Differential and Linear properties of vectorial boolean functions based on chi.

Area of Science:

• Cryptography and Information Security
• Abstract Algebra and Number Theory

Background:

• Modern cryptographic primitives rely on round functions, often comprising non-linear and linear mappings.
• The S-box mapping is a crucial non-linear component in various cryptographic primitives, including Keccak, Subterranean, Ascon, and Simon.
• Understanding the security implications of S-box behavior under differential and linear cryptanalysis is essential.

Purpose of the Study:

• To comprehensively evaluate the security of the S-box mapping against differential and linear cryptanalysis.
• To investigate how the S-box's differential and linear properties vary when applied to strings of different lengths.
• To provide cryptographers with data to optimize the selection of non-linear layers in new cryptographic designs.

Main Methods:

• Revisiting and presenting known differential properties of the S-box mapping.
• Analyzing the linear propagation characteristics of the S-box.
• Comparing the application of parallel S-box instances on small strings versus a single instance on a large string.

Main Results:

• Extended analysis of the S-box's linear properties, complementing existing differential property research.
• Demonstrated how the choice of string length and application method (parallel vs. single instance) impacts the S-box's cryptographic behavior.
• Showcased the applicability of these findings to the non-linear layers of Ascon and Simon due to their affine equivalence with the S-box.

Conclusions:

• The differential and linear properties of the S-box are significantly influenced by its application context (string length and parallelism).
• This research provides valuable insights for selecting and designing robust non-linear layers in symmetric-key cryptography.
• The results are directly transferable to affine-equivalent non-linear layers in widely used cryptographic algorithms like Ascon and Simon.