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

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

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

Studies on error linear complexity measures for multisequences

Ph.DDOCTOR OF PHILOSOPH

ANALYSIS OF SECURITY MEASURES FOR SEQUENCES

Stream ciphers are private key cryptosystems used for security in communication and data transmission systems. Because they are used to encrypt streams of data, it is necessary for stream ciphers to use primitives that are easy to implement and fast to operate. LFSRs and the recently invented FCSRs are two such primitives, which give rise to certain security measures for the cryptographic strength of sequences, which we refer to as complexity measures henceforth following the convention. The linear (resp. N-adic) complexity of a sequence is the length of the shortest LFSR (resp. FCSR) that can generate the sequence. Due to the availability of shift register synthesis algorithms, sequences used for cryptographic purposes should have high values for these complexity measures. It is also essential that the complexity of these sequences does not decrease when a few symbols are changed. The k-error complexity of a sequence is the smallest value of the complexity of a sequence obtained by altering k or fewer symbols in the given sequence. For a sequence to be considered cryptographically ‘strong’ it should have both high complexity and high error complexity values. An important problem regarding sequence complexity measures is to determine good bounds on a specific complexity measure for a given sequence. In this thesis we derive new nontrivial lower bounds on the k-operation complexity of periodic sequences in both the linear and N-adic cases. Here the operations considered are combinations of insertions, deletions, and substitutions. We show that our bounds are tight and also derive several auxiliary results based on them. A second problem on sequence complexity measures useful in the design and analysis of stream ciphers is to determine the number of sequences with a given fixed (error) complexity value. In this thesis we address this problem for the k-error linear complexity of 2n-periodic binary sequences. More specifically: 1. We characterize 2n-periodic binary sequences with fixed 2- or 3-error linear complexity and obtain the counting function for the number of such sequences with fixed k-error linear complexity for k = 2 or 3. 2. We obtain partial results on the number of 2n-periodic binary sequences with fixed k-error linear complexity when k is the minimum number of changes required to lower the linear complexity

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