Application of Residue Arithmetic in Communication and Signal Processing

Application of Residue Arithmetic in Communication and Signal Processing
Application of Residue Arithmetic in
Communication and Signal Processing
by
Pallab Maji
Roll No. # 209EC1109
A Thesis submitted for partial fulfilment for the degree of
Master of Technology
in
Electronics and Communication Engineering
Spcl: Telematics and Signal Processing
Dept. Electronics and Communication Engineering
NATIONAL INSTITUTE OF TECHNOLOGY
Rourkela, Orissa-769008, India
June 2011
Application of Residue Arithmetic in
Communication and Signal Processing
by
Pallab Maji
Roll No. # 209EC1109
A Thesis submitted for partial fulfilment for the degree of
Master of Technology
in
Electronics and Communication Engineering
Spcl: Telematics and Signal Processing
Under the Supervision of
Prof. (Dr.) Girija Sankar Rath
Dept. Electronics and Communication Engineering
National Institute of Technology, Rourkela,
Orissa-769008, India
June 2011
Dedicated
to
my niece
Pauli Maji
National Institute of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, “Application of Residue Arithmetic
in Communication and Signal Processing ” submitted by Pallab Maji in partial
fulfillment of the requirements for the award of Master of Technology Degree in Electronics & Communication Engineering with specialization in Telematics and Signal
Processing during 2010-2011 at the National Institute of Technology, Rourkela (Deemed
University) is an authentic work carried out by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been
submitted to any other University / Institute for the award of any Degree or Diploma.
Date
Prof. (Dr.) Girija Sankar Rath
Dept. of Electronics & Communication Engg.
National Institute of Technology
Rourkela-769008
Orissa, India
Acknowledgements
This dissertation would not have been possible without the guidance and the
help of several individuals who in one way or another contributed and extended their valuable assistance in course of this study.
My utmost gratitude to Prof. Girija Sankar Rath, my dissertation adviser
whose sincerity and encouragement I will never forget. Prof. Rath has been
my inspiration as I hurdle all the obstacles in the completion this research
work and has supported me throughout my project with patience and knowledge whilst allowing me the room to work in my own paradigms.
Sincere thanks to Prof. S. K. Patra, Prof. K. K. Mahapatra, Prof. Samit Ari,
Prof. S. K. Das, Prof. S. K. Behera, Prof. S. Meher, Prof. A. K. Sahoo and Prof.
Poonam Singh for their constant cooperation and encouragement through out
the course.
I also extend my thanks to entire faculty of Dept. of Electronics and Communication Engineering, National Institute of Technology Rourkela, Rourkela
who have encouraged me throughout the course of Master’s Degree.
I would like to thank all my friends, especially Ashish Agarwal, Kapil Parmar, Sanjay Meena, Sujeet Rai, Vaibhab Raj, Dipanjan Bhadra, Prasun Bhattacharya and Venkatesh S. for their help during the course of this work. I also
thank all my classmates for all the thoughtful and mind stimulating discussions we had, which prompted us to think beyond the obvious. I take immense pleasure to thank our seniors namely, Runa Kumari, Yogesh Kumar
Choukiker, Senthilnathan Natarajmani for their endless support in solving
queries and advices for betterment of dissertation work. I would also take this
opportunity to thank Mr. Prasanta Pradhan, Mr. Bijay Muni, Mr. Ayaskanta
Swain and Mr. Jaganath Mohanty for their support during this dissertation.
And finally thanks to my parents, my brother, sister-in-law and Roshni Hazra,
whose faith, patience and teaching had always inspired me to walk upright
in my life. Without all these beautiful people my world would have been an
empty place.
Pallab Maji
[email protected]
Abstract
Residue Number System (RNS) is a non-weighted number system. In RNS,
the arithmetic operations are split into smaller parallel operations which are
independent of each other. There is no carry propagation between these operations. Hence devices operating in this principle inherit property of high
speed and low power consumption. But this property makes overflow detection is very difficult. Hence the moduli set is chosen such that there is
no carry generated. In this thesis, the use of residue number system (RNS)
is portrayed in designing solution to various applications of Communication
and Signal Processing. RNS finds its application where integer arithmetic
is authoritative process, since residue arithmetic operates efficiently on integers. New moduli set selection process, magnitude comparison routine and
sign detection methods were limed on the onset of this dissertation.
A good example of integer arithmetic is digital image. The pixels are represented by 8 bit unsigned number. Thus the operations are primarily unsigned
and restricted to a small range. Hereby, in this thesis, a novel image encryption technique is depicted. The results show the robustness and timeliness of
this technique. This technique is further compared to some of industry standard encryption algorithms for analysis based on robustness, encryption time
and various other paradigms.
Filters are signal conditioners. Each filter functions by accepting an input
signal, blocking pre-specified frequency components, and passing the original
signal minus those components to the output. A lowpass filter allows only low
frequency signals (below some specified cutoff) through to its output, so it can
be used to eliminate high frequencies. A novel design approach for a low pass
filter based on residue arithmetic was also proposed. Some trite techniques
as well as novel approaches were adopted to solve the design challenges. A
technique for mapping the data in another space providing the liberty to work
with floating numbers with a precision was adopted.
PN sequence generator based on residue arithmetic is also formulated. This
algorithm generates a pseudo-noise sequence which further was used to evince
a spread spectrum multiuser communication system. The results are compared with trite techniques like Gold and Kasami sequence generators.
Contents
Contents
vi
List of Figures
ix
List of Tables
xi
List of Acronyms
xii
1 Introduction
1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Overview of RNS Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 Introduction to Residue Arithmetic
7
2.1 Basics of Residue Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.1 Multiplicative Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.2 Reverse Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.1.3 Addition and Multiplication . . . . . . . . . . . . . . . . . . . . . . . . .
9
vi
CONTENTS
2.2 Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Modular Multiplicative Inverse . . . . . . . . . . . . . . . . . . . . . . . 11
2.3.2 Extended Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Limitations and Constraints in Residue Arithmetic . . . . . . . . . . . . . . . 12
2.4.1 Magnitude Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2 Sign Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4.3 Overflow Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Moduli Selection and Mapping
15
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Consecutive Moduli Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Exponential Moduli Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Utility Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.5 Homomorphic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 Application : RNS Based FIR Filter
20
4.1 FIR Filter Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Proposed Filter Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Filter Specification and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4.1 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.4.2 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Application : RNS Based Image Encryption
29
5.1 Overview to Image Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3.1 Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
vii
CONTENTS
5.3.2 Decryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.4 Standard Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5.1 PSNR Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.5.2 Median Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.5.3 Comparision with Blowfish Encryption . . . . . . . . . . . . . . . . . . 36
5.5.4 Bit Error Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5.5 Encryption Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.5.6 Correlation and Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Application : RNS Based PN Sequence
42
6.1 Introduction to Spread Spectrum Communication . . . . . . . . . . . . . . . . 43
6.2 Proposed PN Sequence Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.3 Standard PN Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.5 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7 Conclusion
50
7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.2 Scope for Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Appendix A: Image Encryption Simulation Parameters
52
Appendix B: Spread Spectrum
53
Bibliography
55
Publications
62
7.1 Journal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.2 Conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
