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A theoretical framework for fast correlation attacks on irregularly clocked linear feedback shift registers (LFSRs) based on a recently established linear statistical weakness of decimated LFSR sequences is developed. When the LFSR feedback polynomial is not known, methods for the statistical weakness detection and the feedback polynomial reconstruction are proposed. When the LFSR feedback polynomial is known, an iterative procedure for fast LFSR initial state reconstruction given an observed keystream sequence is introduced. The procedure is based on appropriately defmed parity-check sums and consists in iterative recomputation of the posterior probabilities for unknown elements of the decimation sequence. A convergence condition in terms of the numbers of the parity-check sums needed for successful reconstruction and the required polynomial computational complexity indicate that the proposed fast correlation attack may be realistic, especially in the constrained clocking case. The number of the feedback polynomial multiples of relatively low weight and not too large degree thus proves to be critical for the security of irregularly clocked LFSRs.