Mathematical aspects of selected block ciphers

Block ciphers play a key role in many cryptographic protocols that provide communications security in modern society. The security of such cryptographic protocols is depending basically on the underlying block ciphers that are being used. In order to achieve a secure block cipher, it is important to have a good understanding in how to design and analyse such block cipher. However, the current level of understanding has still not reach the peak, and the progress is active to improve our understanding of how to design and analyse them. Evaluating some common and important mathematical primitives will provide more optimization of block cipher design and analysis. The main objective of this thesis is to analyse the mathematical primitives used in the design of many block ciphers and point out which primitives are essential and important in the fulfilment of confusion and diffusion properties. The findings of the thesis can be divided into the following main contributions: an overview of the different types of block ciphers primitives are given, the block ciphers are explored in terms of their underlying algebraic structure operations, and the algebraic primitives used in block ciphers design are evaluated from their security and efficiency aspects and then compared with random substitution boxes (S-boxes). The main focus is to measure how algebraic primitives are exhibited to meet diffusion and confusion properties. After that, the requirements of Boolean functions and S-boxes are discussed. In addition, an analysis of several Boolean functions and S-boxes is presented in terms of the desired cryptographic properties, and the comparison is drawn in order to show the different strengths and weaknesses.

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