Efficient lattice-based Mordell elliptic curve S-box for secure lightweight cryptography

A key focus in research publications on data encryption algorithms is the Substitution Box (S-Box), a fundamental component. Recently, a predominant approach for S-box generation involves utilizing Mordell elliptic curves, chosen for their high security with small key space attributes. However, prevalent S-box algorithms derived from these ECs exhibit structural and algorithmic limits, rendering them less adept for deployment in small devices and lightweight cryptography applications due to elevated running time complexities and less key space. We present a novel approach to overcome these challenges and craft an S-box suitable for compact devices. Our proposed lattice ordering-based S-box algorithm is designed with efficiency and dynamism in mind, employing a Mordell EC as its foundation. A noteworthy aspect of our methodology involves the expedited generation of elements within the Mordell EC, utilizing an efficient method instead of the conventional group law, and constructing the S-box through lattice ordering applied to these elements, which provides $p-1$ key size and S-Boxes for any prime $p\equiv 2mod3$. This study aims to overcome existing limitations by introducing an alternative S-box with enhanced algorithmic complexity and reduced computation time compared to existing models based on the Mordell elliptic curve. The proposed method generates $p-1$ S-box for a prime $p\equiv 2mod3$, achieves strong cryptographic properties with numerical results: nonlinearity (NL) of 106, differential approximation probability (DAP) of 0.0391, linear approximation probability (LAP) of 0.1328, strict avalanche criterion (SAC) of 0.4958. These results demonstrate superior or comparable performance to contemporary models, making the design well-suited for constrained environments such as IoT and lightweight cryptography applications.