Algebraic attack on the Alternating Step(r,s) Generator

The Alternating Step(r, s) Generator, ASG(r, s), is a clock-controlled sequence generator which is recently proposed by A. Kanso. It consists of three registers of length l, m and n bits. The first register controls the clocking of the two others. The two other registers are clocked r times (or not clocked) (resp. s times or not clocked) depending on the clock-control bit in the first register. The special case r = s = 1 is the original and well known Alternating Step Generator. Kanso claims there is no efficient attack against the ASG(r, s) since r and s are kept secret. In this paper, we present an Alternating Step Generator, ASG, model for the ASG(r, s) and also we present a new and efficient algebraic attack on ASG(r, s) using 3(m + n) bits of the output sequence to find the secret key with O((m²+n²)2^l+1+m³2^m-1+n³2^n-1) computational complexity. We show that this system is no more secure than the original ASG, in contrast to the claim of the ASG(r, s)´s constructor.