Efficient construction of a substitution box based on a Mordell elliptic curve over a finite field

Abstract

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.

An Injective S-Box Design Scheme over an Ordered Isomorphic Elliptic Curve and Its Characterization

“…It is clear from the table that the LAP of the proposed S-boxes is low, while their NL and AC are high enough to resist the linear attacks efficiently. Note that the average value of LAP of the proposed S-boxes is 0.1445 which is less than that of the S-boxes in [21,22,27,31], while their average NL and AC are 106 and 254 which are higher than that of [20,26,27,29,[31][32][33] and [28,30,31], respectively. This implies that the proposed method is capable of generating S-boxes with high security against linear attacks than some of the listed S-boxes.…”

“…Azam et al [22] defined three typical orderings namely natural N, diffusion D, and modulo diffusion M on a given EC for the generation of S-boxes. The main idea behind these ordering is the arrangement of the points with the samecoordinates.…”

“…We generated the S-boxes Tables 2-4, respectively. A comparison of the experimental results with the strongest S-boxes generated by the algorithms in [20][21][22][26][27][28][29][30][31][32][33][34] is also conducted in this section.…”

“…Thus, by choosing some large prime, the proposed method can generate a large number of dynamic S-boxes, and therefore it can easily resist the brute force attacks. For the comparison, the maximum possible number of S-boxes that can be generated by the other schemes [20][21][22] over an EC is also listed in Table 6. It is evident from Table 6 that the proposed method is more suitable for the generation of dynamic S-boxes than the listed schemes.…”

“…Furthermore, the output of these algorithms is uncertain in the sense that it may or may not generate an S-box for each input parameters, and are independent of the underlying EC. Azam et al [22] used some typical type of orderings on a class of Mordell elliptic curve (MEC) over a finite field to design an 8 × 8 S-box in constant time. All these schemes can generate at most one S-box for a given EC.…”