Linear complexity over F_q and over F_{q^m} for linear recurring sequences

Since the \F_q-linear spaces \F_q^m and \F_{q^m} are isomorphic, an $m$-fold multisequence $\mathbf{S}$ over the finite field \F_q with a given characteristic polynomial f \in \F_q[x], can be identified with a single sequence $\mathcal{S}$ over \F_{q^m} with characteristic polynomial $f$. The linear complexity of $\mathcal{S}$, which we call the generalized joint linear complexity of $\mathbf{S}$, can be significantly smaller than the conventional joint linear complexity of $\mathbf{S}$. We determine the expected value and the variance of the generalized joint linear complexity of a random $m$-fold multisequence $\mathbf{S}$ with given minimal polynomial. The result on the expected value generalizes a previous result on periodic $m$-fold multisequences. Finally we determine the expected drop of linear complexity of a random $m$-fold multisequence with given characteristic polynomial $f$, when one switches from conventional joint linear complexity to generalized joint linear complexity

Generalized joint linear complexity of linear recurring multisequences

The joint linear complexity of multisequences is an important security measure for vectorized stream cipher systems. Extensive research has been carried out on the joint linear complexity of $N$-periodic multisequences using tools from Discrete Fourier transform. Each $N$-periodic multisequence can be identified with a single $N$-periodic sequence over an appropriate extension field. It has been demonstrated that the linear complexity of this sequence, the so called generalized joint linear complexity of the multisequence, may be considerably smaller than the joint linear complexity, which is not desirable for vectorized stream ciphers. Recently new methods have been developed and results of greater generality on the joint linear complexity of multisequences consisting of linear recurring sequences have been obtained. In this paper, using these new methods, we investigate the relations between the generalized joint linear complexity and the joint linear complexity of multisequences consisting of linear recurring sequences

Stream ciphers and linear complexity

Master'sMASTER OF SCIENC

Studies on error linear complexity measures for multisequences

Ph.DDOCTOR OF PHILOSOPH

On the calculation of the linear complexity of periodic sequences

Based on a result of Hao Chen in 2006 we present a general procedure how to reduce the determination of the linear complexity of a sequence over a finite field \F_q of period $un$ to the determination of the linear complexities of $u$ sequences over \F_q of period $n$. We apply this procedure to some classes of periodic sequences over a finite field \F_q obtaining efficient algorithms to determine the linear complexity

Fast Algorithms for Finding the Characteristic Polynomial of a Rank-2 Drinfeld Module

This thesis introduces a new Monte Carlo randomized algorithm for computing the characteristic polynomial of a rank-2 Drinfeld module. We also introduce a deterministic algorithm that uses some ideas seen in Schoof's algorithm for counting points on elliptic curves over finite fields. Both approaches are a significant improvement over the current literature