Boolean functions with favorable cryptographic properties, including balancedness, high nonlinearity, high algebraic degree, high algebraic immunity, and high-order resiliency, are essential components of stream ciphers. Despite their importance, constructing Boolean functions that simultaneously exhibit these properties remains a significant challenge. Recently, Carlet introduced a general method for constructing Boolean functions, wherein the support of f equals the image set of an injective vectorial function F, which Carlet termed a “parameterization” of f. Inspired by Carlet’s work, this paper delves into the parametric construction of n-variable balanced Boolean functions, where the support of f coincides with the image set of two-to-one mappings over \({\mathbb {F}}_{2^n}\) . We revisit two prominent balanced Boolean functions–the Carlet-Feng function and the odd variable majority function–and demonstrate that they can be parameterized by linearly equivalent two-to-one mappings of the forms \(P_1(x+x^{-1})\) and \(P_2(x^2+x)\) , respectively. We then establish relationships between two-to-one mappings (especially ones of the forms \(P(x^2+x)\) and \(P(x+x^{-1})\) , where P is a permutation) and the algebraic degree, Walsh transform, or the algebraic immunity of the corresponding balanced Boolean functions. Notably, from two-to-one mappings of the forms \(P(x^2+x)\) or \(P(x+x^{-1})\) , we can readily ascertain whether the resulting balanced Boolean functions possess the optimal algebraic degree by analyzing the terms of P, and if one would like to obtain balanced Boolean functions with the optimal algebraic immunity, the algebraic degree of \(P^{-1}\) should be relatively high. Furthermore, we construct two infinite classes of semi-bent balanced Boolean functions using two-to-one mappings of the form \(P(x^2+x)\) , producing new functions that are EA-inequivalent to quadratic functions and the well-known Maiorana-McFarland class. Using MAGMA and the Linear Transformation method, we obtain n-variable \((n=6,10,14)\) 1-resilient semi-bent balanced Boolean functions. Finally, we present theoretical and experimental results on n-variable \((n\le 14)\) Boolean functions with desirable cryptographic properties, including balancedness, high nonlinearity, high algebraic degree, and near-optimal or optimal algebraic immunity. For one thing, from two-to-one mappings of the form \(P(x^2+x)\) , we can offer some n-variable ( \(n=6,10,14\) ) balanced Boolean functions with high nonlinearity, high algebraic degree, the optimal algebraic immunity, and 1-resiliency. Particularly, when \(n=6\) , our Boolean function reaches the Siegenthaler bound. In addition, the theoretical proof of determining the Walsh spectrum of a class of Boolean functions is also offered. For the other thing, we specifically identify balanced Boolean functions with the optimal algebraic degree, the optimal algebraic immunity, and high nonlinearity from two-to-one mappings of the form \(P(x+x^{-1})\) . The proof of the optimal algebraic degree has been easily provided.
