Bijectivity of a Class of Feedback Shift Register with Memory

The initialization stage and keystream generation stage are special cases of the feedback shift register with memory in many stream ciphers. This paper investigates the bijectivity of a class of FSR with memory, and then presents necessary and sufficient conditions under which the update transformations of inner state are bijective. As applications, the SNOW 3G cipher, ZUC 1.4 cipher and ZUC 1.5 cipher are analyzed. It is shown that the initialization stages of SNOW 3G and ZUC 1.5 are both bijective. However, the initialization stage of ZUC 1.4 is not bijective, which is the essential reason for ZUC 1.4 suffering from the chosen IV attacks.