Bounds on partial correlations of sequences

In a previous paper, No, Golomb, Gong, Lee and Gaal (see ibid., vol.44, p.814-17, 1998) conjectured that certain binary sequences having a simple trace description possess the ideal autocorrelation property. In the present paper it is shown that each such sequence is balanced and, moreover, that the dual of the linear cyclic code generated by the sequence and its cyclic shifts, is a triple-error correcting code having the same weight distribution as the triple-error correcting Bose-Chaudhuri-Hocquenghem (BCH) code. This cyclic code also contains a cyclic subcode that yields a new family of sequences having the same size and correlation parameters as does the family of Gold sequences.

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated.

Autocorrelation Function of a Family of Complex Sequences

Dissatisfaction is expressed with the well-known definitions of partial and multiple correlation. Partial correlation is defined here as the ordinary correlation between two random variables after their linear dependence on the partialed-out variables has been subtracted. This leads to familiar results; it also clarifies the auto-regressivelike assumptions required for the partial autocorrelations to have a cut-off property. Multiple correlation is similarly considered.

In this paper we give a new family of elliptic curve pseudorandom binary sequences, and study the well-distribution, correlation, linear complexity, collision and avalanche effect of the family of sequences, by using the estimate of exponential sums over elliptic curves.

Based on the cyclotomy classes of extension fields, a family of binary cyclotomic sequences are constructed and their pseudorandom measures (i.e., the well-distribution measure and the correlation measure of order k) are estimated using certain exponential sums. A lower bound on the linear complexity profile is also presented in terms of the correlation measure.

In this correspondence, we present a family of binary 2n sequences of period 2(2n-1) where n is an integer, which can be seen as a generalization of nonlinear binary sequences obtained from Z4 sequences and recently constructed GKW (Gold, Kasami, and Welch)-like sequences. The sequences have low correlations and are useful in code-division multiple-access (CDMA) communication systems and cryptography

A new family of binary sequences with low 4-value autocorrelation of length 4p based on cyclotomic classes of order four is proposed, as well as the linear complexity and 2-adic complexity of these sequences are determined. The results show that the linear complexity of s is LC(s) = 4p−1, and the 2-adic complexity of s is Φ2s = log2 ((24p−1)/3), both of which are greater than half of the period. The sequences are safe enough to resist Berlekamp–Massey algorithm attacks and rational approximation algorithm attacks, which indicate that they are good sequences in communication.

Binary sequences produced by a generator should appear as random as possible, that is, have no logical pattern to be used in cryptographic applications. In this paper, we give a detailed analysis of the randomness of a family of binary sequences obtained from generalized self-shrinking generator, an element in the class of decimation-based sequence generators. We have applied the most important batteries of statistical tests to the sequence resulting from the concatenation of the family of generalized sequences obtained from a PN-sequence. This complete study provides good results and allow us to construct a new binary sequence with good cryptographic properties from a family of generalized self-shrunken sequences.KeywordsGeneralized self-shrinking generatorPseudo-random number generatorRandomness

In this paper, for odd n and any k with gcd(n, k) = 1, new binary sequence families Sk of period 2n-1 are constructed. These families have maximum correlation $1+2^{n+3\\over 2}$, family size 22n+2n+1 and maximum linear span $n(n+1)\\over 2$. The correlation distribution of Sk is completely determined as well. Compared with the modified Gold codes with the same family size, the proposed families have the same period and correlation properties, but larger linear span. As good candidates with low correlation and large family size, the new families contain the Gold sequences and the Gold-like sequences. Furthermore, Sk includes a subfamily $\\mathcal{S}^k_1$ which has the same period, correlation distribution, family size and linear span as the family So(2) recently constructed by Yu and Gong. In particular, when k=1, $\\mathcal{S}^k_1$ is exactly So(2).

The Gold sequence family [1] is one of the oldest and most well‐known families of binary sequences with optimal correlation properties. This chapter focuses on the binary Gold sequences and their relatives called Gold‐like sequences. Mathematical representations as well as the family sizes and maximum correlation magnitudes of these sequences are reviewed.

In this paper, a new family of binary sequences of period 2n-1 with low correlation is proposed for integer n = em and even m. The new family has family size 2n + 1 and maximum nontrivial correlation 2n+e/2 + 1 and 2n+e-1/2 + 1 for even and odd e respectively. Especially, for n = 2m and 3m, we obtain a new family of binary sequences with maximum nontrivial correlation 2n/2 + 1 + 1, and the obtained family is one of the binary families with best correlation among the known families with family size no less than their period 2n-1 for even n. Moreover, the correlation distribution of the new family is also determined.

This paper gives a brief description of TCH sequences and compares their most important correlation properties with some known sequence families regarding application in spread spectrum systems such as DS-CDMA. The generation method of TCH sequences is also described and many important correlation parameters to be used in the comparative study are described. For the first time TCH sequences of length 65536 were generated and correlation properties are analyzed and compared with those of Gold-like and Kasami (small set) of similar length. The performance of an asynchronous DS-CDMA system with these sequences is also evaluated and the results shown graphically.

In this paper, the partial Hamming correlation properties of frequency-hopping sequences (FHSs) are discussed. The Peng-Fan bounds on sets of FHSs are generalized to the case of partial correlation. Both individual FHSs with optimal partial autocorrelation and sets of FHSs with optimal partial correlation are presented. The former has more new parameters compared with the known individual FHSs with optimal partial autocorrelation, while the later is obtained in the literature for the first time.

The well known family of binary twin-prime sequences is generalised to the multiple-valued case by employing a polyphase representation of the sequence elements. These polyphase versions exhibit similar periodic and aperiodic auto-correlation properties to their binary counterparts, and are referred to as q-phase related-prime (RP) sequences. These sequences have length L=r·s, for r and s both prime, and with s=r+k. They are constructed by combining two polyphase Legendre sequences of lengths r and s, and modifying the resulting composite sequence at certain points. A two-dimensional array structure is employed in the construction and analysis of these sequences. The original q-phase Legendre sequences are derived by converting the index sequences of lengths r and s to modulo-q form. When q is even, two classes of RP sequence arise, depending on whether L≡q+1 mod 2q or L≡1 mod 2q. For odd q, only a single class is available, and here L≡1 mod 2q. The out-of-phase periodic correlation values of these RP sequences are independent of the sequence length, and depend only on the number of phases q and the difference k between the two related primes. The maximum out-of-phase correlation values is given by 1−k. Tables of available sequences are presented.