一句话：研究模电路计算 AND 的复杂度及其可满足性分类，并应用于有限群方程求解。
论文研究模电路计算 AND 的复杂度如何依赖电路深度与模数的素因子分解。作者构造了深度为 2 的次指数规模模电路来计算 AND，并利用已知的 AND 超线性下界，设计出在概率 2^{O(n/ε(n))} 时间内检查 CC0 电路可满足性的算法，其中 ε(n) 是增长极慢的函数。论文还给出多项式规模、深度 2 且仅使用 O(log n) 随机比特的模电路计算 AND，从而得到一个无法去随机化的概率计算模型。最后，在指数时间假设（ETH）下，作者将方法用于正奇数边多边形对称群上的方程求解，确定其复杂度，并为有限群方程求解的复杂度刻画提供范例。
We study how the complexity of modular circuits computing AND depends on the depth of the circuits and the prime factorization of the modulus they use. In particular our construction of subexponential circuits of depth 2 for AND helps us to classify (modulo Exponential Time Hypothesis) modular circuits with respect to the complexity of their satisfiability. We also study a precise correlation between this complexity and the sizes of modular circuits realizing AND. In particular we use the superlinear lower bound from [10] to check satisfiability of CC0 circuits in probabilistic 2O(n/ε(n)) time, where ε is some extremely slowly increasing function. Moreover we show that AND can be computed by a polynomial size modular circuit of depth 2 (with O(log n) random bits) providing a probabilistic computational model that can not be derandomized. We apply our methods to determine (modulo ETH) the complexity of solving equations over groups of symmetries of regular polygons with an odd number of sides. These groups form a paradigm for some of the remaining cases in characterizing finite groups with respect to the complexity of their equation solving.