viii
List of Figures
1.1 Basic RNS Based Signal Processor . . . . . . . . . . . . . . . . . . . . . . . . .
3
3.1 Bit Distribution of Residue Number with Consecutive Moduli . . . . . . . . . 17
3.2 Bit Distribution of Residue Number with Exponential Moduli . . . . . . . . . 17
4.1 FIR Filter Direct Form Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Frequency Response of FIR Filter with cutoff frequency 0.3π . . . . . . . . . 22
4.3 Architecture of the proposed RNS based FIR filter . . . . . . . . . . . . . . . . 23
4.4 Comparison of Frequency Response of Proposed RNS Filter with Traditional Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Performance Analysis between Traditional and RNS Based FIR Filter Implementation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.6 Performance and Stability of the RNS based FIR Filter . . . . . . . . . . . . . 28
5.1 Flowchart of RNS Based Encryption Technique . . . . . . . . . . . . . . . . . 31
5.2 Flowchart of RNS Based Decryption Technique . . . . . . . . . . . . . . . . . 33
5.3 RNS Based Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.4 BER plot of RNS based encryption technique . . . . . . . . . . . . . . . . . . . 36
ix
LIST OF FIGURES
5.5 RNS Based Encryption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.6 Denoised Images Received At Various SNR . . . . . . . . . . . . . . . . . . . . 39
5.7 BER Plot of RNS Based Encryption and Blowfish Ecryption . . . . . . . . . . 40
6.1 Direct Sequence Spread Spectrum Bit Pattern . . . . . . . . . . . . . . . . . . 44
6.2 Direct Sequence CDMA Technique . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.3 RNS Based PN Sequence Generator . . . . . . . . . . . . . . . . . . . . . . . . 45
6.4 Correlation Matrix for 10 sequence generated by RNS based PN generator . 47
6.5 BER plot for RNS based PN sequence for 2 and 15 User . . . . . . . . . . . . 48
6.6 BER plot for Gold, Kasami and RNS based PN sequence for 5 and 10 Users
x
49
List of Tables
2.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
5.1 PSNR Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Encryption time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Correlation and Entropy Calculation . . . . . . . . . . . . . . . . . . . . . . . . 41
6.1 Correlation Matrix for PN sequence with β = 128 . . . . . . . . . . . . . . . . 46
xi
List of Acronyms
RNS
CRT
DAC
ADC
DSP
CDMA
PN
NCRT
BER
SS
DS-SS
SNR
Residue Number System
Chinese Remainder Theorem
Digital to Analog Converter
Analog to Digital Converter
Digital Signal Processor
Code Division Multiple Access
Pseudo Noise
New Chinese Remainder Theorem
Bit Error Rate
Spread Spectrum
Direct Sequence Spread Spectrum
Signal to Noise Ratio
xii
CHAPTER 1
Introduction
1.1 Background
The residue arithmetic basically originated from China in first century A.D. However
Chinese mathematician Sun Tzu is given the due credit for giving birth to residue number system (RNS) along with Greek scholar Nichomachus and Hsin-Tai-Wei. Even though
this number system is such old, not much work was done before twentieth century. Even
the famous Chinese Remainder Theorem (CRT) was proved by Euler in 1734. This number system did not find much practical application until D. H. Lehmer, A. Svoboda and M.
Valach designed a hardware based on residue arithmetic to reduce complex calculations
in 1955. This triggered a series of research application based on RNS. Major contributions came from M. A. Soderstrand, W. K. Jenkins, W. C. Miller, R. I. Tanaka, B. L. Leon,
N. Szabo and others in more than three decades since then. However they basically
worked on ROM based devices. They mainly worked on Residue number scaling, RNS
to binary converters, binary to RNS converters, error corrections etc. Non-Rom based
devices based on RNS were designed with advent of VLSI technologies. This led to work
1
1.1. BACKGROUND
on special moduli sets [1].
M. A. Soderstrand basically worked on design and implementation DAC, adaptive filters, modulo adders, convolvers and digital filters. He gave implementation design for
complex heterodyne tunable filters. G. A. Jullien worked extensively in developing algorithms for selection of efficient moduli for high speed DSP architectures. He gave various
algorithms for selection of moduli set of three based on various applications and also
proposed modified quadratic RNS. He also worked with W. C. Miller for VLSI implementation of digital filters, adders, convolvers and various DSP architectures based on RNS.
They also worked with M. A. Bayoumi to develop systolic array based on RNS for various
DSP algorithms. They also proposed the famous look-up table methodology for various
RNS based designs [1].
However the availability of inexpensive 16-bit and 32-bit digital signal processors
for extensive work on picture encoding, speech encoding and other digital processing
algorithms befogged the need of residue arithmetic based processors. However, special
purpose processors have been developed for various applications.
During the extensive literature survey the quest for low power designs and high speed
computer engines was evident. High speed devices like adder, multipliers, ALU, accumulators were most sought after designs. The ’Break and Process’ approach of RNS domain
attains the increase in speed of operations with its own limitations which is described
later. Nevertheless the applications in which divisions, comparisons, scaling operations
are less and can be avoided, typical RNS system works satisfactorily. Thus applications
like image encoding and encryption, FIR filter, spread spectrum communication systems
can be designed based on Residue Arithmetic. A RNS based system generally operates
on integer data. Fig. 1.1 shows a basic signal processor based on RNS.
As we learn from the fig. 1.1, the front end has an analog to digital converter and
a binary to RNS converter whose k output words corresponding to k moduli will be processed by the iki parallel processors. At the very end, the k words are converted to binary
number and then passed on to digital to analog converter (DAC) to get processed analog
output.
2
1.2. OVERVIEW OF RNS APPLICATIONS
Residue
Processor
1
Analog
Input
ADC
Binary to
RNS
Converter
Residue
Processor
2
RNS to
Binary
Converter
DAC
Analog
Output
Residue
Processor
k
Figure 1.1: Basic RNS Based Signal Processor
1.2 Overview of RNS Applications
As described in section 1.1, RNS is mostly used in VLSI implementation of DSP architecture for achieving low power and high speed. Among the application RNS, implementation of FIR filters, IIR filters, adaptive filters, digital frequency synthesis, two dimensional filters, image encryption and coding are most significant[1].
Most significant research work in the field of RNS was done in forward and reverse
conversion. Recent research on reverse converters [2; 3; 4] uses New Chinese Remainder Theorem proposed by Y. Wang[5]. One dimensional[6; 7; 8; 9] and two dimensional
Discrete Wavelet Transform architecture[10] based on residue arithmetic made a crucial
impact.
RNS had been effectively used in encryption and coding [11; 12]. In field of communication, RNS were introduced in field of CDMA by many researchers. Frequency hopping
techniques[13], DS-CDMA[14; 15; 16; 17; 18] and PN sequence correlators[19] based on
RNS were proposed and designed.
There were many programmable architectures designed based on residue arithmetic.
Griffin and Taylor[20] proposed RNS based RISC architecture as future research area.
H. T. Vergos in the year 2001, proposed a 200 MHz RNS core[21]. There were many im-
3
1.3. MOTIVATION
plementation of fast digital processors based on RNS in literature[22; 23; 24]. J. Ramirez
developed a 32-bit SIMD RISC processor fully based on RNS[25].
Thus we find numerous applications of RNS in field of Communication and Signal
Processing. In this thesis, some novel techniques have been proposed with reference to
some trite techniques present in literature to develop some application of RNS in field of
Communication and Signal Processing.
1.3 Motivation
The ’break and process’ construct drives the researchers to work with residue arithmetic.
This leads to ’carry-free’ arithmetic operations. However operations viz. division, comparison, scaling etc. are very difficult since there is no carry propagation. But applications
where these operations are not dominant and can be avoided and applications which
are limned by integer arithmetic, RNS can be expended. Hereby, these facts are major
stimuli to explore the potential of RNS in various fields of Communication and Signal
Processing. Basics of residue arithmetic leads to complexity in magnitude calculation,
sign detection and incorporation of fractional numbers. These constraints however made
the implementation of ideas challenging and research worthy. Therefore the approach to
digital FIR filter design was viewed from different perspective.
All digital images are represented by 8 bit unsigned pixel values; this sets perfect
platform for the use of RNS in digital image processing. In this thesis, a image processing
technique based on RNS is portrayed. Various techniques are present in literature for
generation of PN sequences in literature. PN sequences with better autocorrelation and
low cross-correlation are of prime interest in spread spectrum communication. Since
the sequences are noise like and deals with random generation of series of bits [-1,1] of
varied lengths, the scope of RNS based PN sequence was large. These facts, constructs
and analyses were impulsive force for the research work done and this dissertation.
4
1.4. THESIS ORGANIZATION
1.4 Thesis Organization
Chapter-1
Introduction
Chapter-2
Introduction to Residue Arithmetic
The basics of RNS and the operations of residue arithmetic and its advantages are
discussed in this chapter. The famous CRT and NCRT are discussed thoroughly. The
proposed problems which were approached in this dissertation are explained in detail.
Chapter-3
Moduli Selection and Mapping
In this chapter, the techniques of generation of co-prime moduli selection based on
operating bits of any application are limned. There are two techniques proposed for the
same. The concept of utility factor is also given as a standard solution to judge the moduli
based on the bit efficiency. The concept of incorporation of fractional numbers with the
aid of Homomorphic Mapping is also introduced.
Chapter-4
Application: RNS Based FIR Filter
In this chapter the first application in field of communication and signal processing
is depicted. A 32-bit FIR filter based on residue arithmetic is portrayed. Results are
compared to the traditional FIR filter and a compendious study is given in the end. The
specifications of the filter simulated is mentioned along with the simulation environment.
Chapter-5
Application : RNS Based Image Coding
This chapter presents the basic concept image encryption and coding. A succinct study
of the proposed algorithm for image encryption and comparison to industry standard encryption techniques are depicted further. The results of this comparison is tabulated and
analysed. The results are compared to the industry standard BLOWFISH encryption
technique and the performance analysis based on encryption time, output image quality,
correlation between encrypted images with original image, entropy for the same and robustness against channel noise is chalked out in detail.
