One of the important performance metrics by which a wide area network is evaluated is based on the success ratio of the number of requests that are accepted in the network. This metric is usually posed in its alternate form as the blocking probability, which refers to the rejection ratio of the requests in the network. The smaller the rejection ratio is, the better the network performance. Although other performance metrics exist, such as the effective traffic carried in the network, the fairness of request rejections with respect to requests requiring different capacity requirements or different path lengths, the most meaningful way to measure the performance of a wide-area network is through the blocking performance. To some extent the other performance metrics described above can be obtained as functions of the blocking performance.

Analytical models that evaluate the blocking performance of wide-area circuit-switched networks are employed during the design phase of a network. In the design phase these models are typically employed as an elimination test, rather than as an acceptance test. In other words, the analytical models are employed as back of the envelope calculations to evaluate a network design, rejecting those designs that are below a certain threshold.

Blocking model

The following assumptions are made to develop an analytical model for evaluating the blocking performance of a TSN.

• The network has N nodes.

• The call arrival at every node follows a Poisson process with rate λn. The choice of Poisson traffic is to keep the analysis tractable.

• […]

Feedback shift register (FSR) sequences have been widely used as synchronization, masking, or scrambling codes and for white noise signals in communication systems, signal sets in CDMA communications, key stream generators in stream cipher cryptosystems, random number generators in many cryptographic primitive algorithms, and testing vectors in hardware design. Golomb's popular book Shift Register Sequences, first published in 1967 and revised in 1982 is a pioneering book that discusses this type of sequences. In this chapter, we introduce this topic and discuss the synthesis and the analysis of periodicity of linear feedback shift register sequences. We give different (though equivalent) definitions and representations for LFSR sequences and point out which are most suitable for either implementation or analysis. This chapter contains seven sections, which are organized as follows. In Section 4.1, we give a general description for feedback shift registers at the gate level for the binary case and as a finite field configuration for the q-ary case. In Sections 4.2–4.4, we introduce the definition of LFSR sequences from the point of view of polynomial rings and discuss their characteristic polynomials, minimal polynomials, and periods. Then, we show the decomposition of LFSR sequences. We provide the matrix representation of LFSR sequences in Section 4.5 as another historic approach and discuss their trace representation for the irreducible case in detail in Section 4.6, which is a more modern approach. (The general case will be treated in Chapter 6.) LFSRs with primitive minimal polynomials are basic building blocks for nonlinear generators.

Randomness of a sequence refers to the unpredictablity of the sequence. Any deterministically generated sequence used in practical applications is not truly random. The best that can be done here is to single out certain properties as being associated with randomness and to accept any sequence that has these properties as random or more properly, a pseudorandom sequence. In this chapter, we will discuss the randomness of sequences whose elements are taken from a finite field. In Section 5.1, we present Golomb's three randomness postulates for binary sequences, namely the balance property, the run property, and the (ideal) two-level autocorrelation property, and the extension of these randomness postulates to nonbinary sequences. M-sequences over a finite field possess many extraordinary randomness properties except for having the lowest possible linear span, which has stimulated researchers to seek nonlinear sequences with similarly such favorable properties for years. In Section 5.2, we show that m-sequences satisfy Golomb's three randomness postulates. In Section 5.3, we introduce the interleaved structures of m-sequences and the subfield decomposition of m-sequences. In Sections 5.4–5.6, we present the shift-and-add property, constant-on-cosets property, and 2-tuple balance property of m-sequences, respectively. The last section is devoted to the classification of binary sequences of period 2n − 1.

Golomb's randomness postulates and randomness criteria

We discussed some general properties of auto- and crosscorrelation in Chapter 1 for sequences whose elements are taken from the real number field or the complex number field.

