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The paper illustrates the concept of the map with logarithmically invertible decomposition. We introduce families of multivariate cryptosystems such that there security level is connected with discrete logarithm problem in Cremona group. The private key of such cryptosystem is a modification of graph based stream ciphers which use stable multivariate maps. Modified version corresponds to a stable map with single disturbance. If the disturbance (or initial condition) allows fast computation thendoi:10.15439/2014f269 dblp:conf/fedcsis/Ustimenko14 fatcat:ngbjs7ufmfaaxhsq65i3vtoqy4