On the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers

Let M be a square-free odd integer and Z/(M) the integer residue ring modulo M. This paper studies the distinctness of primitive sequences over Z/(M) modulo 2. Recently, for the case of M = pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2k+1 over Z/(M) is distinct modulo 2. Besides as an independent interest, the paper also involves two distribution properties of primitive sequences over Z/(M), which related closely to our main results

A new result on the distinctness of primitive sequences over Z(pq) modulo 2

Let Z/(pq) be the integer residue ring modulo pq with odd prime numbers p and q. This paper studies the distinctness problem of modulo 2 reductions of two primitive sequences over Z/(pq), which has been studied by H.J. Chen and W.F. Qi in 2009. First, it is shown that almost every element in Z/(pq) occurs in a primitive sequence of order n > 2 over Z/(pq). Then based on this element distribution property of primitive sequences over Z/(pq), previous results are greatly improved and the set of primitive sequences over Z/(pq) that are known to be distinct modulo 2 is further enlarged

Low Correlation Sequences over the QAM Constellation

This paper presents the first concerted look at low correlation sequence families over QAM constellations of size M^2=4^m and their potential applicability as spreading sequences in a CDMA setting. Five constructions are presented, and it is shown how such sequence families have the ability to transport a larger amount of data as well as enable variable-rate signalling on the reverse link. Canonical family CQ has period N, normalized maximum-correlation parameter theta_max bounded above by A sqrt(N), where 'A' ranges from 1.8 in the 16-QAM case to 3.0 for large M. In a CDMA setting, each user is enabled to transfer 2m bits of data per period of the spreading sequence which can be increased to 3m bits of data by halving the size of the sequence family. The technique used to construct CQ is easily extended to produce larger sequence families and an example is provided. Selected family SQ has a lower value of theta_max but permits only (m+1)-bit data modulation. The interleaved 16-QAM sequence family IQ has theta_max <= sqrt(2) sqrt(N) and supports 3-bit data modulation. The remaining two families are over a quadrature-PAM (Q-PAM) subset of size 2M of the M^2-QAM constellation. Family P has a lower value of theta_max in comparison with Family SQ, while still permitting (m+1)-bit data modulation. Interleaved family IP, over the 8-ary Q-PAM constellation, permits 3-bit data modulation and interestingly, achieves the Welch lower bound on theta_max.Comment: 21 pages, 3 figures. To appear in IEEE Transactions on Information Theory in February 200

An improved method for predicting truncated multiple recursive generators with unknown parameters

Multiple recursive generators are an important class of pseudorandom number generators which are widely used in cryptography. The predictability of truncated sequences that predict the whole sequences by the truncated high-order bits of the sequences is not only a crucial aspect of evaluating the security of pseudorandom number generators but also serves an important role in the design of pseudorandom number generators. This paper improves the work of Sun et al on the predictability of truncated multiple recursive generators with unknown parameters. Given a few truncated digits of high-order bits output by a multiple recursive generator, we adopt the resultant, the Chinese Remainder Theorem and the idea of recovering $p$-adic coordinates of the coefficients layer by layer, and Kannan\u27s embedding technique to recover the modulus, the coefficients and the initial state, respectively. Experimental results show that our new method is superior to that of the work of Sun et al, no matter in terms of the running time or the number of truncated digits required

Systems of difference equations as a model for the Lorenz system

We consider systems of difference equations as a model for the Lorenz system of differential equations. Using the power series whose coefficients are the solutions of these systems, we define three real functions, that are approximation for the solutions of the Lorenz system