A Hybrid Technique for Generation of Highly Nonlinear Component Based on Elliptic Curves and Algebraic Group Structure

Elliptic curves are considered highly secure due to their complex mathematical structure and are widely used in cryptographic key exchange algorithms. This study aimed to construct a confusion component for a symmetric block cipher using a hybrid approach that involves elliptic curve points and an algebraic group structure. Generally, elliptic curve points do not generate much randomness, so they are not recommended for constructing a highly nonlinear S-box. In this study, we first randomized the elliptic curve points to the maximum possible level and generate initial S-boxes. These initial S-boxes were then evolved through algebraic group action to enhance their nonlinearity. The strength of the proposed substitution box was assessed using various tests, including nonlinearity, bit independence criterion, strict avalanche criterion, and differential and linear probabilities. Using this approach, we designed an S-box with a nonlinearity value of 114, which is higher than the standard value presented by AES. We compared the results of other cryptographic tests with well-known S-boxes to validate the effectiveness of the suggested confusion component.