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A Novel Approach for Bent Functions with Dillon-like Exponents and Characterizing Three Classes of Bent Functions via Kloosterman Sums

Ziran Tu, Sihem Mesnager, Xiangyong Zeng, Nian Li, Yupeng Jiang, Yanan Deng

Abstract

Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from $\mathbb{F}_{2^{2n}}$ to $\mathbb{F}_{2}$, with all the exponents being multiples of $2^n-1$ often called Dillon-like exponents. This paper is devoted to bent functions in which we study the bentness of some classes of Dillon-like Boolean functions connected with rational trace functions. Specifically, we introduce a special infinite family of trace rational functions. We shall use these functions as building blocks and generalise notably a criterion due to Li et al. published in [IEEE Trans. Inf. Theory 59(3), pp. 1818-1831, 2013] on the bentness of Dillon-like functions in the binary case, we explicitly characterize three classes of bent functions. These characterizations are expressed in terms of the well-known binary Kloosterman sums. Furthermore, analysis and experiments indicate that new functions not EA-equivalent to all known classes of monomial functions are included in our classes.

A Novel Approach for Bent Functions with Dillon-like Exponents and Characterizing Three Classes of Bent Functions via Kloosterman Sums

Abstract

Dillon-like Boolean functions are known, in the literature, to be those trace polynomial functions from to , with all the exponents being multiples of often called Dillon-like exponents. This paper is devoted to bent functions in which we study the bentness of some classes of Dillon-like Boolean functions connected with rational trace functions. Specifically, we introduce a special infinite family of trace rational functions. We shall use these functions as building blocks and generalise notably a criterion due to Li et al. published in [IEEE Trans. Inf. Theory 59(3), pp. 1818-1831, 2013] on the bentness of Dillon-like functions in the binary case, we explicitly characterize three classes of bent functions. These characterizations are expressed in terms of the well-known binary Kloosterman sums. Furthermore, analysis and experiments indicate that new functions not EA-equivalent to all known classes of monomial functions are included in our classes.

Paper Structure

This paper contains 8 sections, 12 theorems, 97 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and background
  4. On Dillon-like functions of the form $f=g(x^{2^n-1})$
  5. Some Relative Exponential Sums
  6. Main Results
  7. On EA Equivalence
  8. Conclusion

Key Result

Lemma 2.1

Rosendahl Let $m=2n$ be a positive integer, $A\in \mathbb{F}_{2^{m}}\setminus\mathbb{F}_{2^n}$ be fixed and $U$ be given in Eq.defU. Then

Theorems & Definitions (28)

  • Definition 1
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 18 more