Slicing Boolean Functions with Inner Products

Boolean functions with additional structure play an important role in symmetric cryptography, both for achieving strong cryptographic properties and for enabling efficient implementations. Recent works on homomorphic-friendly symmetric primitives, especially in the context of Hybrid Homomorphic Encryption (\HHE), highlighted the interest of Boolean functions whose evaluation can be decomposed according to structured partitions of the Boolean cube. A classical example is given by Hamming-weight decompositions, which underlie symmetric and weightwise degree-$d$ Boolean functions. In this work, we generalize this viewpoint by replacing the Hamming weight with a general integer linear form. Given a vector $v\in\Z^n$, we partition the Boolean cube $\F_2^n$ by grouping the Boolean vectors $x\in\F_2^n$ according to the value of $\langle v,x\rangle$ into so-called $v$-slices, and study functions that have bounded degree on each of them. We study how many such slices are needed to describe a given function, and provide bounds and structural properties for the partitions induced by integer vectors. We also show how this representation leads to a homomorphic evaluation strategy in a \GSW-like setting, together with noise estimates for the resulting ciphertexts. Finally, we generalize several symmetric and weightwise degree-$d$ Boolean functions using different vector families, and experimentally evaluate their algebraic degree, algebraic immunity, and nonlinearity. The results show that direct generalizations of symmetric functions often lose cryptographic strength, while generalized weightwise degree-$d$ constructions lead to richer and more promising behavior.