Nikov Sequences Research Articles

Balance, pattern distribution and linear complexity of M-ary sequences from Sidel’nikov sequences

Balance, pattern distribution and linear complexity of M-ary sequences from Sidel’nikov sequences

On the k-error Linear Complexity of Subsequences of d-ary Sidel’nikov Sequences Over Prime Field 𝔽d

We study the [Formula: see text]-error linear complexity of subsequences of the [Formula: see text]-ary Sidel’nikov sequences over the prime field [Formula: see text]. A general lower bound for the [Formula: see text]-error linear complexity is given. For several special periods, we show that these sequences have large [Formula: see text]-error linear complexity.

Linear Complexity of Geometric Sequences Defined by Cyclotomic Classes and Balanced Binary Sequences Constructed by the Geometric Sequences

Pseudorandom number generators are required to generate pseudorandom numbers which have good statistical properties as well as unpredictability in cryptography. An m-sequence is a linear feedback shift register sequence with maximal period over a finite field. M-sequences have good statistical properties, however we must nonlinearize m-sequences for cryptographic purposes. A geometric sequence is a sequence given by applying a nonlinear feedforward function to an m-sequence. Nogami, Tada and Uehara proposed a geometric sequence whose nonlinear feedforward function is given by the Legendre symbol, and showed the period, periodic autocorrelation and linear complexity of the sequence. Furthermore, Nogami et al. proposed a generalization of the sequence, and showed the period and periodic autocorrelation. In this paper, we first investigate linear complexity of the geometric sequences. In the case that the Chan--Games formula which describes linear complexity of geometric sequences does not hold, we show the new formula by considering the sequence of complement numbers, Hasse derivative and cyclotomic classes. Under some conditions, we can ensure that the geometric sequences have a large linear complexity from the results on linear complexity of Sidel'nikov sequences. The geometric sequences have a long period and large linear complexity under some conditions, however they do not have the balance property. In order to construct sequences that have the balance property, we propose interleaved sequences of the geometric sequence and its complement. Furthermore, we show the periodic autocorrelation and linear complexity of the proposed sequences. The proposed sequences have the balance property, and have a large linear complexity if the geometric sequences have a large one.

New Polyphase Sequence Families With Low Correlation Derived From the Weil Bound of Exponential Sums

In this paper, the sequence families of which maximum correlation is determined by the Weil bound of exponential sums are revisited. Using the same approach, two new constructions with large family sizes and low maximum correlation are given. The first construction is an analog of one recent result derived from the interleaved structure of Sidel'nikov sequences. For a prime p and an integer M|(p-1), the new M-ary sequence families of period p are obtained from irreducible quadratic polynomials and known power residue-based sequence families. The new sequence families increase family sizes of the known power residue-based sequence families, but keep the maximum correlation unchanged. In the second construction, the sequences derived from the Weil representation are generalized, where each new sequence is the elementwise product of a modulated Sidel'nikov sequence and a modulated trace sequence. For positive integers d <; p and M|(pn-1), the new family consists of (M-1)pnd sequences with period pn-1, alphabet size Mp, and the maximum correlation bounded by (d+1)√{pn}+3.

Erratum for ‘Generation of Sidel'nikov sequences and their spectra via permutations’

Erratum for ‘Generation of Sidel'nikov sequences and their spectra via permutations’

New Families of Frequency-Hopping Sequences of Period 2(2&lt;i&gt;&lt;sup&gt;n&lt;/sup&gt;&lt;/i&gt; - 1)

No nontrivial optimal sets of frequency-hopping sequences (FHSs) of period 2(2n - 1) for a positive integer n ≥ 2 have been found so far, when their frequency set sizes are less than their periods. In this paper, systematic doubling methods to construct new FHS sets are presented under the constraint that the set of frequencies has size less than or equal to 2n. First, optimal FHS sets with respect to the Peng-Fan bound are constructed when frequency set size is either 2n - 1 or 2n. And then, near-optimal FHS sets with frequency set size 2n - 1 are designed by applying the Chinese Remainder Theorem to Sidel'nikov sequences, whose FHSs are optimal with respect to the Lempel-Greenberger bound. Finally, a general construction is given for near-optimal FHS sets whose frequency set size is less than 2n - 1. Our constructions give new parameters not covered in the literature, which are summarized in Table 1.

A Construction of a New Family of $M$-ary Sequences With Low Correlation From Sidel'nikov Sequences

In this paper, a new family of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary sequences of period <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -1 is proposed. The proposed family is constructed by the addition of cyclic shifts of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary Sidel'nikov sequence and its reverse sequence. The number of sequences contained in this family is about ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> times of their period and the maximum magnitude of their correlation values is upper bounded by 4 √( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">)</i> +5.

Generation of Sidel'nikov sequences and their spectra via permutations

Using a reduced number of distinct permutations, an efficient method to generate Sidel&apos;nikov M-ary sequences and their spectra from an initial seed sequence is presented. The unfolded structure allows an all-digital and low complexity implementation of a maximum likelihood sequence detector.

