# An optimized substitution box generator based on cubic pell curves and its application in image encryption

## TL;DR

This paper introduces a new method for generating secure substitution boxes for image encryption using cubic Pell curves, which improves data security and efficiency.

## Contribution

A novel substitution box generator based on cubic Pell curves is proposed, achieving optimal nonlinearity and passing cryptographic tests.

## Key findings

- The proposed S-boxes achieve optimal nonlinearity of 108 and pass NIST randomness tests.
- Image encryption results show robustness against statistical attacks using the generated S-boxes.

## Abstract

In today’s digital landscape, safeguarding confidential data from cyber threats and unauthorized breaches is more crucial than ever. A key component in modern cryptographic systems is the substitution box (S-box), which ensures data security through complex transformations. Designing S-boxes with high nonlinearity and computation efficiency is still challenging. In this paper, we propose a novel S-box generator based on cubic Pell curves. The key idea of our generator is to utilize the randomness of the points over the cubic Pell curves by using their binary strings. The optimized S-boxes are obtained by performing swapping operations on initial S-boxes that ensure high nonlinearity. Cryptographic evaluations including nonlinearity (NL), strict avalanche criterion (SAC), bit independence criterion (BIC), differential approximation probability (DAP), linear approximation probability (LAP) and algebraic complexity (AC) demonstrate the strength of our method. The proposed S-boxes achieve optimal nonlinearity (108), outperform existing schemes in speed and security, and pass NIST randomness tests. Image encryption results demonstrate the robustness of the generated S-boxes against statistical attacks, further validating their cryptographic strength.