Simultaneous Shifted Continued Fraction Expansions in Quadratic Time

Abstract

In the theory of stream ciphers, important measures to assess the randomness of a given bit sequence (a_1,\dots,a_n) are the linear and the jump complexity, both obtained from the continued fraction expansion (c.f.e.) of the generating function of the sequence. This paper describes a way to compute all continued fraction expansions (and thereby the linear complexity profiles, l.c.p.'s) of the n shifted sequences (a_n), (a_{n-1},a_n), \dots , (a_1,\dots,a_n) simultaneously in at most 4.5\cdot (n^2+n) bit operations on n + 3 + 2\cdot\lceil \log_2 n\rceil bit space. If n is not fixed beforehand, but varies during the computation, we have to start from a_1. In this case we obtain the result in 4.5\cdot (n^2+n) bit operations on 2\cdot n \cdot \lceil\log_2 n\rceil + 3\cdot n space or as well in 9.5\cdot (n^2+n) bit operations on linear 4\cdot n + 2\cdot \lceil\log_2 n\rceil + 1 space. In comparison, the well-known Berlekamp--Massey algorithm, when applied iteratively, needs O(n^3) steps. A recent algorithm of Stephens and Lunnon works in O(n^2) integer operations, but only gives the l.c.p., not the complete c.f.e.