5
1.4. THESIS ORGANIZATION
Chapter-6
Application : RNS Based PN Sequence
This chapter deals with an algorithm to generate a RNS based PN sequence.The generation of the PN sequence is based on the key provided for the secured and authentic
transmission along with a set of co-prime numbers generated from the spread factor β. A
compendious study comparing the generated sequence with Gold and Kasami sequence
is limned further. The results are portrayed there after.
Chapter-7
Conclusion and Scope for Future Work
The overall conclusion of the thesis is presented in this chapter. It also contains some
future research topics which need attention and further investigation. Application of
RNS to further areas of Communication and Signal Processing are also sighted under
this chapter.
6
CHAPTER 2
Introduction to Residue Arithmetic
Residue number systems are based on the congruence relation, which is defined as follows. Two integers, a and b are said to be congruent modulo m if m divides exactly the difference of a and b; it is common, especially in mathematics tests, to write a ≡ b(mod m i )
to denote this. Thus, for example, 10 ≡ 7(mod 3), 10 ≡ 4(mod 3), 10 ≡ 1(mod 3) and
10 ≡ −2(mod 3). The number m is a modulus or base, and we shall assume that its
values exclude unity, which produces only trivial congruences. If q and r are the quotient and remainder, respectively, of the integer division of a by m, that is, a = q· m + r
then, by definition, we have a ≡ r(mod m i ). The number r is said to be the residue of a
with respect to m, and we shall usually denote this by r = |a|m . The set of m smallest
values, 0, 1, 2, 3 · · · m − 1 that the residue may assume is called the set of least positive
residues modulo m. Suppose we have a set {m1 , m 2 · · · m N }, of N positive and pairwise
relatively prime moduli. Let M be the product of the moduli. Then every number X < M
has a unique representation in the residue number system, which is the set of residues
{| X |m i : 1 ≤ i ≤ N}. A partial proof of this is as follows[1; 26].
Suppose X 1 and X 2 are two different numbers with the same residue set. Then
7
2.1. BASICS OF RESIDUE ARITHMETIC
| X 1 |m i = | X 2 |m i and so {| X 1 − X 2 |m i = 0}. Therefore X 1 and X 2 is the lcm of m i . But if the
m i are relatively prime, then their LCM is M, and it must be that X 1 and X 2 is a multiple of M. So it cannot be that X 1 < M and X 2 < M. Therefore, the set {| X |m i : 1 ≤ i ≤ N}
is unique and may be taken as the representation of X. The number M is called the dynamic range of the RNS, because the number of numbers that can be represented is M.
For unsigned numbers, that range is [0, M − 1]. Representations in a system in which the
moduli are not pairwise relatively prime will not be unique; two or more numbers will
have the same representation.
2.1 Basics of Residue Arithmetic
Consider two numbers 43 and 29. Consider a moduli set P = [3, 5, 7]. Then RNS representation of these two numbers are as follows: 43 → {1, 3, 1}
29 → {2, 4, 1}
2.1.1 Multiplicative Inverse
M i .M i−1 = 1 (mod p i )
where M i =
R
pi .
(2.1)
Thus for the above case, R = 3 × 5 × 7 = 105 and M 1 = 35, M2 = 21 and
M3 = 15. Hence M1−1 = 2, M3−1 = 1 and M3−1 = 1.
2.1.2 Reverse Conversion
k M i .M −1 n i i
i=0
R
where R is range p i is i th number of moduli set P.
Thus for RNS number {1,3,1},
|(2 × 35 × 1) + (1 × 21 × 3) + (1 × 15 × 1) |105 = |70 + 63 + 15 |105 = |148|105 = 43
Similarly for {2,4,1} gives 29.
8
(2.2)
2.2. ADVANTAGES
2.1.3 Addition and Multiplication
+
=
43||
29||
72||
1
2
0
3
4
2
1
1
2
Table 2.1: Addition
*
=
43||
29||
92||
1
2
2
3
4
2
1
1
1
Table 2.2: Multiplication
Multiplication gives 92 as the result is |1247|105 = 92
2.2 Advantages
Most important advantage of residue arithmetic over conventional arithmetic is the absence of carry propagation, in the two main operations of addition and multiplication, and
the relatively low precisions required ranging to individual prime or co-prime number of
the moduli set, which enables LUT implementations in various operations. In practice,
these may make residue arithmetic worthwhile, even though in terms of practical applications such arithmetic has little fields over conventional arithmetic to cover. However
the concept of ’break and process’ can be very useful in places where integer arithmetic
is predominant. However, the fields of Communication and Signal Processing are yet to
be explored thoroughly for application on RNS[1; 26] .
Basic advantages of residue arithmetic are:
• High Speed
• Low Power
• Superior Fault Tolerance
• Reduction of Computational Load
9
2.3. CHINESE REMAINDER THEOREM
2.3 Chinese Remainder Theorem
Consider two positive numbers, x (the dividend) and y (the divisor). Then x modulo y
(abbreviated as a(mod n)) can be thought of as the remainder, on division of x by y. Now
two simultaneous congruences n = n 1 (mod m1 ) and n = n 2 (mod m2 ) are only solvable
when n 1 = n 2 (mod (gcd(m 1 , m 2 ))). The solution is unique when m 1 and m 2 are co-prime
and their gcd is 1. The Chinese Remainder Theorem (CRT) may be stated as one of the
most important fundamental results in the theory of RNS[1; 26].
Let m2 and m 2 and y be positive integers which are relatively prime. Let n2 and n 2
be any two integers. Then there is an integer N such that
N ≡ n 1 (mod m1 ) and N ≡ n 2 (mod m2 )
(2.3)
Moreover, N is uniquely determined modulo (m1 . m 2 ). An equivalent statement is that if
gcd(m 1 , m 2 ) = 1 then every pair of residue classes modulo m1 and m 2 corresponds to a
simple residue class modulo (m1 . m 2 ).
The theorem can be generalized as follows:
Given a set of simultaneous congruences
n = n i (mod m i ) , for i = 1 to p and for which m i are relatively prime. Then the solution to
the set of congruences is
x = [a1 · b 1
M
M
M
, a2 · b2
··· ap · bp
]
m1
m2
mp
(2.4)
Where M = { m1 , m 2 · · · m p }, and the b i are determined from:
M
≡ 1(mod m i )
bi m
i
10
(2.5)
2.3. CHINESE REMAINDER THEOREM
2.3.1 Modular Multiplicative Inverse
m1 .m 2 = 1(modn)
1
m−
1 = m 2 (mod n)
(2.6)
The multiplicative inverse of m1 modulo n exists iff m1 and n are coprime, i.e,
gcd(m1 , n) = 1
(2.7)
If the modular multiplicative inverse of m1 modulo n exists, the operation of division by
m1 modulo n can be defined as multiplying by the inverse, which is in essence the same
concept as division in the field of real. The modular multiplicative inverse can be found
out using various techniques, one of the most efficient being the Extended Euclidean
Algorithm.
2.3.2 Extended Euclidean Algorithm
The Extended Euclidean Algorithm can find the multiplicative inverse of any number
a and b in Z when x and y are given and their inverse exists. Assume the following
equation[1; 26]:
a.x + b.y = gcd(x, y)
However, if the multiplicative inverse of b exists, then gcd(n,b) = 1. So we have now:
⇒ a.x + b.y = 1
Now we apply modulo operator to both sides. We will have
⇒ (a.x + b.y) mod x = 1
⇒ (a.x) mod x + (b.y) mod x = 1
⇒ 0 + (b.y) mod x = 1
⇒ b−1 ≡ y (mod x)
This means that y is the multiplicative inverse of b in Z.
11
(2.8)
2.4. LIMITATIONS AND CONSTRAINTS IN RESIDUE ARITHMETIC
2.4 Limitations and Constraints in Residue Arithmetic
On completion of extensive literature survey the following problem statements were defined:
• Magnitude Calculation
• Sign Detection
• Overflow Detection and Correction
2.4.1 Magnitude Calculation
Chinese Remainder Theorem (CRT) can be used to convert to the weighted number system of any given number and then can be compared with the other one. However this
technique is very time consuming as well as increases computational load. Hereby, the
approximate CRT method is used to compare magnitude. The theorem[26] is as follow:
N = {n 1, n2 · · · n p } =
k
−1
i=0
N {n1, n2 · · · n p }
=
=
M
M
N {n1, n2 · · · n p }
=
=
M
M
M i 〈 a i · n i 〉m i
M
k
−1
i=0
k
−1
i=0
Mi
〈 a i · n i 〉m i
M
1
m−i 1 〈a i · n i 〉m i
(2.9)
1
N
) are in [0, 1]. These values are stored in the look up table and the
All values of ( M
magnitude of each number is calculated to get a magnitude representation.
