A Truly Dynamic Substitution Box Generator for Block Ciphers Based on Elliptic Curves Over Finite Rings

Several dynamic substitution box (S-box) generators have been proposed in recent years. Some of the existing S-box generators can generate an S-box with good cryptographic properties. However, no analysis has been performed for almost all generators to confirm if they can output highly dynamic S-boxes in a reasonable computational complexity. This article aims to propose an S-box generator that can efficiently generate highly dynamic S-boxes with good cryptographic properties. For this purpose, we use elliptic curves over finite rings. To create randomness in the points on the curves, we define two families of total orders. We then use these ordered curves to generate an S-box of size m using the y-coordinates of an ordered subset of size m of the underlying curve. We performed a rigorous analysis to show that our proposed generator can efficiently generate dynamic S-boxes with good cryptographic properties. Compared to some of the existing generators, it is shown that the proposed S-box generator is more suitable for cryptographic purposes.