Periodicity Detection of the Substitution Box in the CBC Mode of Operation: Experiment and Study

Abstract

This paper presents a technique for investigating the cyclic properties of substitution boxes (S-boxes) in the Cipher Block Chaining (CBC) mode of operation. S-boxes provide nonlinear transformations in encryption algorithms to create confusion and enhance cryptographic strength. The CBC mode design is used in block ciphers to hide periodic patterns and create a diffusion effect. The main objective of this study was to detect the periodicity of the bijective S-boxes in CBC mode to evaluate their cryptographic strength. The study of S-boxes using the presented technique allows us to examine them in a different manner and study their diffusion levels, the metrics of which are the periodicities of the S-box element sequences. To apply the diffusion effect of the CBC mode to the S-boxes, the encryption function used in the cryptographic ciphers was changed to a substitution function for the S-boxes used as an inner nonlinear component of the encryption function. The S-box used in the Advanced Encryption Standard (AES) was selected for experiment and study. In this study, the cyclic properties of the S-box were considered from two different aspects: periodicity detection of the S-box with respect to iterations and blocks. According to our study, the maximal periods of the AES S-box and various other S-boxes were found to be very large, indicating that the influence of the CBC mode spread over many iterations and blocks, thus confirming the high level of cryptographic strength of the S-boxes.

Keywords

• AES
• block cipher
• CBC mode of operation
• cyclic properties
• periodicity
• S-box