두 p진 데시메이션 수열 간의 상호상관도

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2017. 2. 노종선.In this dissertation, the cross-correlation between two differently decimated sequences of a $p$-ary m-sequence is considered. Two main contributions are as follows. First, for an odd prime $p$, $n=2m$, and a $p$-ary m-sequence of period $p^n -1$, the cross-correlation between two decimated sequences by $2$ and $d$ are investigated. Two cases of $d$, $d=\frac{(p^m +1)^2}{2}$ with $p^m \equiv 1 \pmod4$ and $d=\frac{(p^m +1)^2}{p^e +1}$ with odd $m/e$ are considered. The value distribution of the cross-correlation function for each case is completely deterimined. Also, by using these decimated sequences, two new families of $p$-ary sequences of period $\frac{p^n -1}{2}$ with good correlation property are constructed. Second, an upper bound on the magnitude of the cross-correlation function between two decimated sequences of a $p$-ary m-sequence is derived. The two decimation factors are $2$ and $2(p^m +1)$, where $p$ is an odd prime, $n=2m$, and $p^m \equiv 1 \pmod4$. In fact, these two sequences corresponds to the sequences used for the construction of $p$-ary Kasami sequences decimated by $2$. The upper bound is given as $\frac{3}{2}p^m + \frac{1}{2}$. Also, using this result, an upper bound of the cross-correlation magnitude between a $p$-ary m-sequence and its decimated sequence with the decimation factor $d=\frac{(p^m +1)^2}{2}$ is derived.1 Introduction 1 1.1 Background 1 1.2 Overview of This Dissertation 7 2 Preliminaries 9 2.1 Finite Fields 9 2.2 Trace Functions and Sequences 11 2.3 Cross-Correlation Between Two Sequences 13 2.4 Characters and Weils Bound 15 2.5 Trace-Orthogonal Basis 16 2.6 Known Exponential Sums 17 2.7 Cross-Correlation of $p$-ary Kasami Sequence Family 18 2.8 Previous Results on the Cross-Correlation for Decimations with $\gcd(p^n -1, d)=\frac{p^{n/2}+1}{2}$ 20 2.9 Cross-Correlation Between Two Decimated Sequences by $2$ and $d=4$ or $\frac{p^n +1}{2}$ 23 3 New $p$-ary Sequence Families of Period $\frac{p^n -1}{2}$ with Good Correlation Property Using Two Decimated Sequences 26 3.1 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{2}$ 27 3.2 Cross-Correlation for the Case of $d=\frac{(p^m +1)^2}{p^e +1}$ 37 3.3 Construction of New Sequence Families 43 4 Upper Bound on the Cross-Correlation Between Two Decimated $p$-ary Sequences 52 4.1 Cross-Correlation Between $s(2t+i)$ and $s(2(p^m +1)t +j)$ 53 4.2 Cross-Correlation Between $s(t)$ and $s(\frac{(p^m +1)^2}{2} t)$ 66 5 Conclusions 69 Bibliography 72 Abstract (In Korean) 80Docto

On the Rudin-Shapiro transform

AbstractThe Rudin–Shapiro transform (RST) is a linear transform derived from the remarkable Rudin–Shapiro polynomials discovered in 1951. The transform has the notable property of forming a spread spectrum basis for RN, i.e. the basis vectors are sequences with a nearly flat power spectrum. It is also orthogonal and Hadamard, and it can be made symmetric. This presentation is partly a tutorial on the RST, partly some new results on the symmetric RST that makes the transform interesting from an applicational point-of-view. In particular, it is shown how to make a very simple O(NlogN) implementation, which is quite similar to the Haar wavelet packet transform

Construction of pp-ary Sequence Families of Period (pn−1)/2(p^n-1)/2 and Cross-Correlation of pp-ary m-Sequences and Their Decimated Sequences

