This article focuses on the study of nondegeneracy and curvature of some cryptographic Boolean functions.
Nondegeneracy of a Boolean function \(f:\mathbb{F}_{2^n}\rightarrow \mathbb{F}_2\) is the distance between \(f\) and the set of all algebraically degenerate Boolean functions and it is given by the formula \[ \text{nd}(f)=2^{n-1}-\frac{1}{4} \max_{\mathbf{0} \neq u\in \mathbb{F}_2^n} \Delta_f(u). \] Here, \(\Delta_f\) is the autocorrelation of \(f\) with itself which is \[ \Delta_f(u) = \sum_{x \in \mathbb{F}_2^n} (-1)^{f(x+u)\oplus f(x)}. \] The curvature \(\text{curv}(f)\) of a Boolean function \(f\) is calculated as the sum of the absolute values of its Walsh transform values and we always have \[ 2^n\leq \text{curv}(f) \leq 2^{3n/2}. \] In Section 2, the author considers some special classes of Boolean functions, subclasses of Maiorana-McFarland functions and calculate the exact values and bounds for the nondegeneracy and curvature of these special classes. In Section 3, the curvature results are given for the functions which are obtained through other Boolean functions by changing just one or multiple random values in the value set of the function. In Section 4, curvature definition is extended to vectorial Boolean functions from \(\mathbb{F}_{2^n}\) to \(\mathbb{F}_{2^n}\) and then certain S-boxes are analyzed for the curvature.