Generalized bent (gbent) functions from an $n$-variable Boolean space to $\mathbb{Z}_{2^k}$ are central in cryptography and sequence design. Instead of the usual binary decomposition, we introduce a $2^\ell$-adic representation, for $k=\ell r$, writing such functions as linear combinations of $r$ component functions valued in $\mathbb{Z}_{2^\ell}$. We prove a general result on overconstrained character sums over finite abelian groups: under a common-argument hypothesis, sequences with two-level Fourier magnitude spectra must be extremely sparse, with a conditional extension to multi-level spectra. As an application, we derive consequences for generalized plateaued functions under suitable assumptions. We then show that if $f:\mathbb{F}_2^n\to\mathbb{Z}_{2^k}$ is landscape, then under the $2^\ell$-adic decomposition every function in a certain affine space over $\mathbb{Z}_{2^\ell}$ is again landscape with the same Walsh magnitudes. This gives an unconditional necessity result, with no structural assumptions on $f$, together with a complete characterization using only a small subset of these maps. For generalized bent and generalized plateaued functions, sufficiency is also obtained from linear combinations of lower components under natural assumptions; a counterexample shows these assumptions are essential. Our method reduces verification for landscape functions from $2^{2^{k-1}}$ checks to fewer than $2^{k-\ell+1}+1$ conditions; for gbent functions this drops to a single basis function under the common-argument hypothesis, and for generalized plateaued functions, under additional assumptions, to $2^{k-\ell}$ checks. The $2^\ell$-adic framework also preserves key properties, including duality and differential uniformity.

翻译：从$n$变量布尔空间到$\mathbb{Z}_{2^k}$的广义弯曲（gbent）函数在密码学和序列设计中处于核心地位。不同于常规的二进制分解，我们对$k=\ell r$的情况引入$2^\ell$进制表示，将此类函数写为$r$个取值于$\mathbb{Z}_{2^\ell}$的分量函数的线性组合。我们证明了关于有限阿贝尔群上过约束特征和的一个一般性结果：在共同自变量假设下，具有两级傅里叶幅度谱的序列必须极度稀疏，并将该结果条件性地推广至多级谱情形。作为应用，我们在适当假设下推导了广义平台函数的若干推论。进而证明：若$f:\mathbb{F}_2^n\to\mathbb{Z}_{2^k}$是景观函数，则在$2^\ell$进制分解下，$\mathbb{Z}_{2^\ell}$上某个仿射空间中的每个函数仍为具有相同Walsh幅度的景观函数。这给出了一个无结构假设条件的无条件必然性结果，以及仅利用这些映射中一个小子集的完全刻画。对于广义弯曲函数和广义平台函数，在自然假设下也可通过低阶分量的线性组合获得充分性；一个反例表明这些假设是本质性的。我们的方法将景观函数的验证从$2^{2^{k-1}}$次检查缩减至少于$2^{k-\ell+1}+1$个条件；对于广义弯曲函数，在共同自变量假设下可降至单个基函数；对于广义平台函数，在额外假设下可降至$2^{k-\ell}$次检查。该$2^\ell$进制框架还保留了包括对偶性和差分均匀性在内的关键性质。