12
2.5. SUMMARY
2.4.2 Sign Detection
For sign detection a trite technique was used. In this a sign bit was introduced as 32 nd bit.
This results in a signed bit representation of a number. Operating on floating numbers is
a difficult task in residue arithmetic. Hence we convert the floating point number to an
integer in a range ± Z. The mapping is done such that the operating range Z < R. If the
floating point number ± A is in range, and A < Z, then
Z
A
(2.10)
x
P sn
(2.11)
P sn =
X=
where P sn is precision which decides the incorporation of digits after decimal points.
2.4.3 Overflow Detection
The dynamic range of 18 bits is supposed to be adequate for most practical situation as
the bound of 18 bits is the worst outcome possible [27]. Now during the homomorphic
mapping if we can keep the operating range Z is taken around 218 , and the coefficients
are mapped over range of 214 , then scope for overflow is drastically limited.
2.5 Summary
The constraints defined in section 2.4 and the approach obtained to avoid them is enforced throughout in all applications mentioned in this thesis. The basics of all the application and the way residue arithmetic is involved in all the chapters discussed further
is summarized in this chapter. The basics of residue arithmetic and the operations like
multiplication and addition are approached in ’break and process’ methodology. This
reduces complexity and computational load and processes things faster. However the
undiscovered part of RNS is the power of its wide range as specified by the moduli set.
This system has a non-linear distribution of numbers. Two number cannot be compared
13
2.5. SUMMARY
just viewing them. Computation has to be done to analyse them. This concept can be
used in various application. Image encryption as in chapter 5 uses this fundamental of
RNS for encryption. Further the concept has been used in PN sequence generation.
14
CHAPTER 3
Moduli Selection and Mapping
3.1 Introduction
If we consider any prime number, then there
exists at least one primitive root r ≤ p − 1,
i
such that the set r p : i = 0, 1, 2 . . . p − 2 is set of all possible non-zero residues with
respect to p. 5 and 7 are examples of primitive root. Consider r = 3 and p = 5. Then
0 3 = 1, 31 = 3, 32 = 4, 33 = 2
5
5
5
5
. Thus we get {1, 3, 4, 2} where all possible non-zero residues w.r.t. 5 are present [26].
For any digital application moduli set should be chosen such that representation is
efficient, unique and the difference between the moduli be as small as possible to increase
the dynamic range[26], thus improving the utility factor. Utility factor is mentioned in
section 3.4. In this thesis all applications uses either of the selection of moduli set as
following two proposed methods:
• Consecutive Moduli Selection
15
3.2. CONSECUTIVE MODULI SELECTION
• Exponential Moduli Selection
3.2 Consecutive Moduli Selection
Theorem: Let {N1 , N2 , N3 . . . N k } be a set of k consecutive co-prime numbers. Let these
numbers be expressed as
N i = N1 − m i−1
(3.1)
for i=1,2,3 . . . k, where m i−1 > m i−2 > m i−3 > . . . m i−k > 0. Let N k+1 be another number
that can be added to the set of co-prime numbers, i.e. Nk+1 = N1 − m k then N k+1 will be
co-prime if gcd(N i , m k − m i−1 ) = 1, for i=1,2,3 . . . k.
Proof
For Nk+1 to be co-prime to every other numbers in the set {N1 , N2 , N3 . . . N k },
gcd(N i , Nk+1 ) = 1
⇒ gcd (Nk+1 , N i − Nk+1 ) = 1
⇒ gcd (Nk+1 , m i − m i−1 ) = 1
(3.2)
In order to improve dynamic range of RNS with high bit efficiency, N1 must be selected
as 2 m−1 which will be a m bit number. Then by the above method one can generate the
set of co-prime numbers and use them as moduli set for RNS. The figure represents the
bit pattern obtained for consecutive moduli.
16
3.3. EXPONENTIAL MODULI SELECTION
8 bit
8 bit
8 bit
8 bit
32 bit Number
Figure 3.1: Bit Distribution of Residue Number with Consecutive Moduli
t bit
r bit
s bit
q bit
32 bit Number
Figure 3.2: Bit Distribution of Residue Number with Exponential Moduli
3.3 Exponential Moduli Selection
Figure 3.2 shows the bit distribution in exponential moduli system. In figure 3.2, t =
32 − (s + r + q). Consider a set of number P such that
p i = 2 n− k − 1
(3.3)
where k = { k N : ∀ k, gcd(2 n−k , P) is 1}.
3.4 Utility Factor
Consider set P = { p1 , p 2 . . . p n } of n co-prime numbers of x bits each. Hence every number
will be represented in n.x bits. Thus if we have to represent any number into k bits
system, then
x=
k
n
(3.4)
where,
p i < 2x
17
(3.5)
3.4. UTILITY FACTOR
Hence, n and k should be chosen such that | b|n = 0. Eg. A 32bit number can be represented in the RNS as four 8-bit numbers with a range 0 to P i−1 .
The numbers which cannot be represented by this scheme of representation,
e = 2k −
n
−1
pi
(3.6)
i=0
Thus Utility factor will be:
Uf = 1 −
e
2k
(3.7)
Example
Choose co-prime numbers with b=32 and n=4, i.e., close to 28 (= 256), gives us P = {255,
254, 253, 251} and P = {31, 127, 511, 2047} when consecutive moduli selection and exponential moduli selection is used respectively. Since there cannot be any other combination
of co-prime numbers close to 28 , hence this is the most bit efficient moduli set with widest
dynamic range possible in range of 8 bit numbers. The dynamic range R is hence:
R=
n
−1
pi
(3.8)
i=0
Now calculating R as in eq 3.8, R Con = 4113089310 and R Ex p = 4118168929. In literature there are standard moduli set for medium dynamic ranges (less than 22 bits), three
moduli set 2n, 2nÂś1, and for large dynamic range (greater than 22 bits), general moduli
set in form 2n1 , 2n1 + 1, 2 n2 ± 1 . . .2 n i ± 1 with length greater than three are more efficient
[27; 28; 29]. But this moduli set shows a better bit utilization. The dynamic range of
18 bits is supposed to be adequate for most practical situation as the bound of 18 bits is
the worst outcome possible [27]. Hereby the dynamic range R is a huge margin to avoid
overflow. The calculated
U f con = 0.95765 and U f ex p = 0.95884
. The higher the utility factor better is the moduli set in term of bit utilization.
18
3.5. HOMOMORPHIC MAPPING
3.5 Homomorphic Mapping
The next step is to map the number i.e. to convert the floating point numbers to integers. The operating range is suitably taken such that no overflow occurs. Mapping is a
special correspondence between the members (elements) of two fields. Two homomorphic
systems have the same basic structure. Their elements and operations appear different;
results on one system often apply as well to the other system [30]. Operating on floating
numbers is a difficult task in residue arithmetic. Hence we convert the floating point
number to an integer in a range ± Z. The mapping is done such that the operating range
Z < R.
Precisely if,
Z≤R
then overflow of any operation can be easily avoided. If the floating point number x is
in range ± AandA < Z, then
P sn =
Now, mapped output is:
X=
Z
A
x
A
(3.9)
In this mapping the range ± A has predominant effect on the precision with which the
technique can handle the floating point numbers. As value of A increases the precision
decreases and the floating point numbers accuracy deteriorates.
3.6 Summary
Through out this thesis, the moduli selection and homomorphic mapping is used in various applications. The techniques are novel and hereby the new method of comparing
moduli based on bit representation is unique and has been bit-efficient in every applications depicted further in this thesis.
19
CHAPTER 4
Application : RNS Based FIR Filter
Residue number system (RNS) is generally an integer number system as described in
chapter 2. The foremost canonical reason for implementation of filter in residue arithmetic is the inherent property of carry-free addition, subtraction and multiplication. As
a result we add, subtract and multiply in unison regardless to the numbers. Hereby,
devices operating in this principle are fast and ingest low power. However, principal
limitation of Residue Number System is the slow and complex nature for arithmetic operations viz. division, comparison, sign detection and overflow detection and rejection.