This book is the product of a fruitful collaboration between one of the earliest developers of the theory and applications of binary sequences with favorable correlation properties and one of the currently most active younger contributors to research in this area. Each of us has taught university courses based on this material and benefited from the feedback obtained from the students in those courses. Our goal has been to produce a book that achieves a balance between the theoretical aspects of binary sequences with nearly ideal autocorrelation functions and the applications of these sequences to signal design for communications, radar, cryptography, and so on. This book is intended for use as a reference work for engineers and computer scientists in the applications areas just mentioned, as well as to serve as a textbook for a course in this important area of digital communications. Enough material has been included to enable an instructor to make some choices about what to cover in a one-semester course. However, we have referred the reader to the literature on those occasions when the inclusion of further detail would have resulted in a book of inordinate length.

We plan to maintain a Web site at http://calliope.uwaterloo.ca/∼ggong/book/book.htm for additions, corrections, and the continual updating of the material in this book.

Binary sequences of period N with 2-level autocorrelation have many important applications in communications and cryptology. From Section 7.1, 2-level autocorrelation sequences are in natural correspondence with cyclic Hadamard difference sets with ν = N, κ = (N − 1)/2, and λ = (N − 3)/4. For this reason, they are named cyclic Hadamard sequences. In this chapter, 2-level autocorrelation always means ideal 2-level autocorrelation. There are three classic constructions for binary 2-level autocorrelation sequences that were known before 1997 (including some generalizations along these lines after 1997). One is m-sequences, described in Chapter 5, with period N = 2n − 1. The second construction is based on a number theory approach, including three types of sequences in Chapter 2, which are the quadratic residue sequences, Hall sextic residue sequences, and twin prime sequences. The period of such a sequence is either a prime or a product of twin primes. The third construction is associated with intermediate subfields. The resulting sequences have subfield decompositions and period N = 2n − 1. They include GMW sequences, cascaded GMW sequences, and generalized GMW sequences. Although the resulting sequences are binary, this construction relies heavily on intermediate fields and compositions of functions. As a consequence, it involves sequences over intermediate fields that are not binary sequences. The content of this chapter is organized as follows.

Overview

In the first three of the applications mentioned in the title of this chapter, one of the objectives (often the major objective) is to determine a point in time with great accuracy. In radar and sonar, we want to determine the round-trip time from transmitter to target to receiver very accurately, because the one-way time (half of the round-trip time) is a measure of the distance to the target (called the range of the target).

The simplest approach would be to send out a pure impulse of energy and measure the time until it returns. The ideal impulse would be virtually instantaneous in duration, but with such high amplitude that the total energy contained in the pulse would be significant, much like a Dirac delta function. However, the Dirac delta function not only fails to exist as a mathematical function, but it is also unrealizable as a physical signal. Close approximations to it – very brief signals with very large amplitudes – may be valid mathematically, but are impractical to generate physically. Any actual transmitter will have an upper limit on peak power output, and hence a short pulse will have a very restricted amount of total energy: at most, the peak power times the pulse duration. More total energy can be transmitted if we extend the duration; but if we transmit at uniform power over an extended duration, we do not get a sharp determination of the round-trip time. This dilemma is illustrated in Figure 12.1.

Finite fields are used in most of the known constructions of pseudorandom sequences and analysis of periods, correlations, and linear spans of linear feedback shift register (LFSR) sequences and nonlinear generated sequences. They are also important in many cryptographic primitive algorithms, such as the Diffie-Hellman key exchange, the Digital Signature Standard (DSS), the El Gamal public-key encryption, elliptic curve public-key cryptography, and LFSR (or Torus) based public-key cryptography. Finite fields and shift register sequences are also used in algebraic error-correcting codes, in code-division multiple-access (CDMA) communications, and in many other applications beyond the scope of this book. This chapter gives a description of these fields and some properties that are frequently used in sequence design and cryptography. Section 3.1 introduces definitions of algebraic structures of groups, rings and fields, and polynomials. Section 3.2 shows the construction of the finite field GF(pn). Section 3.3 presents the basic theory of finite fields. Section 3.4 discusses minimal polynomials. Section 3.5 introduces subfields, trace functions, bases, and computation of the minimal polynomials over intermediate subfields. Computation of a power of a trace function is shown in Section 3.6. And, the last section presents some counting numbers related to finite fields.

Algebraic structures

In this section, we give the definitions of the algebraic structures of groups, rings and fields, polynomials, and some concepts that will be needed for the study of finite fields in the later sections.

