Reduction of the key space of the cipher A5/1 and invertibility of the next-state function for a stream generator

We study the stream ciphers that are based on the feedback shift registers. For a stream generator (in general form), we prove a theorem which allows us to equate the concept of invertibility of the next-state function and the concept of recurrency of the shift control function. Then we study a generator for the stream cipher A5/1 used in the GSM cellular telephone standard to ensure the confidentiality of conversations. For this generator, we count the number of states that can be obtained after t clock cycles from the initial states without predecessors and cannot be obtained in this way after the smaller number of cycles. We show how to exponentially reduce the key space of A5/1 while clocking. The results can be directly used in cryptanalysis of A5/1. © 2012 Pleiades Publishing, Ltd.