In this paper we have described some novel approaches to grapple with the limitations of
comparison, sign detection and averting overflow. The selection of moduli in RNS is most
important in attaining to solutions of problems as described earlier. Accordingly, a set of
moduli is selected. Further in this paper we have used this set of moduli to successfully
depict a design approach for 32-bit lowpass finite impulse response (FIR) filter. A transposed form of FIR filter is preferred always when larger filters are used. For an 8-tap,
16 bit FIR filter the device utilization and performance obtained are almost identical for
both direct form as well as transposed form . But when large filters are deployed across
20
4.1. FIR FILTER BASICS
multiple devices, the traditional approach provides a lethargic response as the input to
output latency is reduced [31]. This filter architecture allows parallel processing of the
input signal thereby, sample rate is increased. RNS arithmetic is carry free and each
modulo arithmetic operation is independent to each other. This causes overflow detection
to be indocile. In literature there is no approach to detect overflow during any operation. However, if a moduli set is chosen such that the range of the moduli set limits the
magnitude of any practical signal, overflow can be significantly obviated.
4.1 FIR Filter Basics
-1
hk
-1
+
z
hk-1
X
+
z
h0
X
y(n)
X
x(n)
Figure 4.1: FIR Filter Direct Form Transpose
The input to a digital filter is usually a sequence of numbers obtained from sampling
an analog signal. The FIR filter may be represented by a block diagram of the type
shown in the figure 4.1. The z−1 block represents a unit delay. If the input is x(n),
then the output of the delay element is x(n-1), which is the value of x(n) one time-period
before now, or, put simply, the previous input. Similarly, x(n-2) simply means the value
of the input two sampling periods before now. The Filter shown in the figure 4.1 may be
represented by a difference equation.
y(n) = a 0 x(n) + a 1 x(n − 1) + a 2 x(n − 2) . . . a n x(0)
(4.1)
The number of branches in the filter is known as the taps of the filter. The filter can be
described in terms of its impulse response, which is a series of weighted impulses. With
the impulse response, it is easy to compute the output of the filter simply by multiplying
21
4.1. FIR FILTER BASICS
Magnitude (dB)
20
0
−20
−40
−60
−80
−100
−120
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
Normalized Frequency (×π rad/sample)
Phase (degrees)
0
−500
−1000
−1500
−2000
−2500
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency (×π rad/sample)
Figure 4.2: Frequency Response of FIR Filter with cutoff frequency 0.3π
the impulse response and the input train of sampled pulses present at the desired time.
We may use the idea of delay elements to rewrite the difference equation and so obtain
the transfer function. Since z-1 stands for a delay element we have
y(n) =
N
−1
k=0
h(k)x(n − k)
(4.2)
where n = 0, 1 . . . N − 1. Now the coefficients h k determine the characteristic of the fir
filter viz. low pass, high pass, band pass, band reject etc. There are several techniques
to determine the coefficients. The filter coefficients are determined using a Hammingwindow based, linear-phase filter with normalized cutoff frequency 0.3π rads. By default
the filter is normalized so that the magnitude response of the filter at the center frequency of the pass band is 0 dB. Thus the frequency response of the filter is as in figure
4.2; the frequency response of the lowpass fir filter with normalized cutoff frequency 0.3π
rads/sec. Matlab function ’fir1’ uses the window method of FIR filter design. If w(n) denotes a window, where 1 ≤ n ≤ N, and the impulse response of the ideal filter is h(n),
where h(n) is the inverse Fourier transform of the ideal frequency response, then the
22
4.2. PROPOSED FILTER ARCHITECTURE
windowed digital filter coefficients are given by following equation.
b(n) = w(n).h(n)
(4.3)
4.2 Proposed Filter Architecture
(a) Architecture
-1
-1
+
z
z
X
y p (n)
i
h0 mod pi
hk-1 mod pi
hk mod pi
+
X
X
x p (n)
i
(b) Mod p i Filter
Figure 4.3: Architecture of the proposed RNS based FIR filter
The figure 4.3 shows the proposed architecture of the FIR filter based on residue
arithmetic. The moduli selection as discussed in chapter 3 is the basic and foremost
23
4.3. FILTER SPECIFICATION AND DESIGN
step of this process. The next step is to map the number i.e. to convert the floating
point numbers to integers. The operating range is suitably taken such that no overflow
occurs. Mapping is a special correspondence between the members (elements) of two
fields. Two homomorphic systems have the same basic structure. Their elements and
operations appear different; results on one system often apply as well to the other system
[30]. Operating on floating numbers is a difficult task in residue arithmetic. Hence we
convert the floating point number to an integer in a range ± Z. The mapping is done
such that the operating range Z < R. Thereafter the filter process goes on as in the
figure. The architecture is a 32 bit filter architecture. However, the processing of RNS
conversion adds a 33 rd sign bit. The introduction of magnitude comparison technique for
the addition operation caused a reduction in time consumption. The multipliers used for
this filter is based on the IDEA cryptosystem using a recursive multiplication approach
[32]. This reduces the time as the time drastically. The performance analysis of the
proposed RNS based FIR filter is as portrayed further.
4.3 Filter Specification and Design
The coefficients of the filter decide the nature of the filter. There are several techniques
available in literature that generates the coefficients of a filter for a particular behaviour.
In RNS, since operations are split into smaller parallel operations which are independent
of each other, there exists n number of filters operating simultaneously. This provides
very high speed architecture. Figure 4.3 show the proposed RNS based filter. The residue
numbers and converted back by the Chinese remainder theorem based reverse conversion
method [26]. The performance analysis is with respect to the traditional FIR filter implementation technique as done in many papers present in the literature [26; 28; 29; 33; 34].
However, the simulation was done in matlab and no hardware implementation was done.
24
4.4. RESULTS
4.4 Results
4.4.1 Simulation
(a) Magnitude Response
(b) Phase Response
Figure 4.4: Comparison of Frequency Response of Proposed RNS Filter with Traditional
Filter
The filter coefficients are determined using a Hamming-window based linear-phase
64 tap lowpass FIR filter with normalized cutoff frequency 0.3 rads. By default the filter
is normalized so that the magnitude response of the filter at the center frequency of the
pass band is 0 dB. The filter is designed in Matlab ver. 7.9.0. This simulation depicts
the effects of this design approach on the filter. The magnitude response is compared
25
4.4. RESULTS
with the original response to find out the error as in figure 4.4(a). The phase response
figure 4.4(b) is linear over the filter pass band, but it loses its piecewise linearity after
the cut-off region, unlike the original FIR filter. However, this does not affect the filter
performance.
4.4.2 Performance Analysis
The filter generated with RNS moduli set of P=[31,127,511,2047] and provides the maximum range R = 4118168929 possible. Hereby the filter can operate between these ranges
and provides better bit efficiency than existing RNS based filters. The poles zero diagram, in figure 7, of the RNS based filter shows the filter stability. The pole zero plot
is generally used to analyze the stability of the system, in this case the designed filter.
The poles and the zeros of the RNS filter designed are almost same as the general FIR
filter. The stability of the designed filter is not affected at all by the proposed design
methodology as suggested by the figure 4.5 and 4.6.
Figure 4.5: Performance Analysis between Traditional and RNS Based FIR Filter Implementation Techniques
26
4.5. SUMMARY
The performance analysis is done with respect to the generic traditional FIR filter
[33; 34; 35]. However the phase response shows deviation as because there are few data
as well as coefficients those are truncated to their higher value during the homomorphic
mapping and RNS conversion. This creates an error which propagates through the filter
to its output. The detailed report of the comparison drawn between the designed filter
after simulation and the traditional FIR filter is shown in figure 4.5 .
4.5 Summary
The magnitude response and the pole zero plot analyses the filter designed with the coefficients obtained from hamming window technique and implemented on residue arithmetic
with a homomorphic mapping. The results are satisfactory with the moduli set proposed.
The RNS based filter design can produce very high speed FIR filter. A FIR filter
based on the residue number system is exhibited and the filter performance is analysed
extensively. The new proposed lowpass FIR filter operates over a high range and the
homomorphic mapping provide the incorporation of fractional part of the signal with a
calculated precision. Hereby the bit efficiency of the residue number based FIR filter is
concluded to have a satisfactory performance. However, the major problem with the RNS
based filter is the detection of overflow. If the overflow detection can be done without
any use of lookup tables unlike many methods cited in literature viz. [36; 37], then these
filters can be optimized to be highly accurate filter which will inherit the property of high
speed and low power consumption.
