There are 16^16-factorial 16 Numbers of non-crypto 4-bit S-boxes can be found in crypto-literature. The non-crypto 4-bit S-boxes can be generated from 4-bit Boolean Functions (BFs) in the same manner as crypto 4-bit S-boxes are generated in [1]. But to generate crypto 4-bit S-boxes the security of the generated 4-bit S-boxes is sacrificed into some extend. Since 12870 4-bit balanced BFs are responsible for factorial 16 crypto 4-bit S-boxes and the nonlinearity of the balanced 4-bit BFs are at most 4. So the 4-bit BFs with highest nonlinearity 6 are left abandoned. These 4-bit BFs are called as 4-bit Bent BFs. Here in this paper we generate non-crypto 4-bit S-boxes from 4-bit Bent BFs. The generated non-crypto 4-bit S-boxes are analyzed with the existing cryptanalysis techniques to prove them much secure 4-bit S-boxes from crypto angle.

Note: This publication is a landmark in cryptography. Here the non-crypto S-boxes are advised to use in block ciphers. The publication is under review in the Journal of Institute of Engineers- Series B of Springer India.