### Upper bounds on the numbers of resilient functions and of bent
functions

To appear in Special Issue Dedicated to
Philippe Delsarte, Springer Verlag LNCS.
Authors:

Andrew Klapper, 779A Anderson Hall, Dept. of Computer Science,
University of Kentucky, Lexington, KY, 40506-0046, klapper at cs.uky.edu.
www.cs.uky.edu/~klapper/andy.html

Claude Carlet, INRIA

**Abstract**
Bent and resilient functions play significant roles in cryptography, coding
theory, and combinatorics. However, the numbers of bent and resilient
functions on a given number of variables are not known. Even a reasonable
bound on the number of bent functions is not known and the best known bound
on the number of resilient functions seems weak for functions of high
orders. In this paper we present new bounds which significantly
improve upon those which can be directly deduced from the restrictions on
the degrees of these functions. In the case of bent functions, it is the
first one of this type. In the case of m-resilient functions, it improves
upon the known bounds for m large.

**Index Terms --**
Boolean function, nonlinearity, bent, resilient, correlation-immune.