Constructing Dynamic S-boxes Based on Chaos and Irreducible Polynomials for Image Encryption

abstract

Substitution boxes (S-boxes) lie at the heart of modern symmetric ciphers, and the cryptographic strength of a cipher depends heavily on the nonlinearity, differential uniformity, and unpredictability of its S-box. Static S-boxes, however, are vulnerable to algebraic and side-channel attacks once their structure is fixed. This work proposes a dynamic S-box construction scheme that combines a two-dimensional chaotic map with irreducible polynomials over the finite field .

The chaotic map provides key-dependent sensitivity and a vast parameter space, so that each key induces a distinct S-box; irreducible polynomials over supply algebraic structure that bounds worst-case cryptographic metrics. We conduct ablation experiments to evaluate the generated S-boxes on the standard battery of cryptographic criteria — nonlinearity, strict avalanche criterion (SAC), bit independence criterion (BIC), and differential / linear approximation probabilities — and on generation efficiency. The constructed S-boxes consistently meet or exceed the requirements for cryptographic use.

We further integrate the dynamic S-box into a block-cipher image encryption pipeline, and show on standard test images that it delivers strong pixel-level confusion and diffusion, near-uniform ciphertext histograms, high NPCR/UACI scores against plaintext-sensitivity attacks, and robust resistance to differential, statistical, and brute-force attacks.

@article{luo2024dynamic, title={Constructing Dynamic S-boxes Based on Chaos and Irreducible Polynomials for Image Encryption}, author={Luo, Chunlei and Wang, Yong and Fu, Yongji and others}, journal={Nonlinear Dynamics}, year={2024}, publisher={Springer} }

Background

Symmetric ciphers rely on S-boxes to provide the confusion layer that resists linear and differential cryptanalysis. A static, key-independent S-box leaks the same algebraic structure across every ciphertext it produces, so once the structure is known it becomes the natural attack surface.

What this paper proposes

Chaos-driven keying. A 2-D chaotic map is used to produce, for every key, a fresh S-box. Sensitivity to initial conditions means a small change in the key yields a structurally different S-box.
Algebraic anchoring in . Candidate S-boxes are built from irreducible polynomials over , which bounds the worst-case nonlinearity and differential uniformity so that every instance meets the cryptographic criteria.
Standard-battery evaluation. Ablations on nonlinearity, SAC, BIC, and differential/linear approximation probability, plus generation-cost measurements.
Image-encryption pipeline. The dynamic S-box is dropped into a block-cipher image encryptor and evaluated on standard test images: near-uniform histograms, high NPCR/UACI, and resistance to differential / statistical / brute-force attacks.

Figure

Dynamic S-box construction