Correlation of the two-prime Sidel’nikov sequence

Motivated by the concepts of Sidel'nikov sequences and two-prime generator (or Jacobi sequences) we introduce and analyze some new binary sequences called two-prime Sidel'nikov sequences. In the cases of twin primes and cousin primes equivalent 3 modulo 4 we show that these sequences are balanced. In the general case, besides balancedness we also study the autocorrelation, the correlation measure of order k and the linear complexity profile of these sequences showing that they have many nice pseudorandom features.

On the Sidel'nikov Sequences as Frequency-Hopping Sequences

A (v,l, lambda)-FHS denotes a frequency-hopping sequence of length v over a frequency set of size l with maximum out-of-phase Hamming autocorrelation lambda. Recently, Ding and Yin constructed two FHS families for a prime power q satisfying q=ef+1 with positive integers e and f . Theorems 4 and 5 in their paper claim that these two FHS families include optimal (q-1, e, f)-FHSs and (q-1, e+1, f-1)-FHSs with respect to the Lempel-Greenberger bound, respectively. In this paper, we give counterexamples and make corrections to them. Furthermore, we observe that these FHSs are closely related to Sidel'nikov sequences. Based on our results on the spectrum of their Hamming autocorrelation values, we also correct the theorem on the spectrum of Hamming distances of nearly equidistant codes derived by Sidel'nikov.

On the Cross-Correlation Distributions of $M$-ary Multiplicative Character Sequences

It is well known that the magnitude of the cross correlation between any distinct constant multiple sequences of an M-ary power residue sequence of period p is upper bounded by radicp +2 and that of an M -ary Sidel'nikov sequence of period p m-1 is upper bounded by radic{p m} +3, where p is a prime and m is a positive integer. In this paper, we first show that their cross-correlation functions are closely related to Jacobi sums and cyclotomic numbers. We then derive the cross-correlation distribution of constant multiple sequences of an M -ary power residue sequence. In the case of constant multiple sequences of an M-ary Sidel'nikov sequence, we get the possible cross-correlation values whose occurrence numbers are expressed in terms of the cyclotomic numbers of order M and are possibly zero.

On Some Properties of M-Ary Sidel'nikov Sequences

On Some Properties of M-Ary Sidel'nikov Sequences

New Families of $M$-Ary Sequences With Low Correlation Constructed From Sidel'nikov Sequences

In this correspondence, for a positive integer <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> and a prime p such that M|p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1, three families of M-ary sequences using the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary Sidel'nikov sequences with period p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1 are constructed. Two small families contain [(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 1)/2]+ <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -2 or p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -3 <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -ary sequences, and both of their maximum magnitudes of correlation values are upper bounded by 2 radic{p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> } + 6. The largest family has its maximum magnitude of correlation values upper bounded by 3 radic{p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> }+5 and the family size is ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> -1) + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1 for p = 2 or ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> - 3)/2 + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> -1)/2 for an odd prime p.

On the linear complexity of Sidel’nikov sequences over nonprime fields

We introduce a generalization of Sidel’nikov sequences for arbitrary finite fields. We show that several classes of Sidel’nikov sequences over arbitrary finite fields exhibit a large linear complexity. For Sidel’nikov sequences over F 8 we provide exact values for their linear complexity.

On the Linear Complexity and $k$-Error Linear Complexity Over $ {\BBF }_{p}$ of the $d$-ary Sidel'nikov Sequence

The d-ary Sidel'nikov sequence S = s0, s1... of period q-1 for a prime power q=pm is a frequently analyzed sequence in the literature. Recently, it turned out that the linear complexity over Fp of the d-ary Sidel'nikov sequence is considerably smaller than the period if the sequence element s(q-1)/2mod(q-1) is chosen adequately. In this paper this work is continued and tight lower bounds on the linear complexity over Fp of the d-ary Sidel'nikov sequence are given. For certain cases exact values are provided. Finally, results on the k-error linear complexity over Fp of the d-ary Sidel'nikov sequence are presented.

Cross Correlation of Sidel'nikov Sequences and Their Constant Multiples

In this correspondence, we prove that the complex cross-correlation of a k-ary Sidel'nikov sequence of period q-1 and its constant multiple sequence is upper bounded by radicq+3, where q=p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and here p is an odd prime and m is a positive integer

On the Autocorrelation Distributions of Sidel'nikov Sequences

For a prime p and positive integers M and n such that M|p/sup n/-1, Sidel'nikov introduced M-ary sequences (called Sidel'nikov sequences) of period p/sup n/-1, the out-of-phase autocorrelation magnitude of which is upper bounded by 4. In this correspondence, we derived the autocorrelation distributions, i.e., the values and the number of occurrences of each value of the autocorrelation function of Sidel'nikov sequences. The frequency of each autocorrelation value of an M-ary Sidel'nikov sequence is expressed in terms of the cyclotomic numbers of order M. It is also pointed out that the total number of distinct autocorrelation values is dependent not only on M but also on the period of the sequence, but always less than or equal to (M/2)+1.

Linear Complexity Over&amp;lt;tex&amp;gt;$F_p$&amp;lt;/tex&amp;gt;of Sidel'nikov Sequences

Helleseth, Kim, and No (2003) described the linear complexity over F/sub p/ of Sidel'nikov sequences of length p/sup m/-1 for p=3, 5, and 7. In this correspondence, the result is generalized to all odd primes.