Computing the biases of parity-check relations

TL;DR

This paper introduces exact formulas and a new algorithm for computing the bias of parity-check relations in cryptanalysis, improving the ability to distinguish keystream generators by analyzing their biases.

Contribution

The paper provides the first exact expressions for the bias of parity-check relations and a novel algorithm for their computation, enhancing cryptanalysis techniques.

Findings

Two exact formulas for bias calculation

A new algorithm for bias estimation

Simplified formulas for common cryptographic cases

Abstract

A divide-and-conquer cryptanalysis can often be mounted against some keystream generators composed of several (nonlinear) independent devices combined by a Boolean function. In particular, any parity-check relation derived from the periods of some constituent sequences usually leads to a distinguishing attack whose complexity is determined by the bias of the relation. However, estimating this bias is a difficult problem since the piling-up lemma cannot be used. Here, we give two exact expressions for this bias. Most notably, these expressions lead to a new algorithm for computing the bias of a parity-check relation, and they also provide some simple formulae for this bias in some particular cases which are commonly used in cryptography.