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2015. 2. 노종선.This dissertation includes three main contributions: a construction of a new family of $p$-ary sequences of period $\frac{p^n-1}{2}$ with low correlation, a derivation of the cross-correlation values of decimated $p$-ary m-sequences and their decimations, and an upper bound on the cross-correlation values of ternary m-sequences and their decimations. First, for an odd prime $p = 3 \mod 4$ and an odd integer $n$, a new family of $p$-ary sequences of period $N = \frac{p^n-1}{2}$ with low correlation is proposed. The family is constructed by shifts and additions of two decimated m-sequences with the decimation factors 2 and $d = N-p^{n-1}$. The upper bound on the maximum value of the magnitude of the correlation of the family is shown to be $2\sqrt{N+1/2} = \sqrt{2p^n}$ by using the generalized Kloosterman sums. The family size is four times the period of sequences, $2(p^n-1)$. Second, based on the work by Helleseth \cite{Helleseth1}, the cross-correlation values between two decimated m-sequences by 2 and $4p^{n/2}-2$ are derived, where $p$ is an odd prime and $n = 2m$ is an integer. The cross-correlation is at most 4-valued and their values are $\{\frac{-1\pm p^{n/2}}{2}, \frac{-1+3p^{n/2}}{2}, \frac{-1+5p^{n/2}}{2}\}$. As a result, for $p^m \neq 2 \mod 3$, a new sequence family with the maximum correlation value $\frac{5}{\sqrt{2}} \sqrt{N}$ and the family size $4N$ is obtained, where $N = \frac{p^n-1}{2}$ is the period of sequences in the family. Lastly, the upper bound on the cross-correlation values of ternary m-sequences and their decimations by $d = \frac{3^{4k+2}-3^{2k+1}+2}{4}+3^{2k+1}$ is investigated, where $k$ is an integer and the period of m-sequences is $N = 3^{4k+2}-1$. The magnitude of the cross-correlation is upper bounded by $\frac{1}{2} \cdot 3^{2k+3}+1 = 4.5 \sqrt{N+1}+1$. To show this, the quadratic form technique and Bluher's results \cite{Bluher} are employed. While many previous results using quadratic form technique consider two quadratic forms, four quadratic forms are involved in this case. It is proved that quadratic forms have only even ranks and at most one of four quadratic forms has the lowest rank $4k-2$.Abstract i Contents iii List of Tables vi List of Figures vii 1. Introduction 1 1.1. Background 1 1.2. Overview of Dissertation 9 2. Sequences with Low Correlation 11 2.1. Trace Functions and Sequences 11 2.2. Sequences with Low Autocorrelation 13 2.3. Sequence Families with Low Correlation 17 3. A New Family of p-ary Sequences of Period (p^n−1)/2 with Low Correlation 21 3.1. Introduction 22 3.2. Characters 24 3.3. Gaussian Sums and Kloosterman Sums 26 3.4. Notations 28 3.5. Definition of Sequence Family 29 3.6. Correlation Bound 30 3.7. Size of Sequence Family 35 3.8. An Example 38 3.9. Related Work 40 3.10. Conclusion 41 4. On the Cross-Correlation between Two Decimated p-ary m-Sequences by 2 and 4p^{n/2}−2 44 4.1. Introduction 44 4.2. Decimated Sequences of Period (p^n−1)/2 49 4.3. Correlation Bound 53 4.4. Examples 59 4.5. A New Sequence Family of Period (p^n−1)/2 60 4.6. Discussions 61 4.7. Conclusion 67 5. On the Cross-Correlation of Ternary m-Sequences of Period 3^{4k+2} − 1 with Decimation (3^{4k+2}−3^{2k+1}+2)/4 + 3^{2k+1} 69 5.1. Introduction 69 5.2. Quadratic Forms and Linearized Polynomials 71 5.3. Number of Solutions of x^{p^s+1} − cx + c 78 5.4. Notations 79 5.5. Quadratic Form Expression of the Cross-Correlation Function 80 5.6. Ranks of Quadratic Forms 83 5.7. Upper Bound on the Cross-Correlation Function 89 5.8. Examples 93 5.9. Related Works 94 5.10. Conclusion 94 6. Conclusions 96 Bibliography 98 초록 109Docto

Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

An Assessment of Indoor Geolocation Systems

Currently there is a need to design, develop, and deploy autonomous and portable indoor geolocation systems to fulfil the needs of military, civilian, governmental and commercial customers where GPS and GLONASS signals are not available due to the limitations of both GPS and GLONASS signal structure designs. The goal of this dissertation is (1) to introduce geolocation systems; (2) to classify the state of the art geolocation systems; (3) to identify the issues with the state of the art indoor geolocation systems; and (4) to propose and assess four WPI indoor geolocation systems. It is assessed that the current GPS and GLONASS signal structures are inadequate to overcome two main design concerns; namely, (1) the near-far effect and (2) the multipath effect. We propose four WPI indoor geolocation systems as an alternative solution to near-far and multipath effects. The WPI indoor geolocation systems are (1) a DSSS/CDMA indoor geolocation system, (2) a DSSS/CDMA/FDMA indoor geolocation system, (3) a DSSS/OFDM/CDMA/FDMA indoor geolocation system, and (4) an OFDM/FDMA indoor geolocation system. Each system is researched, discussed, and analyzed based on its principle of operation, its transmitter, the indoor channel, and its receiver design and issues associated with obtaining an observable to achieve indoor navigation. Our assessment of these systems concludes the following. First, a DSSS/CDMA indoor geolocation system is inadequate to neither overcome the near-far effect not mitigate cross-channel interference due to the multipath. Second, a DSSS/CDMA/FDMA indoor geolocation system is a potential candidate for indoor positioning, with data rate up to 3.2 KBPS, pseudorange error, less than to 2 m and phase error less than 5 mm. Third, a DSSS/OFDM/CDMA/FDMA indoor geolocation system is a potential candidate to achieve similar or better navigation accuracy than a DSSS/CDMA indoor geolocation system and data rate up to 5 MBPS. Fourth, an OFDM/FDMA indoor geolocation system is another potential candidate with a totally different signal structure than the pervious three WPI indoor geolocation systems, but with similar pseudorange error performance

On Cryptographic Properties of LFSR-based Pseudorandom Generators

Pseudorandom Generators (PRGs) werden in der modernen Kryptographie verwendet, um einen kleinen Startwert in eine lange Folge scheinbar zufälliger Bits umzuwandeln. Viele Designs für PRGs basieren auf linear feedback shift registers (LFSRs), die so gewählt sind, dass sie optimale statistische und periodische Eigenschaften besitzen. Diese Arbeit diskutiert Konstruktionsprinzipien und kryptanalytische Angriffe gegen LFSR-basierte PRGs. Nachdem wir einen vollständigen Überblick über existierende kryptanalytische Ergebnisse gegeben haben, führen wir den dynamic linear consistency test (DLCT) ein und analysieren ihn. Der DLCT ist eine suchbaum-basierte Methode, die den inneren Zustand eines PRGs rekonstruiert. Wir beschließen die Arbeit mit der Diskussion der erforderlichen Zustandsgröße für PRGs, geben untere Schranken an und Beispiele aus der Praxis, die veranschaulichen, welche Größe sichere PRGs haben müssen

Statistical methods for the identification and control of multivariate stochastic systems

Imperial Users onl