  Elliptic curve cryptography (ECC) is used in many security systems due to its
small key size and high security as compared to the other cryptosystems. In
many well-known security systems substitution box (S-box) is the only
non-linear component. Recently, it is shown that the security of a cryptosystem
can be improved by using dynamic S-boxes instead of a static S-box. This fact
necessitates the construction of new secure S-boxes. In this paper, we propose
an efficient method for the generation of S-boxes based on a class of Mordell
elliptic curves (MECs) over prime fields by defining different total orders.
The proposed scheme is developed in such a way that for each input it outputs
an S-box in linear time and constant space. Due to this property, our method
takes less time and space as compared to all existing S-box construction
methods over elliptic curve. Furthermore, it is shown by the computational
results that the proposed method is capable of generating cryptographically
strong S-boxes with comparable security to some of the existing S-boxes
constructed over different mathematical structures.
