Strong s-box construction approach based on Josephus problem

There are two basic requirements for symmetric encryption algorithms. The first of these is diffusion. The second and most important is confusion. In these algorithms, this requirement is usually met by s-box structures. Therefore, s-box structures must be strong. So, a cryptographically good s-box will make the encryption algorithm difficult to crack. However, obtaining a strong s-box is a rather difficult problem. In this study, Josephus circle logic is used to solve this problem. Initially, with a random s-box structure, the elements are replaced according to their Josephus positions, and the s-box is made stronger. In the proposed algorithm, according to the Josephus logic, the elements that kill each other are replaced and this process continues until one element remains. The last 30 surviving elements are replaced with all elements. In this way, three different s-boxes were obtained. In two of them, the nonlinearity value was 110.5, and in one of them, the nonlinearity value was 110.75. Fixed points in the proposed s-box structures were identified and eliminated. In addition, it has been proven because of the analysis that the obtained s-box structures also meet other cryptographic requirements. In addition to leaving behind most studies in the literature, the proposed method will also provide strong s-box structures for encryption algorithms to be built in the future.