  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.
