  In this paper two algorithms solving circuit satisfiability problem over
supernilpotent algebras are presented. The first one is deterministic and is
faster than fastest previous algorithm presented by Aichinger. The second one
is probabilistic with linear time complexity. Application of the former
algorithm to finite groups provides time complexity that is usually lower than
in previously best (given by F\"oldv\'ari) and application of the latter leads
to corollary, that circuit satisfiability problem for group G is either
tractable in probabilistic linear time if G is nilpotent or is NP-complete if G
fails to be nilpotent. The results are obtained, by translating equations
between polynomials over supernilpotent algebras to bounded degree polynomial
equations over finite fields.
