On the normality of p-ary bent functions

Area of Science:

• Cryptography
• Abstract Algebra
• Number Theory

Background:

• Bent functions are crucial in cryptography, particularly in constructing secure systems.
• Understanding the structure of bent functions, specifically their normality, is key to analyzing their cryptographic properties.
• Existing research shows many Boolean and p-ary bent functions exhibit a high degree of normality, often being affine on large subspaces.

Purpose of the Study:

• To develop a general algorithm for testing the normality of p-ary bent functions.
• To investigate whether p-ary bent functions can deviate from the established bounds on their normality.
• To construct new classes of bent functions with specific normality properties.

Main Methods:

• Development of a novel algorithm to test the normality of functions from F_p^n to F_p.
• Application of the algorithm to analyze specific p-ary bent functions in small dimensions.
• Utilizing the direct sum construction to generate bent functions with novel properties.

Main Results:

• The study demonstrates that not all p-ary bent functions adhere to the theoretical bounds on normality.
• The developed algorithm successfully identified bent functions that are not affine on subspaces up to the expected maximal dimension.
• New examples of p-ary bent functions exhibiting lower normality were constructed, extending to infinitely many dimensions.

Conclusions:

• The findings challenge the assumption that most p-ary bent functions are highly normal.
• The new algorithm provides a valuable tool for further cryptanalysis and the design of cryptographic primitives.
• The discovery of bent functions with different normality behaviors opens new avenues for research in coding theory and cryptography.