27
4.5. SUMMARY
(a) Impulse Response
(b) Step Responce
(c) Pole-Zero Plot
Figure 4.6: Performance and Stability of the RNS based FIR Filter
28
CHAPTER 5
Application : RNS Based Image Encryption
5.1 Overview to Image Encryption
Since image is an important and trite form of information, secured and authentic transmission is crucial and requisite. Many applications are based on confidential and authentic transmission over unsecured and noisy channel. Thus protection of data, in this
case image, from foreign attacks and external channel noise is of prime interest. The
basic requirement of encryption is to assure confidentiality, integrity and availability
[38]. There are many security mechanism like encipherment, authentic exchange, traffic
padding, notarization, digital signature and many more. However, single mechanism for
encryption of data is not optimal. Multiple mechanisms are applied together on a data to
encrypt is before transmission. An authentic transmission may be achieved by combining
, digital signature and authentication exchange [38]. However, this increases computational complexity and are very difficult to encrypt data in real-time online transmission.
Hence cryptography, a part of encipherment, was explored to achieve encoding data with
a key where the extraction of this key by unauthorised user becomes practically impossi-
29
5.2. INTRODUCTION
ble. If a key take substantially a large amount of time to decipher a key for the extraction
of the data, the data is said to be well encrypted. Numerous techniques are developed
and present in literature which claims better encryption of any image. However very few
viewed the transmission channel noise as a threat to loss of data. Robustness of any encryption technique to channel noise is also of prime importance. In this chapter a robust
encryption technique is proposed.
5.2 Introduction
The major security goals for which image, needs to be encoded and encrypted before
transmission through a channel are confidentiality, integrity and availability[38]. These
are serious issues with respect to data security and transmission and has been challenging for researchers to develop secured encoding techniques. There have been multiple
proposed techniques in literature [39; 40; 41; 42; 43; 44; 45]. However there is no specific
absolutely reliable technique. In this chapter a novel, secured and low computational
image coding scheme based on residue arithmetic is adduced. Basics of residue arithmetic is explained in chapter 2. The concept of ’break and process is again expended
to operate on the data in parallel such that the computational load is minimum as well
as shared. Moreover the number system works effectively with integer arithmetic. The
proposed technique has very low. The robustness and performance of this scheme under
adverse channel conditions is exemplified further. This technique was found to be robust
and a detailed analysis is given further in results. Standard encryption techniques like
Blowfish encryption techniques are compared to the proposed technique for performance
analysis.
30
5.3. PROPOSED ALGORITHM
Input Image
n, m
key
break image
into n x m
blocks
z=mxnx8
bits
Zigzag Scan of blocks and
form them into a word
Generate moduli
R
set P of Z bits and
calculate range R
Scale
q pixel to x
Convert into RNS numbers
and combine to a word
Word broken into
m x n blocks
Map blocks to the image
KEY
n
m
p1
p2
.....
Output Encrypted
Image
px
Figure 5.1: Flowchart of RNS Based Encryption Technique
5.3 Proposed Algorithm
5.3.1 Encryption
As in the flowchart described in figure 5.1, we find there are two parts of the key used in
this encryption technique. The first part is used to break the image into (m x n) blocks.
The second part of the key holds Z bit moduli set. The algorithm for the same is as
follows:
Step I: Input image and block size m and n.
31
5.3. PROPOSED ALGORITHM
Step II: Break image into m x n blocks and Zigzag Scan the intermediate blocks.
Step III: Z = m x n x 8 bits. Generate a moduli set P, where P = { p 1 , p 2 . . . p k } corresponding to Z bits.
Step IV: Convert the intermediate zigzag scanned block into one word and scale to avoid
overflow.
2z
·X
q=
R
(5.1)
Step V: Use the set P to convert the word into RNS.
Step VI: Regroup the RNS number to form a word and further break them into m x n
blocks.
Step VII: Map the encrypted block to image.
5.3.2 Decryption
As in the flowchart described in figure 5.2, we find there are two parts of the key used in
this decryption technique as in encryption. The first part is used to break the image into
(m x n) blocks. The second part of the key holds Z bit moduli set. The algorithm for the
decryption of the encrypted image is as follows:
Step I: Input encrypted image and key.
Step II: The first two numbers of the key represents m and n block size. The next part
represents the moduli set P.
Step III: Break encrypted image into m x n blocks and the intermediate blocks are converted to one word.
Step IV: Bit length pf every p i is calculated and the word is then broken according to
the bit length into separate numbers.
32
5.3. PROPOSED ALGORITHM
KEY
n
m
p1
p2
.....
Input Encrypted
Image
px
Decode
the key
First 2
digits?
break image in n x
m blocks and
combine to a word
Yes
Bit Len of pi.
Word broken
accordingly.
No
Moduli set
P
RNS to Decimal
conversion
Scale
x pixel to q
Map the block
to the image
Word broken into
m x n blocks
Output Image
Figure 5.2: Flowchart of RNS Based Decryption Technique
Step V: Convert these RNS numbers to decimal or binary and scale accordingly.
2z
·q
X=
R
Step VI: Decimal or binary number is further broken into m x n blocks.
Step VII: Zigzag inverse the blocks and map them to form the image.
33
(5.2)
5.4. STANDARD ALGORITHM
5.4 Standard Algorithm
Blowfish is a symmetric block cipher designed by Bruce Schneier in 1993 and is commercially used in large number of encryption products. It is also used in internet widely for
secured data transmission. It designed based on DES and Fiestel Network structure. It
has 64 bit block size and key length varying from 1 to 448 bits. Blowfish is open source
and free license [46]. Twofish is recently discovered encryption algorithm and is preferred over Blowfish in many ways. It is also a symmetric key block cipher with block
size of 128 bits and key length 128, 192 or 256 bits. It is derived from Blowfish and
hence follows Fiestel Network structure [45; 47]. Threefish is another very new concept
of encipherment published in 2008. In this algorithm, key size is always same as block
size. Block sizes are 256, 512 or 1024; hence the key sizes are selected respectively. 266
and 512 block sizes uses 72 rounds. However, 1024 block size uses 80 rounds. Threefish
algorithm takes 6.1 cycles per byte on core 2 processor.
5.5 Results
The proposed algorithm was simulated in Matlab in Core 2 duo 2.33 Ghz processor. Lena
image of 256x256 as in figure 5.3(a) was used for simulation. The encrypted image is
shown in figure 5.3(b).
The encrypted image figure 5.3(b) in was transmitted through an AWGN channel and
the output image was analysed for error. The bit error plot is given in figure 5.4. The
transmitted image at 8db, 10db, 12db and 15db is shown in figure 5.5. These noisy images
are denoised by simple median filter and the output is shown in figure 5.6.
5.5.1 PSNR Analysis
Peak signal-to-noise ratio (PSNR) has traditionally been used in analog systems as a
trusty quality metric. Due to its low complexity, PSNR is used widely for analysis of
quality image and video processing metric for evaluating algorithms for de-noising, encryption and compression methods and is considered to be a reference benchmark for
34
5.5. RESULTS
(a) Original image
(b) RNS Encrypted Image
Figure 5.3: RNS Based Encryption
developing perceptual image or video procesing [48]. However, correlation of PSNR subjective to equality have been shown to perform below average [49].
MSE is defined as follows [50]:
MSE =
−1 N
−1 2
1 M
O im g (i, j) − A im g (i, j)
MN i=0 j=0
(5.3)
PSNR is hereby defined as [50]:
PSNR = 10 log 10
MaxV al 2
MSE
(5.4)
where MaxVal is maximum pixel value possible in the image. Table 5.1 gives the PNSR
performance for the proposed algorithm when applied on the image in figure 5.3(a) and
AWGN channel SNR from 1 to 20db.
35
5.5. RESULTS
Plot of BER for SNR 1−20 db in AWGN Channel
0
10
−1
BER −−−−>
10
−2
10
−3
10
−4
10
−5
10
0
2
4
6
8
10
12
14
16
18
SNR (in db) −−−−>
Figure 5.4: BER plot of RNS based encryption technique
5.5.2 Median Filter
The median filter is a non-linear digital filter. It is just like a mean filter. It is a spatial
filter and is mostly used to reduce noise in image. Median filter looks at the surrounding
neighbours to decide whether or not its is representative of its surroundings. The median
of those values are calculated and then replaced. Median filter is also a smoothing technique hence the sharpness of the output image is marginally lost. The received images
are denoised by passing through a median filter. Hence the denoised images are seemed
to lose the sharpness as evident in figure 5.6
5.5.3 Comparision with Blowfish Encryption
As mentioned in section 5.4, Blowfish is an encryption technique and is widely used
commercially. Thus the proposed technique is compared to it with respect to
• Bit Error Rate
• Encryption time
• Correlation and Entropy
36
5.5. RESULTS
(a) Recieved Image at 8db SNR
(b) Recieved Image at 10db SNR
(c) Recieved Image at 12db SNR
(d) Recieved Image at 15db SNR
Figure 5.5: RNS Based Encryption
5.5.4 Bit Error Rate
The figure 5.7 shows the BER plot of two encryption technique. The plot of Blowfish is
better than the proposed technique. However both starts to be noise immune from 18db.
