In this work, we close the aforementioned gap by introducing a parametrized leakage-abuse attack that applies to practical response-hiding structured encryption schemes. The use of non-parametric estimation techniques makes our attack agnostic to both the data and the query distribution. At the very core of our technique lies the newly defined concept of a counting function with respect to a range scheme. We propose a two-phase framework to approximate the counting function for any range scheme. By simply switching one counting function for another, i.e., the so-called “parameter” of our modular attack, an adversary can attack different encrypted range schemes. We propose a constrained optimization formulation for the attack algorithm that is based on the counting functions. We demonstrate the effectiveness of our leakage-abuse attack on synthetic and real-world data under various scenarios.