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Large families of sequences with near-optimal correlations and large linear span
Abstract
In order to build spread-spectrum communication systems based on
the CDMA paradigm, it is necessary to have large families of binary
sequences with low pairwise correlation values. For these systems to
have resistance to certain cryptanalytic attacks and resistance to
jamming, the sequences must have large linear span. We describe certain
families of sequences that have these desirable properties. The
sequences are based on families of quadratic forms over finite fields
... In brief, the process of previous generating method is as follows: firstly, it uses primitive polynomial of degree m over odd characteristic field F p to generate maximum length vector sequence as elements in F p m , then applies trace function to map the vectors to F p scalars, and finally applies Legendre symbol to map the scalars to binary values such as {1, −1}. Such a binarized sequence over finite field is often called geometric sequence and it has been well studied [11]- [14]. The pseudo random binary sequence of our previous work is characterized with two parameters m and p, where m is the degree of primitive polynomial and p is the characteristics. ...
... Binary geometric sequences have been well studied [11]- [14]. The authors [10] have also proposed a binary sequence that uses primitive element in extension field, trace function for mapping vectors to scalars, and Legendre symbol for binarizing the trace values. ...
This paper proposes a new multi-value sequence generated by utilizing primitive element, trace, and power residue symbol over odd characteristic finite field. In detail, let p and k be an odd prime number as the characteristic and a prime factor of p-1, respectively. Our proposal generates k-value sequence T={ti | ti=fk(Tr(ωⁱ)+A)}, where ω is a primitive element in the extension field $\F{p}{m}$, Tr(⋅) is the trace function that maps $\F{p}{m} \rightarrow \f{p}$, A is a non-zero scalar in the prime field $\f{p}$, and fk(⋅) is a certain mapping function based on k-th power residue symbol. Thus, the proposed sequence has four parameters as p, m, k, and A. Then, this paper theoretically shows its period, autocorrelation, and cross-correlation. In addition, this paper discusses its linear complexity based on experimental results. Then, these features of the proposed sequence are observed with some examples.
... Klapper introduces d -form sequences [ 1,2] by using homogeneous function. d -form sequences with d = 2 include No sequences [3] as a special case. ...
... QF sequences [2] are another special case of the d-form sequences. By this method presented in this paper, we can construct extended QF sequences that have the same crosscorrelation as the QF sequences. ...
In this paper, a construction method to generate binary extended d-form sequences with the same correlation properties as d-form sequences is proposed and using the TN sequences (a special case of d-form sequences), the extended TN sequences with the optimal correlation are constructed. Finally, we give an example of the families of the extended TN sequences, which are constructed from Legendre sequences.
Bent sequence families are mainly generated by a linear onto mapping and a bent function. Nevertheless, for a long time, the linear onto mapping has not an explicit expression. Firstly analyzing the conditions that the linear onto mapping must obey, then based on the theory of finite field and the distribution of m-sequences, the explicit expression of linear onto mapping is obtained. Therefore the trace representation of Bent sequences is gotten. Finally a family of Bent sequences is constructed which can be rapidly generated by the trace representation of Bent sequences and some Bent functions.
Let p and m be an odd characteristic and extension degree, respectively. Then, let ! be a primitive element in F∗pm, a geometric sequence S = Lsir, si =(Tr(ωi)/p) for i = 0, 1, 2,... has been proposed together with its some properties, where Tr (·) is the trace function over Fp and (a/p) is Legendre symbol for a in Fp. This paper shows an efficient method for generating the sequence when the base field Fpm is defined by an irreducible binomial over Fp with small odd characteristic p.
Pseudo-random sequence with good correlation property and large linear span is widely used in code division multiple access (CDMA) communication systems and cryptology for reliable and secure information transmission. In this paper, sequences with long period, large complexity, balance statistics, and low cross-correlation property are constructed from the addition of m-sequences with pairwise-prime linear spans (AMPLS). Using m-sequences as building blocks, the proposed method proved to be an efficient and flexible approach to construct long period pseudo-random sequences with desirable properties from short period sequences. Applying the proposed method to double struct F sign2, a signal set ((2n - 1)(2m - 1), (2n + 1)(2 m + 1), (2(n+1)/2 + 1)(2(m+1)/2 + 1)) is constructed.
Let p be an odd characteristic and m be the degree of a primitive polynomial f(x) over the prime field Fp. Let ω be its zero, that is a primitive element in F*pm, the sequence S={si}, si=Tr(ωi) for i=0,1,2,… becomes a non-binary maximum length sequence, where Tr(·) is the trace function over Fp. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period, autocorrelation, and linear complexity have not been discussed. Then, this paper shows that the generated binary sequence (geometric sequence by Legendre symbol) has the period n=2(pm-1)/(p-1) and a typical periodic autocorrelation. Moreover, it is experimentally observed that its linear complexity becomes the maximum, that is the period n. Among such experimental observations, especially in the case of m=2, it is shown that the maximum linear complexity is theoretically proven. After that, this paper also demonstrates these properties with a small example.
