The Alternating Step Generator (ASG) is a PRNG combining 3 LFSRs. Output of the ASG is the XOR of the output of two clock-gated LFSRs. At each step, a single one of these LFSRs is clocked, according to the output of the control LFSR.
The best (AFAIK) claimed attack on the ASG is Reduced Complexity Attacks on the ASG (Slides). However it attacks a variant ASG', which output is the output of the clock-gated LFSR that was clocked.
Justification given in that paper is: It is known [13, 8, 12] that instead of working with the original definition of ASG we can consider a slightly different description (..). [13] only states It is not hard to show that ASG and ASG’ are equivalent (..) with no argument. I fail to find a discussion about ASG' in [8]. [12] has a constructive proof, but I fail to get it (unless I assume that one of the LFSR is known/enumerated).
Any clue on a proof/sketch/argument that an attack on ASG' reconstructing the initial states of the LFSRs can be turned into an equally efficient attack on ASG?