Continued Fraction Expansion as Isometry: The Law of the Iterated Logarithm for Linear, Jump, and 2--Adic Complexity

TL;DR

This paper introduces an isometry-based approach to analyze the complexity profiles of sequences used in cryptography, applying probabilistic bounds like the Law of the Iterated Logarithm to model their randomness.

Contribution

It defines new isometries for formal power series and dyadic integers that enable precise probabilistic analysis of sequence complexities in cryptography.

Findings

Linear and jump complexities can be modeled as Bernoulli experiments.

The complexity profiles follow the Law of the Iterated Logarithm.

Average behavior of 2-adic complexity resembles coin tossing.

Abstract

In the cryptanalysis of stream ciphers and pseudorandom sequences, the notions of linear, jump, and 2-adic complexity arise naturally to measure the (non)randomness of a given string. We define an isometry K on F_q^\infty that is the precise equivalent to Euclid's algorithm over the reals to calculate the continued fraction expansion of a formal power series. The continued fraction expansion allows to deduce the linear and jump complexity profiles of the input sequence. Since K is an isometry, the resulting F_q^\infty-sequence is i.i.d. for i.i.d. input. Hence the linear and jump complexity profiles may be modelled via Bernoulli experiments (for F_2: coin tossing), and we can apply the very precise bounds as collected by Revesz, among others the Law of the Iterated Logarithm. The second topic is the 2-adic span and complexity, as defined by Goresky and Klapper. We derive again an isometry, this time on the dyadic integers Z_2 which induces an isometry A on F_2}^\infty. The corresponding jump complexity behaves on average exactly like coin tossing. Index terms: Formal power series, isometry, linear complexity, jump complexity, 2-adic complexity, 2-adic span, law of the iterated logarithm, Levy classes, stream ciphers, pseudorandom sequences