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Dyadic-Chaotic Lifting S-Boxes for Enhanced Physical-Layer Security within 6G Networks

Ilias Cherkaoui, Indrakshi Dey

TL;DR

The paper tackles the need for lightweight, reconfigurable confusion at the 6G physical layer to defend against large-scale precomputation and side-channel attacks. It proposes a chaos-driven, $\beta$-transformation–based $8\times8$ S-box with dyadic conditional sampling to produce time-varying, seedable permutations while preserving uniformity. Key results show per-bit algebraic degree $7$, average nonlinearity $102.5$, max differential probability $10/256$, and max linear probability $0.648$, enabling strong diffusion with a compact, table-free implementation. Hardware validation demonstrates sub-millisecond generation latency at $200$ MHz and favorable area/power trade-offs, supporting slice-level agility for URLLC/mMTC; future work will address finite-precision effects, online parameter adaptation, and integration with slice controllers for over-the-air deployment.

Abstract

Sixth-Generation (6G) wireless networks will interconnect billions of resource-constrained devices and time-critical services, where classical, fixed, and heavy cryptography strains latency and energy budgets and struggles against large-scale, pre-computation attacks. Physical-Layer Security (PLS) is therefore pivotal to deliver lightweight, information-theoretic protection, but still requires strong, reconfigurable confusion components that can be diversified per slice, session, or device to blunt large-scale precomputation and side-channel attacks. In order to address the above requirement, we introduce the first-ever chaos-lifted substitution box (S-box) for PLS that couples a $β$-transformation-driven dynamical system with dyadic conditional sampling to generate time-varying, seedable 8-bit permutations on demand. This construction preserves uniformity via ergodicity, yields full 8-bit bijections, and supports on-the-fly diversification across sessions. The resulting S-box attains optimal algebraic degree 7 on every output bit and high average nonlinearity 102.5 (85% of the 8-bit bound), strengthening resistance to algebraic and linear cryptanalysis. Differential and linear profiling report max DDT entry 10 (probability 0.039) and max linear probability 0.648, motivating deployment within a multi-round cipher with a strong diffusion layer, where the security-to-efficiency trade-off is compelling. Our proposed reconfigurable, lightweight S-box directly fulfills key PLS requirements of 6G networks by delivering fast, hardware-amenable confusion components with built-in agility against evolving threats.

Dyadic-Chaotic Lifting S-Boxes for Enhanced Physical-Layer Security within 6G Networks

TL;DR

The paper tackles the need for lightweight, reconfigurable confusion at the 6G physical layer to defend against large-scale precomputation and side-channel attacks. It proposes a chaos-driven, -transformation–based S-box with dyadic conditional sampling to produce time-varying, seedable permutations while preserving uniformity. Key results show per-bit algebraic degree , average nonlinearity , max differential probability , and max linear probability , enabling strong diffusion with a compact, table-free implementation. Hardware validation demonstrates sub-millisecond generation latency at MHz and favorable area/power trade-offs, supporting slice-level agility for URLLC/mMTC; future work will address finite-precision effects, online parameter adaptation, and integration with slice controllers for over-the-air deployment.

Abstract

Sixth-Generation (6G) wireless networks will interconnect billions of resource-constrained devices and time-critical services, where classical, fixed, and heavy cryptography strains latency and energy budgets and struggles against large-scale, pre-computation attacks. Physical-Layer Security (PLS) is therefore pivotal to deliver lightweight, information-theoretic protection, but still requires strong, reconfigurable confusion components that can be diversified per slice, session, or device to blunt large-scale precomputation and side-channel attacks. In order to address the above requirement, we introduce the first-ever chaos-lifted substitution box (S-box) for PLS that couples a -transformation-driven dynamical system with dyadic conditional sampling to generate time-varying, seedable 8-bit permutations on demand. This construction preserves uniformity via ergodicity, yields full 8-bit bijections, and supports on-the-fly diversification across sessions. The resulting S-box attains optimal algebraic degree 7 on every output bit and high average nonlinearity 102.5 (85% of the 8-bit bound), strengthening resistance to algebraic and linear cryptanalysis. Differential and linear profiling report max DDT entry 10 (probability 0.039) and max linear probability 0.648, motivating deployment within a multi-round cipher with a strong diffusion layer, where the security-to-efficiency trade-off is compelling. Our proposed reconfigurable, lightweight S-box directly fulfills key PLS requirements of 6G networks by delivering fast, hardware-amenable confusion components with built-in agility against evolving threats.

Paper Structure

This paper contains 14 sections, 4 theorems, 2 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Mathematical foundation
  3. Cryptographic Design
  4. Theoretical Performance Analysis
  5. Method and Metrics
  6. Differential behavior
  7. Linear behavior
  8. Algebraic structure
  9. Hardware Verification Metrics and Closed-Loop Validation
  10. Metric 1: S-Box Generation Latency ($\tau_{\text{gen}}$)
  11. Metric 2: Hardware Area and Power (A/P)
  12. Closed-Loop Reporting and Acceptance
  13. Parameter Tuning and Trade-offs
  14. Conclusion

Key Result

Theorem 2.2

For $\beta\in\mathcal{B}$, the system $([0,1[,T_\beta)$ is chaotic in the sense of Devaney.

Figures (5)

  • Figure 1: Three-Stage S-Box Construction
  • Figure 2: Nonlinearity per Output Bit: The S-box has very strong nonlinearity and the nonlinearity of the output bits has been measured, thus the average nonlinearity of $102.5$ is obtained. This amount is about $85.4\%$ of the theoretical maximum ($\approx$120 for $8$-bit S-boxes), so it can be said that the S-box has a very strong resistance against linear cryptanalysis.
  • Figure 3: Differential Distribution Table Analysis: The DDT maximum entry value of $10$ is equivalent to a differential probability of $10/256 = 0.039062$, which is quite a bit above the optimal limit of 4 (probability $4/256 = 0.015625$).
  • Figure 4: Linear Approximation Table Distribution: The maximum absolute bias of $76$ means the linear probability of $0.648438$, which is remarkably higher than the bent limit of $16$ (probability $0.531250$). The bias corresponding to occurrence numbers presents a pattern of decreasing from $12.386$ at bias $4$ down to just $1.049$ at bias $36$.
  • Figure 5: Algebraic Degree and ANF Structure: All output bits reach the highest algebraic degree of $7$, which is the highest possible algebraic complexity for the S-box.

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Definition 2.5
  • Corollary 2.6
  • proof