New results on the state cycles of Trivium

Trivium is a well-known lightweight synchronous stream cipher that was submitted to the European eSTREAM project in April 2005. Thus far, it has been difficult to determine cycles of the internal state values of Trivium, remaining an open problem to date. In this paper, the internal structure of Trivium is exploited to generate an algorithm called Trivium-model. Based on Trivium-model, some new results are obtained through the state cycles of Trivium. It is noted that there exists one cycle of size 10, one cycle of size 11, two cycles of size 12, one cycle of size 15 and at least 16 cycles of size \(3(2^{93}-1)\). Therefore, the distribution of cycle lengths is not equiprobable, and small cycles exist. These cycles may cause the weak initial states and be the vulnerability of Trivium.