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Elliptic curve cryptography (ECC) is used in many security systems due to itssmall key size and high security as compared to the other cryptosystems. Inmany well-known security systems substitution box (S-box) is the onlynon-linear component. Recently, it is shown that the security of a cryptosystemcan be improved by using dynamic S-boxes instead of a static S-box. This factnecessitates the construction of new secure S-boxes. In this paper, we proposean efficient method for the generation of S-boxes based on a class of Mordellelliptic curves (MECs) over prime fields by defining different total orders.The proposed scheme is developed in such a way that for each input it outputsan S-box in linear time and constant space. Due to this property, our methodtakes less time and space as compared to all existing S-box constructionmethods over elliptic curve. Furthermore, it is shown by the computationalresults that the proposed method is capable of generating cryptographicallystrong S-boxes with comparable security to some of the existing S-boxesconstructed over different mathematical structures.

Efficient Construction of a Substitution Box Based on a Mordell Elliptic Curve Over a Finite Field