Discrete logarithm for nilpotent groups and cryptanalysis of polylinear cryptographic system

We present an efficient algorithm to compute a discrete logarithm in a finite nilpotent group, or more generally, in a finitely generated nilpotent group. Special cases of a finite $p$-group ($p$ is a prime) and a finitely generated torsion free nilpotent group are considered. Then we show how the derived algorithm can be generalized to an arbitrary finite or finitely generated nilpotent group respectively. We suppose that group is presented by generating elements and defining relators or like a subgroup of a triangular matrix group over a prime finite field (in finite case) or over the ring of integers (in torsion-free case). On the base of the derived algorithm we give a cryptanalysis of some schemes of polylinear cryptography known in the literature.