Overconstrained Character Sums and Tower Decompositions: A Local–Global Perspective on Generalized Boolean Functions

Generalized Boolean functions with highly regular Walsh–Hadamard spectra, such as generalized bent functions, play an important role in cryptography, coding theory, and sequence design. Their structure is commonly studied via binary decompositions, but this viewpoint rapidly becomes unwieldy as the modulus grows. In this talk, we develop a \(2^\ell\)-adic tower decomposition perspective motivated by rigidity phenomena for character sums over finite abelian 2-groups. A recurring theme is that certain spectral constraints are severely overdetermined: character sums supported on small subsets and restricted to few magnitudes exhibit strong sparsity and vanishing behavior. This can be interpreted as a local–global principle, whereby global spectral regularity is forced by limited local information. These ideas lead to flexible decomposition results for generalized Boolean functions taking values in \(\mathbb{Z}_{2^k}\) , relating their spectral behavior to that of families of lower-modulus components. Rather than relying on exhaustive spectral checks, the tower viewpoint isolates structural conditions that propagate across levels. Variants of this approach also apply to plateaued and related classes of functions. The methods draw on tools from additive combinatorics and algebraic number theory, including uncertainty principles for finite abelian groups, character-theoretic rigidity, and Galoistheoretic symmetry considerations. Applications include potential explanations for observed spectral deficiencies in cryptographic constructions and new avenues for systematic design. Overall, the tower decomposition viewpoint provides a unifying lens for understanding spectral regularity phenomena in generalized Boolean functions.