Several Classes of Niho Type Boolean Functions with Few Walsh Transform Values

Abstract

Boolean functions with n variables are functions from $\mathbb {f}_{2^n}$ to $\mathbb {f}_2$. They play an important role in both cryptographic and error correcting coding activities. The important information about the cryptographic properties of Boolean functions can be obtained from the study of the Walsh transform. Generally speaking, it is difficult to construct functions with few Walsh transform values and determine the Walsh transform value completely due to the difficulty in solving equations. In this paper, we study the Walsh transform of the Niho type Boolean function with the form

$$ f(x)=\sum _{i=l}^{k}\mathrm{{Tr}}_{1}^{n}(ax^{s_{i}(2^m-1)+1}), $$

where k, l, m, n are positive integers satisfying $1\le l \le k<2^m$, $n=2m$ and $a+a^{2^m} \ne 0$. By choosing $s_i$ properly, three classes of such functions with at most 5-valued Walsh transform are obtained. Besides, by using particular techniques in solving equations over finite fields, the value distributions of the Walsh transform for these functions are also completely determined.