What question did this study set out to answer?

This paper aims to identify and characterize two new classes of complete permutation polynomials over finite fields.

May 27, 2026

Two New Classes of Complete Permutation Polynomials Over Finite Fields 𝔽33n

Key Points

This paper aims to identify and characterize two new classes of complete permutation polynomials over finite fields.
Investigated two specific polynomial forms for completeness in permutation properties.
Analyzed multiple values of variables to establish conditions for completeness.
Characterized the second class of polynomials for specific variable instances.
Identified four specific values for the first class where completeness is achieved for all polynomials.
Characterized the second class under various conditions, providing comprehensive forms.
Demonstrated the conditions necessary for completeness in both classes.

Abstract

In this paper, we investigate two classes of complete permutation polynomials over finite fields Formula: see text. The first class of complete permutation polynomial is of the form Formula: see text. In this class, for a fixed Formula: see text, we find four values of Formula: see text Formula: see text for which Formula: see text is a complete permutation polynomial over Formula: see text for every polynomial Formula: see text. The second class of complete permutation polynomial is of the form Formula: see text. We characterize this class for several specific values of Formula: see text and Formula: see text.

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