Edit distances and probabilities for correlation attacks on clock-controlled combiners with memory

A theoretical framework for correlation attacks based on edit distances and edit probabilities on binary keystream generators consisting of clock-controlled shift registers combined by a function with memory is introduced. Recursive algorithms for efficient computation of the proposed many-to-one string edit distances and statistically optimal edit probabilities are derived for both constrained and unconstrained irregular clocking. The distances and probabilities are based on mutually correlated linear transforms of input and output sequences in the corresponding regularly clocked combiner with memory. Linear transforms can also incorporate linear models of clock-controlled shift registers. The complexity of the recursive algorithms is exponential in the memory size of the input linear transform which can be considerably smaller than the memory size of combining function. This is demonstrated for a special type of combiners with memory based on a time-varying memoryless function. In addition, a decimation method for reducing the memory size of the input linear transform is proposed. The design criteria with respect to the introduced correlation attacks are also discussed.