Let p be an odd characteristic and m be the degree of primitive polynomial f(x). Let ω be its zero, that is a primitive element in Fpm*, then the sequence S = {si}, si = Tr (ωi) for i = 0, 1, 2, ... becomes a maximum length sequence, where Tr (·) is the trace function over Fp. On this fact, this paper proposes to binarize the sequence by using Legendre symbol. It will be a class of geometric sequences but its properties such as the period and autocorrelation has not been discussed. Then, it is shown that the obtained binary sequence (geometric sequence with Legendre symbol) has the period L given by 2(pm - 1)/(p-1) and a certain periodic autocorrelation. After that, this paper also shows the numbers of ones and minus ones in the proposed binary sequence per a period together with some small examples.
p-ary sequences of period are widely used in many areas of engineering and sciences. Some well-known applications include coding theory, code-division multiple-access (CDMA) communications, and stream cipher systems. The analysis of cross-correlations of these sequences is a very important problem in p-ary sequences research. In this paper, we analyze cross-correlations of p-ary sequences which is associated with the equation over finite fields.
In this paper, a construction method to generate binary extended d-form sequences with the same correlation properties as d-form sequences is proposed. Using the QF sequences (a special case of d-form sequences), the extended QF sequences with the optimal correlation are constructed. Finally, we give an example of the families of the extended QF sequences, which are constructed from Legendre sequences
Cross-correlation functions are determined for a large class of geometric sequences based on m-sequences in characteristic two. These sequences are shown to have low cross-correlation values in certain cases. They are also shown to have significantly higher linear complexities than previously studied geometric sequences. These results show that geometric sequences are candidates for use in spread-spectrum communications systems in which cryptographic security is a factor.
Let P(x) be a function from GF(2n) to GF(2). P(x) is called “bent” if all Fourier coefficients of (−1)P(x) are ±1. The polynomial degree of a bent function P(x) is studied, as are the properties of the Fourier transform of (−1)P(x), and a connection with Hadamard matrices.
In a recent paper Olsen, Scholtz, and Welch (OSW) describe the use of bent functions to construct families of sequences with asymptotically optimal correlation properties. We show that the sequences produced by the OSW construction possess the claimed correlation properties if and only if the underlying bent functions are pairwise orthogonal. Consequently, the cardinality of a family of OSW sequences of length 2^{n}-1 cannot exceed 2^{n/2} . We also describe methods of selecting a maximal set of orthogonal bent functions.
In this paper we construct a new family of nonlinear binary signal sets which achieve Welch's lower bound on simultaneous cross correlation and autocorrelation magnitudes. Given a parameter n with n=0 pmod{4} , the period of the sequences is 2^{n}-1 , the number of sequences in the set is 2^{n/2} , and the cross/auto correlation function has three values with magnitudes leq 2^{n/2}+1 . The equivalent linear span of the codes is bound above by sum_{i=1}^{n/4}left(stackrel{n}{i} right) . These new signal sets have the same size and correlation properties as the small set of Kasami codes, but they have important advantages for use in spread spectrum multiple access communications systems. First, the sequences are "balances," which represents only a slight advantage. Second, the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly "hopped" from sequence to sequence for code division multiple access operation. Most importantly, the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it. This high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing.
The design and operation of spread-spectrum (SS) communication systems are examined in an introductory text intended for graduate engineering students and practicing engineers. Chapters are devoted to an overview of SS systems, the historical origins of SS, basic concepts and system models, antijam communication systems, pseudonoise generators, coherent direct-sequence systems, noncoherent frequency-hopped systems, coherent and differentially coherent modulation techniques, pseudonoise acquisition and tracking in direct-sequence receivers, time and frequency synchronization of frequency-hopped receivers, low-probability-of-intercept communication, and multiple-access communication. Graphs, diagrams, and photographs are provided.
A collection of families of binary {0,1} pseudorandom sequences (referred to as the families of No sequences) is introduced. Each sequence within a family has period N =2<sup>n</sup>-1, where n =2 m is an even integer. There are 2<sup>m</sup> sequences within a family, and the maximum overall (nontrivial) auto- and cross-correlation values equal 2<sup>m</sup>+1. Thus these sequences are optimum with respect to the Welch bound on the maximum correlation value. Each family contains a Gordon-Mils-Welch (GMW) sequence and the collection of families includes as a special case, the small set of Kasami sequences. The linear span of these sequences is large. The balance properties of such families are evaluated, and a count of the number of distinct families of given period N that can be constructed is provided
Recently, Olsen, Scholtz, and Welch presented families of binary sequences called bent-function sequences which can be generated through nonlinear operations on m -sequences. These families of sequences possess asymptotically optimum correlation properties and large equivalent linear span (ELS). Upper and lower bounds to the ELS of bent-function sequences are derived. The upper bound improves upon Key's upper bound and the lower bound, obtained through construction, and exceeds left(stackrel{n/2}{n/4}right)cdot 2^{n/4} , where n is the length of the shift register generating the m -sequence. An interesting general result contained in the derivation is the exhibition of a class of nonlinear sequences whose ELS is guaranteed to be large.