Substitution Box Generation and Enumeration Based on a Montgomery Elliptic Curve

Abstract

Cryptographic primitives, including substitution boxes (S-boxes), are essen tial for protecting sensitive data from attackers in the constantly changing world of digital security. Robust S-box construction and design are critical to strengthen encryption methods because they provide non-linearity and confusion that are necessary to counter a variety of cryptographic attacks. This thesis explores the complex field of S-box generation and enumeration, concentrating on using the intrinsic qualities of Montgomery elliptic curves to create S-boxes that have better cryptographic features. Montgomery curves, a notable subclass of elliptic curves, have special benefits like effective arith metic operations and resistance to particular kinds of cryptographic assaults, making them a compelling option for cryptographic applications. Furthermore, the utilization of Linear Fractional Transformation enhances the quality of the generated S-boxes, making the system more dynamic. Through its application, a large number of distinct S-boxes can be generated by the proposed scheme. Additionally, by employing a newly introduced theorem based on Linear Fractional Transformation, the exact number of total S-boxes can be determined. Time complexity is a key component of S-boxes design that is examined in this thesis. The temporal complexity study includes a thorough examination of different S-box generation techniques, taking into account variables like the size of the underlying elliptic curve, the desired cryptographic characteristics of the generated S-boxes and the available computing power. Time complexity analysis provides important insights into the scalability and effectiveness of various S-box generation methods, making it possible to identify strategies that strike the best possible balance between computational overhead and cryptographic strength. In addition, this thesis explores the counting of S-boxes generated from fix Montgomery elliptic curve over the imaginary extension field of fixed prime f ield, offering techniques and methods for methodically producing a significant quantity of unique S-boxes. The enumeration procedure not only reveals the extent of S-box variety that can be achieved with Montgomery curves, but it also makes the investigation of cryptographic features easier over a wide range of designs