Complexity Measures For Sequences Research Articles

The Eigenvalue Complexity of Sequences in the Real Domain

The eigenvalue is one of the important cryptographic complexity measures for sequences. However, the eigenvalue can only evaluate sequences with finite symbols—it is not applicable for real number sequences. Recently, chaos-based cryptography has received widespread attention for its perfect dynamical characteristics. However, dynamical complexity does not completely equate to cryptographic complexity. The security of the chaos-based cryptographic algorithm is not fully guaranteed unless it can be proven or measured by cryptographic standards. Therefore, in this paper, we extended the eigenvalue complexity measure from the finite field to the real number field to make it applicable for the complexity measurement of real number sequences. The probability distribution, expectation, and variance of the eigenvalue of real number sequences are discussed both theoretically and experimentally. With the extension of eigenvalue, we can evaluate the cryptographic complexity of real number sequences, which have a great advantage for cryptographic usage, especially for chaos-based cryptography.

Distribution of one-error linear complexity of binary sequences for arbitrary prime period

Complexity measures for sequences, such as the linear complexity and the k-error linear complexity, play an important role in stream ciphers. This contribution studies the distribution of 1-error linear complexity of binary sequences with arbitrary prime period. For any odd prime N, the authors present all possible values of 1-error linear complexity of N-periodic binary sequences, and derive the exact formulas to count the number of N-periodic binary sequences with any given 1-error linear complexity.

Error linear complexity measures for multisequences

Complexity measures for sequences over finite fields, such as the linear complexity and the k-error linear complexity, play an important role in cryptology. Recent developments in stream ciphers point towards an interest in word-based stream ciphers, which require the study of the complexity of multisequences. We introduce various options for error linear complexity measures for multisequences. For finite multisequences as well as for periodic multisequences with prime period, we present formulas for the number of multisequences with given error linear complexity for several cases, and we present lower bounds for the expected error linear complexity.

Visualization and analysis of DNA sequences using DNA walks

Visual methods illustrate how DNA sequences are read along a single DNA strand from the 5′ end to the 3′ end and they provide the hopes of gaining an understanding of the underlying genomic language. By handling genomic sequence residues as elements of a discrete-time signal, digital signal processing techniques can be employed for the analysis of genomic information. Using these representations and applying frequency domain transformations, it is shown that structures, or seemingly nonrandom behavior, may be readily identified in nucleotide sequences. We review the basic method of DNA walks and we show how these representations can be used to extract useful knowledge from the genomic data; namely long-range correlation information, sequence periodicities, and other sequence characteristics. Further information is elucidated through wavelet transform analysis. This work finally relates a measure of sequence complexity to these visual findings and offers conclusions regarding quantifying DNA sequence behavior or structure.

The expected value of the joint linear complexity of periodic multisequences

Complexity measures for sequences of elements of a finite field, such as the linear complexity, play an important role in cryptology. Recent developments in stream ciphers point towards an interest in word-based (or vectorized) stream ciphers, which require the study of the complexity of multisequences. We extend a well-known relationship between the linear complexity of an N-periodic sequence and the (generalized) discrete Fourier transform of N-tuples to the case of multisequences. Using the concept of the generalized discrete Fourier transform for multisequences, we compute the expected value of the joint linear complexity of random periodic multisequences, and for some types of period lengths N we determine the number N N t(c) of t N-periodic sequences with given joint linear complexity c.

Counting Functions and Expected Values for the k-Error Linear Complexity

In cryptology, complexity measures for sequences of elements of a finite field, such as the linear complexity, play an important role. Cryptographically strong sequences or finite strings must not only have a large linear complexity, but also the change of a few terms must not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity Ln,k(S) of a string S with terms in a finite field Fq and length n. In this article, bounds for the number of strings S of length n with k-error linear complexity Ln,k(S)=c or Ln,k(S)≤c for a given c are established. Under certain conditions on n, k, and c, exact formulas are also determined. On the basis of these results we derive bounds for the expected value of Ln,k(S) for random strings S of length n.

Linear Complexity, k-Error Linear Complexity, and the Discrete Fourier Transform

Complexity measures for sequences of elements of a finite field play an important role in cryptology. We focus first on the linear complexity of periodic sequences. By means of the discrete Fourier transform, we determine the number of periodic sequences S with given prime period length N and linear complexity LN, 0(S)=c as well as the expected value of the linear complexity of N-periodic sequences. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity LN, k(S) of sequences S with period length N. For some k and c we determine the number of periodic sequences S with given period length N and LN, k(S)=c. For prime N we establish a lower bound on the expected value of the k-error linear complexity.

A measurement of the sequence complexity of polysomal messenger RNA in sea urchin embryos

The first measurement has been made of the number of diverse mRNA sequences (mRNA sequence complexity) in the total polysomes of a eucaryotic system, the sea urchin gastrula. mRNA was purified of nuclear RNA and any other heterogeneous RNA contaminants by release from polysomes with puromycin. Trace quantities of labeled nonrepetitive DNA fragments were hybridized with an excess of mRNA. The hybridization reaction followed ideal first order kinetics in mRNA concentration. At completion of the hybridization reaction, 1.35% of the nonrepetitive DNA was present as mRNA-DNA hybrid. The hybridized DNA was extracted and was at least 70% hybridizable with mRNA, demonstrating a 50-fold purification of the expressed sequences. This purified DNA fraction reassociated with excess unfractionated sea urchin DNA at a rate identical to that of the total nonrepetitive DNA tracer. The mRNA had therefore been hybridized to nonrepetitive DNA sequence, and the amount of hybrid could be used as a direct measure of the mRNA sequence complexity. The complexity of the gastrula mRNA can be calculated as about 17 million nucleotides, sufficient to comprise some 14,000 distinct structural genes. This result also provides an estimate of the number of diverse proteins being translated in the gastrula. From the rate of mRNA-DNA hybrid formation, we estimate that about 8% of the mRNA belongs to this complex class, and that less than 500 copies of each species of message in this class exist per embryo. Most of the mRNA population consists of a relatively small number of diverse species represented a much larger number of times.