On Ordered Mordell Elliptic Curves and Their Applications in Cryptography

Abstract

This chapter summarizes the work presented in this thesis and some future directions are also discussed. The goals achieved as a result of this work are: (1) New mathematical structures such as total orders are defined on ECs, which are then employed in developing new algorithms for the construction of non-linear components. (2) Existing structure such as isomorphism on ECs is utilized to design secure S-box construc tion technique and image encryption technique. (3) For a prime p and m ≤ p, the concept of mp-complete set is introduced on an EC, which is further used in designing highly non-linear S-boxes and random numbers generator with strong cryptographic properties. (4) The newly designed S-box and random number generators are used in proposing image encryption technique with the desired security. 120 121 (5) A new image encryption scheme based on quasi-resonant Rossby/drift wave triads and ordered ECs is presented and experimentally its security is verified. While working on the current schemes, several ideas came in way which are not worked yet. Some of them are given by: (1) Enhancing the algebraic and statistical strength of the suggested schemes by minimizing their time and space complexity. (2) To develop new schemes for generating n × n, n ≥ 9 S-boxes with low time and space complexity based on the newly defined total orders on ECs. (3) To design a new software for evaluating the cryptographic strength of n×n, n ≥ 9 S-boxes. (4) To propose new RGB image encryption algorithms utilizing 24 × 24 S-boxes. (5) Utilizing the existing and new mathematical structures on ECs to develop techniques for the generation of random bit strings with the desired security.