Changing almost perfect nonlinear functions on affine subspaces of small codimensions (2501.03922v1)

Abstract: In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function yield new ones. Furthermore, we show that some of the newly constructed functions are extended-affine inequivalent to the original ones.

• The paper introduces a switching construction method to alter coordinate functions of APN functions, yielding extended-affine inequivalent mappings.
• The study fully characterizes modifications of quadratic APN functions on hyperplanes, establishing new H-equivalence classes using trace-based techniques.
• The paper formulates explicit conditions via exponential sums for constructing novel APN mappings, enhancing cryptographic and coding applications.

Understanding Modifications of Almost Perfect Nonlinear Functions on Affine Subspaces

The paper "Changing almost perfect nonlinear functions on affine subspaces of small codimensions" by Hiroaki Taniguchi, Alexandr Polujan, Alexander Pott, and Razi Arshad investigates algebraic decompositions and secondary constructions of Almost Perfect Nonlinear (APN) functions. The authors examine the conditions under which modifications of a given APN function yield new APN functions, some of which are extended-affine inequivalent to the original function.

APN functions, particularly valuable for cryptographic applications, exhibit optimal differential properties. They minimize the cardinality of solution sets for the equation $F(x+a) + F(x) = b$, typically not exceeding two solutions for non-zero $a$ and any $b$. These functions play a crucial role in the construction of cryptographic primitives like S-Boxes and in coding theory, given their relationship with optimal codes and large Sidon sets.

Contributions and Findings

The paper makes substantial contributions in the domain of secondary constructions of (n, m)-APN functions, especially focusing on constructions involving affine subspaces of small codimensions:

1. Switching Construction and Variants: The paper revisits and extends the switching construction method for (n, m)-APN functions by altering one coordinate function while maintaining the APN property. It generalizes conditions under which APN functions constructed by concatenating two (n-1, m)-APN functions defined on complementary hyperplanes remain APN.
2. Quadratic Functions and Hyperplane Modifications: The paper provides a complete characterization of new APN functions derived by modifying quadratic APN functions on hyperplanes, using linear functions. It defines a notion of "H-equivalence," indicating that all quadratic APN functions on $F_{2^6}$ can be classified into equivalence classes formed by modifying trace-based functions with linear terms.
3. Necessary and Sufficient Conditions: Explicit conditions using exponential sums are put forth for determining the APN nature of functions of the form $G(x) = x^3 + \text{Tr}(x)L(x)$. This provides an efficient approach for generating new APN mappings under different spectral conditions.
4. Extended Techniques: By focusing on APN functions over finite fields with constant modifications on codimension-two affine subspaces, the authors identify conditions for obtaining new extended-affine inequal APN functions, increasing the diversity of potential cryptographic primitives.

Implications and Future Directions

This work enhances the theoretical framework and practical methods for generating and categorizing APN functions, a domain critical for secure cryptographic systems. By unveiling structurally new APN forms and providing complete characterization conditions, the paper sets a foundation for further explorations in both the construction of APN functions and their applications in cryptography, including encryption algorithm optimization and error-correcting code design.

Future exploration can expand on the comparative benefits of these secondary constructions under various metrics beyond classical differential properties. Pursuing the identification of families of APN functions with non-classical Walsh spectra remains a promising direction, potentially yielding new insights into secure function configurations.

The analytical and empirical findings underscore the relevance of continued research into sophisticated algebraic methods for APN functions across different dimensional configurations and field characteristics. Integrating these APN construction methods with advanced computational tools may improve the explorative efficiency in cryptographic research, increasing resilience against new classes of cryptanalytic attacks.