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. On the other hand showing that AND can be computed by a polynomial size probabilistic 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.

中文翻译：

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模块化电路的复杂性

我们研究模块化电路计算的复杂性如何取决于电路的深度和它们使用的模数的素因数分解。特别是我们为 AND 构建的深度为 2 的次指数电路有助于我们根据其可满足性的复杂性对（模指数时间假设）模块化电路进行分类。我们还研究了这种复杂性与实现 AND 的模块化电路大小之间的精确相关性。另一方面，表明 AND 可以通过深度为 2 的多项式大小概率模块化电路（具有 O(log n) 个随机位）计算，从而提供无法去随机化的概率计算模型。我们应用我们的方法来确定（模 ETH）在具有奇数边的正多边形对称组上求解方程的复杂性。