5.5.5 Encryption Time
The encryption time for the proposed algorithm is less then that of Blowfish encryption
as depicted in table 5.2.
37
5.5. RESULTS
MSE
SNR
PSNR
8.524 × 10 3
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
8.8242
8.9268
9.0649
9.2745
9.4465
9.9002
10.4117
11.0685
12.0293
13.1042
14.8547
16.6835
19.2325
22.2243
26.0141
30.9006
37.4252
47.5283
Inf
Inf
Table 5.1: PSNR Analysis
5.5.6 Correlation and Entropy
Correlation is defined as:
m n (I mn − Ī)(Jmn − J̄)
r= ( m n (I mn − Ī)2 )( m n (Jmn − J̄)2 )
(5.5)
where I and J are two images and I¯ and J̄ are mean of those images respectively.
Entropy is again defined as [51]
En = −
M
−1
P k log 2 (P k )
i=0
38
(5.6)
5.6. SUMMARY
(a) Denoised Image at 8db SNR
(b) Denoised Image at 10db SNR
(c) Denoised Image at 12db SNR
(d) Denoised Image at 15db SNR
Figure 5.6: Denoised Images Received At Various SNR
The encrypted images in both techniques are correlated to the original image to calculate pixel to pixel relation. The correlation is found for the proposed algorithm is less
then that of Blowfish encryption as depicted in table 5.3.
5.6 Summary
Thus we find that the proposed algorithm of RNS based image encryption is not as robust
as Blowfish, however the fact that is of interest is the encryption time. The encryption
time is appreciably less compared to Blowfish. Further research can be done to improve
39
5.6. SUMMARY
Plot of BER for SNR 1−20 db in AWGN Channel
0
10
RNS Based Encryption
Blowfish Encryption
−1
BER −−−−>
10
−2
10
−3
10
−4
10
−5
10
0
2
4
6
8
10
12
14
16
18
20
SNR (in db) −−−−>
Figure 5.7: BER Plot of RNS Based Encryption and Blowfish Ecryption
Tr. No.
1
2
3
4
5
RNS Encryption (in sec.)
0.14314
0.14269
0.14363
0.14632
0.14364
Blowfish (in sec.)
0.18736
0.18964
0.18767
0.18811
0.18722
Avg.
0.14389
0.1880
Table 5.2: Encryption time
the quality of the output image.
40
5.6. SUMMARY
Perf. Factors
RNS Encryption
Blowfish
Correlation
Entropy
.0066
7.9874
0.0089
7.9971
Table 5.3: Correlation and Entropy Calculation
41
CHAPTER 6
Application : RNS Based PN Sequence
Before beginning with spread spectrum system, we focus on to concept of spectrum. Every system that modulates or transmits a signal has two basic cardinals, namely the
frequency at which the signal is centred and bandwidth of the signal when modulated.
Spectrum gives the range of a signal; in communication and signal processing, the range
of the signal is spoken in terms of frequencies covered by the signal. Thus by spectrum of
a signal we mean frequency spectrum. The difference between the maximum frequency
and minimum frequency of the signal in view is called bandwidth of the signal. As mentioned in chapter 4, any signal in time domain can also be presented in frequency domain
and vice versa by various transform techniques. In spread spectrum we therefore just
spread the spectrum i.e. a signal covering one or narrow band of frequencies are spread
over a large band of frequencies, quite large in fact than frequency required to transmit
signal consummately, hence converting a small bandwidth to a very large bandwidth.
42
6.1. INTRODUCTION TO SPREAD SPECTRUM COMMUNICATION
6.1 Introduction to Spread Spectrum Communication
The basis of spread spectrum technology is expressed Shannon in terms of channel capacity:
C = ω log 2
S
1+
N
(6.1)
where
C is capacity in bps
ω is bandwidth in Hz
S is signal power
N is noise power
Equation 6.1. limns the competency of a channel to transmit information devoid of
any error compared to SNR in the channel and bandwidth used to transmit the information [52]. The bit duration is T b as in figure 6.1(a). The bit duration of the PN sequence is
T c as in fig. 6.1(b) and is called as chip period. Now when the bit is multiplied to the PN
sequence, the output is in fig. 6.1(c) which also has a bit duration of T c . Since T b >> T c ,
the frequency of the transmitted bit becomes very large compared to the signal. This is
called Direct Sequence Spread Spectrum (DS-SS). Frequency hopping is also a concept
widely used in spread spectrum communication. However this chapter only deals with
DS-SS as frequency hopping is out of scope.
DS-SS is primarily used in CDMA for multiple access. The structure of the spread
DS-CDMA is portrayed in figure 6.2. This is similar to MA-ACSK scheme except the fact
that the spreading sequences used in the system are vectors generated based on residue
arithmetic.The transmitter generates N no. of RNS based orthogonal PN sequences for
N users. The details of the proposed RNS based PN sequence generator and comparison
to standard DS-CDMA techniques are chalked out in the rest of the chapter.
6.2 Proposed PN Sequence Algorithm
The proposed PN sequence is generated based on residue arithmetic. There is a key input
to the generator along with the spread factor β. The figure 6.3 shows the block diagram
43
6.2. PROPOSED PN SEQUENCE ALGORITHM
(a) Bit to Transmit
(b) PN Sequence
(c) Transmitted Symbol
Figure 6.1: Direct Sequence Spread Spectrum Bit Pattern
of the PN sequence generated. As depicted in the chapter 3, moduli set are selected either by consecutive method or exponential method. The input k in figure also known as
spread factor β is responsible for the no. of moduli to be generated. The bits of moduli
set should be same as that of k. The keys in separate phase are selected such that the orthogonality of the output vector holds. The generated numbers in RNS are concatenated
and converted into binary sequence of 1 and 0. This sequence is passed through the NRZ
encoder to make a PN sequence of -1 and 1. This RNS based PN sequence generator is
then used in DS-CDMA.
44
6.3. STANDARD PN SEQUENCES
b1
R1(x)
Correlator
X
q1'
b1'
b2
R2(x)
+
X
q2'
Decision
Circuit
b2'
n(t)
bN
RN(x)
Correlator
+
X
Correlator
qN'
bN'
Figure 6.2: Direct Sequence CDMA Technique
k
Moduli
Selection
Dec to RNS
Converter
Selection
of keys
Keys
RNS to Bin
Converter
NRZ
Encoder
PN
Sequence
Figure 6.3: RNS Based PN Sequence Generator
6.3 Standard PN Sequences
In this chapter, performance of the RNS based PN sequence is compared to Gold Code sequence and Kasami Code sequence. Gold Sequence was proposed by Robert Gold. These
are constructed by EXOR-ing two m-sequences of the same length with each other. Thus,
for a Gold sequence of length m = 2l − 1, one uses two LFSR, each of length m = 2 l − 1.
Choosing LFSRs appropriately, Gold sequences give better cross-correlation properties
than maximum length LSFR sequences [53; 54]. Kasami sequences are binary sequences
of length 2 N where N is an even integer. Kasami sequences have good cross-correlation
values and approaches close the Welch lower bound. Kasami sequences commonly have
two classes, namely the small set and the large set. The process of generating a Kasami
sequence starts by generating a maximum length sequence x(n), where n = 1, 2 . . .2 N − 1.
Maximum length sequences are periodic sequences so a(n) is repeated periodically for
45
6.4. RESULTS
N larger than 2 N − 1. Henceforth another sequence x(n) = y(q.n) is generated where
N
q = 2 2 + 1. Kasami sequences are formed by adding x(n) and a time shifted version of
y(n)modulo two. All sequences generated by different time shifts of y(n) plus the x(n) and
N
y(n) sequence constitute the Kasami sequence. This set has 2 2 different sequences.
