Bent functions are maximally nonlinear Boolean functions. They are important functions introduced by Rothaus and studied firstly by Dillon and next by many researchers for more than five decades. Rotation symmetric Boolean functions, introduced by Pieprzyk and Qu, are those Boolean functions which are invariant under the cyclic shifts of inputs. In this paper, we first give two flexible construction methods of bent functions on n variables by defining two subsets T ’s of F 2 n to modify the support of Maiorana–McFarland’s bent function f 0 ( x , y ) = x ⋅ π ( y ) ⊕ h ( y ) , where n = 2 m , x , y ∈ F 2 m , π is a permutation over F 2 m , and h is a Boolean function on m variables. These methods have corresponding constraints on π and h . Then, we deduce the dual functions of the newly constructed bent functions. Lastly, we propose the methods of constructing 2-rotation symmetric bent functions by redefining the two subsets T ’s of F 2 n and by imposing more restrictive constraints on π and h .

The notion of correlation immune functions has been introduced by Siegenthaler (1984) in symmetric cryptography in the framework of stream ciphers. At the conference CRYPTO’91 by Camion et al., it has been pointed out that this notion existed in statistics and combinatorics. It has recently been highlighted that such functions also play an important role in a new framework related to side-channel attack counter-measures. Since then, the interest in correlation immune Boolean functions has been renewed, and new challenges regarding these functions have appeared. Specifically, low Hamming weight correlation immune functions have been selected as useful for counter-measures to side-channel attacks. Despite their importance, the literature is not abundant in this research direction. Two very interesting articles in which such correlation immune functions were nicely explored, given this novel use of them. Carlet initiated the first one in 2013, and the second one is due to Carlet and Chen (2018). This paper deals with correlation immune Boolean functions aiming to produce more candidates of those processing low Hamming weights. We shall focus on correlation immune Boolean functions with Hamming weights power of 2 (which offer a flexibility to control the correlation immunity aspects) and present several methods of designing them. Some design methods are efficient and could be employed to derive such functions. Consequently, given two positive integers$n$and$m$, we derive new effective constructions of correlation immune Boolean functions with Hamming weight power of 2. Furthermore, an upper bound on the correlation immunity of the newly constructed$n$-variable Boolean functions with Hamming weight$2^{m}$was determined for$n-m\ge 0$. Besides, exact values and lower bounds on the maximum correlation immunity of those functions are explored and discussed, mainly when the values of$n$and$m$are very close. This paper also exhibits explicit examples of those correlation immune functions that illustrate our methods.

In this paper, we first present two systematic constructions of bent functions by modifying the truth tables of Rothaus's bent function and Maiorana-McFarland's bent function respectively. The number of the newly constructed bent functions by modifying the truth table of Rothaus's bent function is also determined. The methods of constructing self-dual bent functions are then given after the dual functions of these bent functions being determined. Finally, as an application, two constructions of 2-rotation symmetric bent functions are presented in this paper.

Rotation symmetric bent functions and their generation two-rotation symmetric bent functions are two classes of cryptographically significant Boolean functions. However, few constructions have been presented in the literature, which either have restriction on integer n or have algebraic degree no more than 4. In this paper, for any even integer n ≥ 4, three classes of bent functions are presented respectively. Most notably, the proposed n-variable rotation symmetric bent functions and two-rotation symmetric bent functions can have any possible algebraic degree ranging from 2 to n/2. Besides, we obtain bent idempotent functions with the maximal algebraic degree n/2.