Non-existence results for vectorial bent functions with Dillon exponent

TL;DR

This paper establishes new non-existence results for certain vectorial bent functions with Dillon exponents over specific finite fields, and provides conditions for their coefficients, aiding in the construction of examples.

Contribution

It proves the non-existence of vectorial Dillon-type bent functions for odd m > 3 and derives conditions for bent coefficients when m is even, facilitating example construction.

Findings

No such functions exist for odd m > 3.

Derived a condition for the bent coefficient when m is even.

Provided a method to find examples for m=6.

Abstract

We prove new non-existence results for vectorial monomial Dillon type bent functions mapping the field of order $2^{2m}$ to the field of order $2^{m/3}$. When $m$ is odd and $m>3$ we show that there are no such functions. When $m$ is even we derive a condition for the bent coefficient. The latter result allows us to find examples of bent functions with $m=6$ in a simple way.