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

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

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

Studies on error linear complexity measures for multisequences

Ph.DDOCTOR OF PHILOSOPH

Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

Generalized joint linear complexity of linear recurring sequences, in: S.W. Golomb, et al

Abstract. 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 Nperiodic 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