Results on cubic bent and weakly regular bent $p$-ary functions leading to a class of cubic ternary non-weakly regular bent functions

2025-12-22

AI Summary

A significant gap exists between binary and non-binary bent function theory over odd-characteristic fields, particularly concerning the relationship between cubic-like bent functions and weak regularity. Method: This paper systematically investigates the structure and construction of bent functions over arbitrary finite fields of odd prime characteristic. We introduce a novel bentness criterion based on the second-order derivative being identically a nonzero constant—generalizing the binary cubic-like bent concept—and develop a higher-order derivative analysis framework applicable to all odd prime characteristics. Contribution/Results: We present the first elementary construction of infinite families of ternary cubic bent functions that are provably non-weakly regular. Rigorous proofs establish both their bentness and non-weak regularity. These results extend the theoretical boundaries of cubic bent functions and bridge a critical gap in the study of non-binary bent functions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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• Security and Privacy: Applications of cryptography
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• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

Much work has been devoted to bent functions in odd characteristic, but a gap remains between our knowledge of binary and nonbinary bent functions. In the first part of this paper, we attempt to partially bridge this gap by generalizing to any characteristic important properties known in characteristic 2 concerning the Walsh transform of derivatives of bent functions. Some of these properties generalize to all bent functions, while others appear to apply only to weakly regular bent functions. We deduce a method to obtain a bent function by adding a quadratic function to a weakly regular bent function. We also identify a particular class of bent functions possessing the property that every first-order derivative in a nonzero direction has a derivative (which is then a second-order derivative of the function) equal to a nonzero constant. We show that this property implies bentness and is shared in particular by all cubic bent functions. It generalizes the notion of cubic-like bent function, that was introduced and studied for binary functions by Irene Villa and the first author. In the second part of the paper, we provide (for the first time) a primary construction leading to an infinite class of cubic bent functions that are not weakly regular. We show the bentness of the functions by two approaches: by calculating the Walsh transform directly and by considering the second-order derivatives (and applying the results from the first part of the paper).

Problem

• Generalizes properties of binary bent functions to nonbinary odd characteristic cases.
• Develops a method to construct bent functions from weakly regular ones.
• Provides the first construction of cubic non-weakly regular bent functions.

Innovation

• Generalizing Walsh transform properties to odd characteristic
• Constructing bent functions via quadratic additions to weakly regular ones
• Creating infinite class of non-weakly regular cubic bent functions