An optimal and efficient framework for the construction of nonlinear components based on a polynomial ring

Abstract

The widespread use of internet makes it essential to protect the private data from the intruders via developing highly secure cryptographic systems. The Substitution box (s-box) is the only nonlinear component of any security system. It plays a crucial role in securing data from an unauthorized access by inverting it into an unreadable form. The algebraic structures are mostly utilized to develop two types of s-box generators, namely the randomized and the optimal generators. The one type outputs dynamic, while the other one is responsible to design s-boxes with high cryptographic properties. However, the generator which gives highly dynamic and optimal s-boxes needs high computational overhead, which limits the encryption throughput of a large useful data set. This fact implies the need of developing an s-box generator that is capable to construct an s-box with high cryptographic properties at a low computational cost. This study presents an innovative algebraic method for constructing optimal s-boxes using a finite field and a polynomial ring. Our approach offers flexibility in selecting unconstrained primes and polynomials, allowing the generation of highly dynamic and nonlinear s-boxes. By evaluating polynomials over a finite field and introducing a new total ordering, we effectively diffuse input elements and derive optimal s-boxes with a nonlinearity of 112. We determine the total number of s-boxes based on the proposed scheme. The performance of the proposed s-box is assessed using standard metrics. We compare the attained results with that of the s-boxes constructed by the recent and state of the art algorithms. Apart from this, we compare the proposed s-box generator with an efficient one regarding the execution time and other cryptographic properties. We showed that the proposed scheme attains the highly nonlinear component approximately 5 times faster than the existing one, and the experimental results indicate that the current method outperforms than others across the standard cryptographic metrics.