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Analysis and Comparison of Known and Randomly Generated S-boxes for Block Ciphers

James Kim

TL;DR

This work addresses the challenge of designing robust S-boxes for SPN block ciphers by comparing mathematically constructed $8 imes8$ S-boxes with randomly generated ones. It systematically evaluates a broad set of cryptographic properties, including differential uniformity, nonlinearity, SAC, BIC, cycle structure, and algebraic characteristics, using both unconstrained and cycle-constrained random generation. The authors conduct a large-scale experiment (100000 random S-boxes) and embed selected S-boxes into a simple SPN to measure diffusion through avalanche metrics, finding that mathematical constructions generally offer stronger, provable guarantees while random methods serve as useful baselines for lightweight contexts, albeit with weaker overall security guarantees. The findings provide practical guidance for S-box design and establish a baseline for future improvements in random S-box generation and hybrid construction methods, with implications for standards and cryptanalytic resilience in SPN-based ciphers.

Abstract

Mathematically constructed S-boxes arise from algebraic structures and finite field theory to ensure strong, provable cryptographic properties. These mathematically grounded constructions allow for generation of thousands of S-Boxes with high nonlinearity, APN properties, and balanced avalanche characteristics, unlike fully random methods, which lack such theoretical guarantees in exchange for low complexity and more varied results. In this work, we compare mathematically constructed constructions with randomly generated ones to evaluate the relative weakness of the latter. We also establish an average measure of performance for randomly generated permutations, as well as random with forced cycle constraints, and compare them to well-established designs in a simple SPN setting.

Analysis and Comparison of Known and Randomly Generated S-boxes for Block Ciphers

TL;DR

This work addresses the challenge of designing robust S-boxes for SPN block ciphers by comparing mathematically constructed S-boxes with randomly generated ones. It systematically evaluates a broad set of cryptographic properties, including differential uniformity, nonlinearity, SAC, BIC, cycle structure, and algebraic characteristics, using both unconstrained and cycle-constrained random generation. The authors conduct a large-scale experiment (100000 random S-boxes) and embed selected S-boxes into a simple SPN to measure diffusion through avalanche metrics, finding that mathematical constructions generally offer stronger, provable guarantees while random methods serve as useful baselines for lightweight contexts, albeit with weaker overall security guarantees. The findings provide practical guidance for S-box design and establish a baseline for future improvements in random S-box generation and hybrid construction methods, with implications for standards and cryptanalytic resilience in SPN-based ciphers.

Abstract

Mathematically constructed S-boxes arise from algebraic structures and finite field theory to ensure strong, provable cryptographic properties. These mathematically grounded constructions allow for generation of thousands of S-Boxes with high nonlinearity, APN properties, and balanced avalanche characteristics, unlike fully random methods, which lack such theoretical guarantees in exchange for low complexity and more varied results. In this work, we compare mathematically constructed constructions with randomly generated ones to evaluate the relative weakness of the latter. We also establish an average measure of performance for randomly generated permutations, as well as random with forced cycle constraints, and compare them to well-established designs in a simple SPN setting.
Paper Structure (34 sections, 20 equations, 7 figures, 4 tables, 2 algorithms)

This paper contains 34 sections, 20 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Theoretical Background
  2. Notations and Abbreviations Used
  3. Introduction to Cryptography and Symmetric Block Ciphers
  4. The Importance of S-boxes
  5. Motivation
  6. Desirable Properties
  7. Differential Uniformity
  8. Almost Perfect Nonlinear (APN) Functions
  9. Extended Affine and Carlet Charpin Zinoviev equivalence
  10. APN permutations and small even dimensions 4, 6, 8
  11. Nonlinearity and Bias
  12. Strict Avalanche Criterion
  13. Bit Independence Criterion
  14. Cycle Structure
  15. Algebraic Degree and Algebraic Immunity
  16. ...and 19 more sections

Figures (7)

  • Figure 1: Rijndael S-box used in AES in decimal representation
  • Figure 2: a 4x4 S-box and its DDT
  • Figure 3: PRESENT S-box and its DDT
  • Figure 4: LAT visualisation of AES S-box
  • Figure 5: LAT visualisation of DBIC_Rij_Cyc
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 8 more