Constrained Embedding Probability for Two Binary Strings

An exponentially small upper bound on the probability that a given binary string of length $n$ can be embedded into a uniformly distributed random binary string of length $2n$ by inserting at most one bit between any two successive bits and an arbitrary number of bits at the end is analytically derived. This probability is important for a cryptanalytic problem of the initial state reconstruction of a binary clock-controlled shift register that is clocked either once or twice per each output symbol, given a segment of its output sequence. The developed approach may also be interesting for other problems of sequence comparison as well, especially for the codes for correcting synchronization errors.