Construction of $(n,n)$-functions with low differential-linear uniformity

2025-08-02

AI Summary

This work addresses the cryptographic requirement for $(n,n)$-functions to resist differential-linear attacks by systematically investigating theoretical bounds and construction methods for their differential-linear uniformity (DLU). We propose a novel construction framework integrating exponential sums and refined analysis, generalized cyclotomic mappings, power functions, and low-degree polynomials (e.g., quadratic and cubic functions). For the first time, we construct multiple new function families achieving optimal or near-optimal DLU; notably, several attain strictly lower DLU than previously known optimal constructions. Our approach overcomes the limitations of conventional methods relying solely on power functions or affine equivalence, thereby significantly expanding the design space for low-DLU functions. These results provide both new theoretical foundations and practical tools for designing highly secure S-boxes in block ciphers and hash functions.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Feature Construction/Reformulation

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The differential-linear connectivity table (DLCT), introduced by Bar-On et al. at EUROCRYPT'19, is a novel tool that captures the dependency between the two subciphers involved in differential-linear attacks. This paper is devoted to exploring the differential-linear properties of $(n,n)$-functions. First, by refining specific exponential sums, we propose two classes of power functions over $mathbb{F}_{2^n}$ with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of $(n,n)$-functions that are polynomials by utilizing power functions with known DLU. Specifically, by combining a cubic function with quadratic functions, and employing generalized cyclotomic mappings, we construct several classes of $(n,n)$-functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results.

Problem

• Study (n,n)-functions with low differential-linear uniformity
• Propose power functions with refined exponential sums
• Construct optimal DLU functions using cubic and quadratic combinations

Innovation

• Refining exponential sums for power functions
• Combining cubic and quadratic functions
• Using generalized cyclotomic mappings