Symmetry Analysis in Construction Two Dynamic Lightweight S-Boxes Based on the 2D Tinkerbell Map and the 2D Duffing Map

Abstract

The lack of an S-Box in some lightweight cryptography algorithms, like Speck and Tiny Encryption Algorithm, or the presence of a fixed S-Box in others, like Advanced Encryption Standard, makes them more vulnerable to attacks. This proposal presents a novel approach to creating two dynamic 8-bit S-Boxes (16 × 16). The generation process for each S-Box consists of two phases. Initially, the number initialization phase involves generating sequence numbers 1, sequence numbers 2, and shift values for S-Box1 using the 2D Tinkerbell map. Additionally, sequence numbers 3, sequence numbers 4, and shift values for S-Box2 are generated using the 2D Duffing map. Subsequently, the S-Box construction phase involves the construction of S-Box1 and S-Box2. The effectiveness of the newly proposed S-Boxes was evaluated based on various criteria, including the bijective property, balance, fixed points, and strict avalanche criteria. It was observed that S-Box1 achieved a remarkable linear and differential branch number of 4, surpassing any previous studies. Furthermore, it exhibited a non-linearity of 105.50, a differential uniformity of 12, and an algebraic degree of 7. Similarly, S-Box2 also achieved a linear and differential branch number of 4, a non-linearity of 105.25, a differential uniformity of 14, and an algebraic degree of 7. Moreover, the reduction in the number of linear and nonlinear operations for both S-Boxes makes them suitable for lightweight algorithms. The architecture of the proposed S-Boxes demonstrates robustness, with a total of 3.35 × 10504 possible S-Boxes, providing protection against algebraic attacks.

Keywords

• lightweight algorithms
• two dynamic S-Boxes
• 2D Tinkerbell map
• 2D Duffing map