Shrinking generators and statistical leakage

Shrinking is a newly proposed technique for combining a pair of pseudo random binary sequences, ( , ), to form a new sequence, , with better randomness, where randomness here stands for difficulty of prediction. The ones in the second sequence are used to point out the bits in the sequence to be included in . The generator that performs this process is known as the shrinking generator (SG). In this paper, it is shown for the existing combining method that deviation from randomness in the statistics of leads to the leakage of this statistics into . We also show that it is sufficient for constructing a statistically balanced SG to at least have one statistically balanced generator. A new shrinking rule that yields statistically balanced output, even if and are not balanced, is then proposed. Self-shrinking in which a single pseudo random bit generator (PRBG) shrinks itself is also investigated and a modification of the existing shrinking rule is proposed. Simulation results show the robustness of the proposed methods. For self-shrinking, in particular, results show that the proposed shrinking rule yields sequences with balanced statistics even for extremely biased generators. This suggests possible application of the new rule to strengthen running key generators.