Nonlinear Component of a Block Cipher over Mordell Elliptic Curve Using Linear Congruent Generator

Abstract

These privacy, security, and accessibility remain unchanged but have evolved to include additional criteria. Not only should data be encrypted while it is stored on a computer, but also when it is being transmitted to and from other machines. Cryptology refers to the study of techniques for securing communications against third parties (referred to as adversaries). It is further divided into two subfields called Cryp tography and Cryptanalysis. The former relates to the use and practice of the technolo gies needed to create secure communication protocols, while the latter pertains to the study of how to access encrypted information without gaining access to the key, which is equivalent to learning how to break cryptographic protocols. It is evident that neither area would exist without the other, and the interaction between the two is extremely vital. Historically, this subject was viewed exclusively through the lens of privacy, and the words ‘cryptography’ and ‘encryption’ were considered synonymous. The intended purpose of encryption was always to exchange keys between two parties who possess the same key (symmetric cryptography). With the development of computers, modern cryptology evolved in numerous directions, offering today a wide range of protocols, including public-key cryptography, authentication schemes, zero-knowledge methods of identification, and so on. Cryptography has received a lot of attention in the last several decades, and many new areas of study have been created as a result of concerns about the safety of sensitive in formation. Numerous proposals for data security techniques were made by researchers, each one built on a unique mathematical framework. The goal of these methods is to prevent unauthorized parties from accessing sensitive information by rendering it un intelligible. To create confusion in the input data up to a certain degree, most classic symmetric cryptosystems, such as Advance Encryption Standard (AES), International 7 Data Encryption Algorithm (IDEA), and Data Encryption Standard (DES), effectively depend on the use of substitution boxes (S-boxes). This means that the effectiveness of these systems is almost entirely dependent on the cryptographic characteristics of their S-boxes. When it comes to improving encryption’s robustness, the S-box is crucial. Elliptic curves (ECs) have been receiving a lot of attention in the cryptography commu nity as of late, and are being included in some of the most secure cryptosystems available. Algorithms for building S-boxes using elliptic curves have been devised by certain cryp tographers. built 8x8 S-boxes using an elliptic curve over an ordered isomorphic elliptic curve and typical orderings on a class of Mordell elliptic curves over a finite field. For a particular elliptic curve, all of these elliptic curve-based techniques can only produce a single S-box, in either x or y coordinates. This thesis is arranged in a way that the definitions and introductory ideas introduced in Chapter 1, are crucial to understanding the overall argument of the thesis. Chapter 2, contains the research behind the construction of S-boxes using Mordell elliptic curves. Chapter 3, has explained a model approach for building S-boxes via elliptic curves over a Galois field. Lastly, Chapter 4, contains a detailed comparison of the new S-boxes to various already-in-place schemes and conducts a thorough security study of their design