Loop Based S-box Constructions and Data Hiding Implementation

Abstract

Cryptography is used for protection and transmitting information in such a way that only specified persons can read and process it. The basic cryptographic techniques are used for thousands of years in different areas. In history, many governments or organized groups mostly used cryptography to conceal secret messages from enemies. About 100 BC Julius Caesar made use of certain encryptions so that he may send secret messages to his army. This type of encryption is known as Caesar cipher, which is a type of substitution cipher (A cipher is an encryption or decryption algorithm and substitution means that each character of a message is replaced by another character to form the message unreadable). Even though over the last 40 years, Modern cryptography is considered as a mature branch of science but it is still relatively new field of study compared to other subjects and every new day brings so many developments. But now a days, millions of secure and encoded transmissions occur online every day. Cryptographic standards are used to protect data, banking data, images, videos, health information and much more. In all these, the online security threats evolve so quickly, so there is a need for network security, which is the study of methods for protecting data in communication systems and computers from unauthorized authorities. Network security or data security progressed rapidly since 1975. Modern cryptography and Network security techniques are mostly based on mathematical theory and computer science practices. For few years, finite Galois rings have great importance in cryptography and coding theory. In 1979, Priti Shankar established a relationship between BCH- codes over Galois ring 𝐺𝑅(𝑝𝑘,𝑚) and Galois field 𝐺𝐹(𝑝𝑚) through a p-reduction map. In the construction of these BCH-codes, the maximal cyclic subgroup 𝐺𝑝𝑚−1 of a group of units of Galois ring 𝐺𝑅(𝑝𝑘,𝑚) plays a pivotal role. The maximal cyclic subgroup 𝐺𝑝𝑚−1 is isomorphic to Galois group 𝐺𝐹(𝑝𝑘)∗ and this provides a way to use it in cryptography. The Galois rings were firstly used in cryptography by Shah et al. In cryptography, the substitution box (S-box) is the vital component of almost all symmetric cryptosystem. The process of encryption creates confusion and diffusion in data and the S-box plays a key role to make confusion in data because it is the only non-linear and invertible part in the encryption process. The strength of encryption technique depends on the ability of Sbox in twisting the data hence, the process of finding new and powerful S-boxes is of great importance in the field of cryptography. Firstly, S-boxes are constructed only by using Galois [2]

fields. But Shah et al. give method of construction of S-boxes by using elements of Galois rings 𝐺𝑅(4,2) 𝑎𝑛𝑑 𝐺𝑅(4,4). Here Shah et al. used the maximal cyclic subgroup of the group of units of Galois ring 𝐺𝑅(4,4), which has 16 elements and so that 4 × 4 S-box is formed. The maximal cyclic subgroup of a group of units of commutative chain ring is also used to construct healthier S-box. The purpose of the research is to develop a good understanding of some basic concepts of cryptography but mainly focused on the construction of S-boxes on inverse property loops of order 16 and 256. The newly constructed S-boxes are then analyzed by some algebraic analyses to determine the strength of the proposed S-boxes and by statistical analyses of their application in image encryption algorithms. The details of the dissertation are here under:  The first chapter consists of three section. In the first section, we discussed some basic algebraic concepts, which were necessary to understand cryptography. In the second section, we discussed some basic components of cryptography and lastly we discussed some examples of classical and modern cryptography. In the last section, we discuss some basic definitions of loop theory.  In the second chapter, the concept of S-box is discussed. In addition, the construction techniques of S-box on a maximal cyclic subgroup of a group of Galois ring 𝐺𝑅(2𝑘,4) for 1 ≤ k ≤ 5.  In the third chapter, a novel technique to construct S-boxes based on inverse property loops of order 16 and order 256. Some algebraic analysis to determine the strength of these Sboxes are also given in this chapter  In the fourth chapter, we discuss the application of these S-boxes in watermarking and image encryption.  The last chapter consists the conclusion of this work.

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Contents NOTATIONS........................................................................................................................................................ 5 1