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

| 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 de ne an isometry K on Fq 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 proles of the input sequence. Since K is an isometry, the resulting Fq {sequence is i.i.d. for i.i.d. input. Hence the linear and jump complexity pro les may be modelled via Bernoulli experiments (for F2: coin tossing), and we can apply the very precise bounds as collected by R ev esz, among others the Law of the Iterated Logarithm. The second topic is the 2{adic span and complexity, as de ned by Goresky and Klapper. We derive again an isometry, this time on the dyadic integers Z2 which induces an isometry A on F2. 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, L evy classes, stream ciphers, pseudorandom sequences Supported by Project FONDECYT 2001, No. 1010533 of CONICYT, Chile