In this chapter, we introduce constructions for signal sets with low crosscorrelation. These sequences have important applications in wireless CDMA communications. There are three classic constructions for signal sets with low correlation, namely, the Gold-pair construction, the Kasami (small) set construction, and the bent function signal set construction. In Section 10.1, we introduce some basic concepts and properties for crosscorrelation of sequences or functions, signal sets, and one-to-one correspondences among sequences, polynomial functions, and boolean functions. After that, three classic constructions will be presented in Sections 9.2, 9.3, and 9.4 respectively. With the development of new technologies, the demand for constraints on other parameters, such as linear spans of sequences, and the sizes of the signal sets has increased. Here, we will provide two examples of constructions that sacrifice ideal correlation in order to improve other properties, in Sections 9.5 and 9.6, respectively. One example is the interleaved construction for large linear spans, and the other is ℤ4 sequences to obtain large sizes of signal sets.

Crosscorrelation, signal sets, and boolean functions

In this section, we discuss some basic properties of crosscorrelation of sequences (some of them have been discussed in Chapter 1), refine the concept of signal sets, and develop the one-to-one correspondence between sequences and boolean functions. (Note that the one-to-one correspondence between sequences and functions is discussed in Chapter 6.)

We will keep the following notation in this section.

Before 1997, only two essentially different constructions that were not based on a number theory approach were known for cyclic Hadamard difference sets with parameter (2n − 1, 2n−1 − 1, 2n−2 − 1) or, equivalently, for binary 2-level autocorrelation sequences of period 2n − 1 for arbitrary n. One is the Singer construction, which gives m-sequences, and the other is the GMW construction, which produces four types of GMW sequences. Exhaustive searches had been done for n = 7, 8, and 9 in 1971, 1983, and 1992, respectively. However, there was no explanation for several of the sequences found for these lengths that did not follow from then-known constructions. In this chapter, we will describe the remarkable progress in finding new constructions for 2-level autocorrelation sequences of period 2n − 1 since 1997. (An exhaustive search was also done for n = 10 in 1998.) The order of presentation of these remarkable constructions will follow the history of the developments of this research. Section 9.1 presents constructions of 2-level autocorrelation sequences having multiple trace terms. In Section 9.2, the hyper-oval constructions are introduced. Section 9.3 shows the Kasami power construction. In the last section, we introduce the iterative decimation-Hadamard transform, a method of searching for new sequences with 2-level autocorrelation.

Multiple trace term sequences

In this section, we present 3-term sequences, 5-term sequences, and the Welch-Gong transformation sequences.

The prehistory of our subject can be backdated to 1202, with the appearance of Leonardo Pisano's Liber Abaci (Fibonacci 1202), containing the famous problem about breeding rabbits that leads to the linear recursion fn+1 = fn + fn−1 for n ≥ 2, f1 = f2 = 1, which yields the Fibonacci sequence. Additional background can be attributed to Euler, Gauss, Kummer, and especially Edouard Lucas (Lucas 1876). For the history proper, the earliest milestones are papers by O. Ore (Ore 1934), R.E.A.C. Paley (Paley 1933), and J. Singer (Singer 1938). Ore started the systematic study of linear recursions over finite fields (including GF(2)), Paley inaugurated the search for constructions yielding Hadamard matrices, and Singer discovered the Singer difference sets that are mathematically equivalent to binary maximum length linear shift register sequences (also known as pseudorandom sequences, pseudonoise (PN) sequences, or m-sequences).

It appears that by the early 1950s devices that performed the modulo 2 sum of two positions on a binary delay line were being considered as key generators for stream ciphers in cryptographical applications. The question of what the periodicity of the resulting output sequence would be seemed initially mysterious. This question was explored outside the cryptographic community by researchers at a number of locations in the 1953–1956 time period, resulting in company reports by E. N. Gilbert at Bell Laboratories, by N. Zierler at Lincoln Laboratories, by L. R. Welch at the Jet Propulsion Laboratory, by S.W. Golomb at the Glenn L. Martin Company (now part of Lockheed-Martin), and probably by others as well.