A Note on the Periodicity and the Output Rate of Bit Search Type Generators

TL;DR

This paper analyzes the periodicity and output rate of bit-search type generators, specifically BSG and ABSG, providing bounds, classifications, and probabilistic distributions for their output sequences in different input scenarios.

Contribution

It offers the first bounds on expected output periods and characterizes the output distribution for BSG and ABSG under both deterministic and probabilistic inputs.

Findings

Classified output sequences into two shift-equivalent sets for m-sequence inputs.

Derived bounds on the expected output period assuming no subperiods.

Obtained closed-form expressions for output length distribution and proved asymptotic Gaussianity.

Abstract

We investigate the bit-search type irregular decimation algorithms that are used within LFSR-based stream ciphers. In particular, we concentrate on BSG and ABSG, and consider two different setups for the analysis. In the first case, the input is assumed to be a m-sequence; we show that all possible output sequences can be classified into two sets, each of which is characterized by the equivalence of their elements up to shifts. Furthermore, we prove that the cardinality of each of these sets is equal to the period of one of its elements and subsequently derive the first known bounds on the expected output period (assuming that no subperiods exist). In the second setup, we work in a probabilistic framework and assume that the input sequence is evenly distributed (i.e., independent identically distributed Bernoulli process with probability 1/2). Under these assumptions, we derive closed-form expressions for the distribution of the output length and the output rate, which is shown to be asymptotically Gaussian-distributed and concentrated around the mean with exponential tightness.