6.4 Results
Pn Seq
P N1
P N2
P N3
P N4
P N5
P N6
P N1
P N2
P N3
P N4
P N5
P N6
1
0.0691
0.0020
0.0796
0
0.1463
0.0691
1
0.0544
-0.0866
0.1411
0.0713
0.0020
0.0544
1
0.0023
-0.0157
-0.0577
0.0796
-0.0866
0.0023
1
0.1459
-0.2050
0
0.1411
-0.0157
0.1459
1
-0.0314
0.1463
0.0713
-0.0577
-0.2050
-0.0314
1
Table 6.1: Correlation Matrix for PN sequence with β = 128
The table 6.1 represents correlation matrix of six RNS based PN sequences generated
from key matrix Q=[1536958 1536965 1235688 4321531 2315312 2447683]. The spread
factor β = 128 and hence the generated moduli set P=[255 254 253 251 247 241 239 233
229 227 223 217 211 199 197 193].
The figure 6.5 shows the BER performance of the proposed sequence over a AWGN
channel.
6.5 Performance Comparison
The figure 6.6 shows the comparison between the performance of Gold sequence, Kasami
sequence and RNS based PN sequence over a AWGN channel based on BER. The plots
shows the nature of the PN sequences with increase in user. As the user increases the
proposed PN sequence is not affected as much as Gold and Kasami Sequence. All the
simulations are done for spread factor β = 128.
46
6.6. SUMMARY
1
CorMat(1:10,1)
0.8
CorMat(1:10,2)
CorMat(1:10,3)
0.6
Correlation
CorMat(1:10,4)
0.4
CorMat(1:10,5)
CorMat(1:10,6)
0.2
CorMat(1:10,7)
0
CorMat(1:10,8)
CorMat(1:10,9)
−0.2
CorMat(1:10,10)
−0.4
1
2
3
4
5
6
7
8
9
10
PN Sequences
Figure 6.4: Correlation Matrix for 10 sequence generated by RNS based PN generator
6.6 Summary
The proposed PN sequence based on RNS has a low computational complexity and has
a very high dynamic key range. Hence, the secured and authentic transmission is possible. The performance based on BER provides adequate data about the robustness of the
technique.
47
6.6. SUMMARY
Plot of BER for SNR 1−20 db for 2User
0
10
−1
10
BER −−−−>
−2
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
Eb/No −−−−−−−−−−−>
(a) 2 User
Plot of BER for SNR 1−20 db for 15User
−1
10
BER −−−−>
−2
10
−3
10
−4
10
0
2
4
6
8
10
12
14
16
18
20
Eb/No −−−−−−−−−−−>
(b) 15 User
Figure 6.5: BER plot for RNS based PN sequence for 2 and 15 User
48
6.6. SUMMARY
Plot of BER for SNR 1−20 db for 5User
−1
10
PNS Based PN Sequence
Gold PN Sequence
Kasami PN Sequence
−2
BER −−−−>
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
14
Eb/No −−−−−−−−−−−>
(a) 5 User
Plot of BER for SNR 1−20 db for 10User
−1
10
RNS Based PN Sequence
Gold Sequence
Kasami Sequence
−2
BER −−−−>
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8
10
12
14
16
18
20
Eb/No −−−−−−−−−−−>
(b) 10 User
Figure 6.6: BER plot for Gold, Kasami and RNS based PN sequence for 5 and 10 Users
49
CHAPTER 7
Conclusion
7.1 Conclusion
In this thesis, focus was on exploring the potential of Residue Arithmetic in the field of
Communication and Signal Processing. The most important aspect of Residue Number
System, ’break and process’, was widely used in this dissertation. Apart from that, large
dynamic range and the non-linearity in the number system was put to its use in generating PN sequence and image encryption. This chapter further describes summary of the
thesis, work done, contributions and scope of future work.
Major contribution was generating moduli set by consecutive and exponential method.
The approach made mor with bit efficiency of a particular application be it a digital FIR
filter, image encryption or PN sequence based on utility factor.
The introduction of homomorphic mapping for incorporation of fractional numbers
with a precision and by large avoiding overflow was another contribution to literature.
Further a design approach for RNS based FIR filter was discussed and the performance
analysis shows credibility of the system designed. The introduction of RNS on image en-
50
7.2. SCOPE FOR FUTURE WORK
cryption was not new. However the idea of grouping of image pixels and combine them
together finally to convert them to RNS and represent the same pixels was a novel idea.
The application of RNS to develop a PN sequence and generate the orthogonal sequence
such that the performance is good, was another major contribution. Sequence generated by this technique, produces good performance as it was shown in the performance
analysis of chapter 6.
7.2 Scope for Future Work
• VLSI implementation of the propose RNS based FIR filter architecture
• Developing results obtained image encryption such that the performance level is
better than existing algorithm keep the encryption time constant if not improved.
• The performance of the proposed PN sequence in fading channel can be analysed
and necessary amendments can be done to improve performance.
• RNS can be introduced in other applications of Communication and Signal Processing like adaptive filtering, neural networks etc.
• RISC processors based on RNS can be developed as lots of research is going on
around the globe.
51
Appendix A: Image Encryption Simulation Parameters
The results of the simulation in section 5.5 shows the performance of the RNS based novel
encryption technique and a performance analysis with Blowfish encryption technique.
The simulation uses:
Parameters
Block Size for Zigzag Scan
Block Size for Encryption
Key Length
Moduli set (P)
Values
4
4
32
Consecutive moduli set selection
52
Appendix B: Spread Spectrum
The RNS based PN sequence generator used to limn the DS-SS communication system
in section 6.4 for generation of several PN sequences uses a PIN that is used by the user.
Hence the simulation results in this section shows results for 15 different users at the
maximum as in figures 6.5 and 6.6. Hence 15 different user uses 15 different PIN (Personal Identification Number). The simulation uses the following PIN for generation of
PN sequences. The PIN can be in range of R as in equation 3.8.
P=[255 254 253 251 247 241 239 233 229 227 223 217 211 199 197 193]
The Gold and Kasami Sequence is generated by polynomial as below (as per Matlab
simulation):
poly1 = [10 6 5 0] and
poly2 = [10 4 2 0]
53
User
User 1
User 2
User 3
User 4
User 5
User 6
User 7
User 8
User 9
User 10
User 11
User 12
User 13
User 14
User 15
PIN
1536958
1536965
1235688
4321531
2315312
2447683
1010157
3051173
2002007
1003765
1065895
2056586
3256581
2325652
0252145
54
Bibliography
[1] Residue number systems: algorithms and architectures.
Kluwer Academic Pub-
lisher, 2002. 2, 3, 7, 9, 10, 11
[2] T. Stouraitis and V. Paliouras, “Considering the alternatives in low-power design,”
in IEEE Electron Devices Society IEEE Lasers and Electro-Optics Society. 3
[3] C.-L. Wang, “New bit-serial vlsi implementation of rns fir digital filters,” in IEEE
Transactions on Circuits and Systems II: Analog and Digital Signal Processing. 3
[4] A. Skavantzos and Y. Wang, “The applications of the new chinese remainder theorems for three moduli sets,” in IEEE Pacific Rim Conference on Communications,
Computers and Signal Processing, 1999. 3
[5] Y. Wang, “New chinese remainder theorems,” in Conference Record of the ThirtySecond Asilomar Conference on Signals, Systems and Computers, 1998. 3
[6] P. . M.-B. U. . T. F. . G. A. . L. A. Ramirez, J. ; Fernandez, “Index-based rns dwt architectures for custom ic designs,” in IEEE Workshop on Signal Processing Systems,
2001. 3
55
BIBLIOGRAPHY
[7] ——, “Implementation of canonical and retimed rns architecture for orthogonal 1-d
dwt over fpl devices,” in Conference Record of the 34th Asilomar Conf. on SSC. 3
[8] A. A. Author, B. B. Author, and C. Author, “Title of article,” Title of Journal, vol. 10,
no. 2, pp. 49–53, 2005. 3
[9] G. A. B. U. T.-F. F. P. L. A. Rama rez, J., “Design of rns-based distributed arithmetic
dwt filterbanks,” vol. 2, 2001, pp. 1193–1196, cited By (since 1996) 2. 3
[10] Y. Liu and E. M. Lai, “Design and implementation of an rns-based 2-d dwt
processor,” IEEE Transactions on Consumer Electronics, vol. 50, no. 1, pp. 376–385,
2004, cited By (since 1996): 8. [Online]. Available: www.scopus.com 3
[11] J. . Bajard and L. Imbert, “A full rns implementation of rsa,” IEEE Transactions on
Computers, vol. 53, no. 6, pp. 769–774, 2004, cited By (since 1996): 43. [Online].
Available: www.scopus.com 3
[12] D. M. Schinianakis, A. P. Kakarountas, and T. Stouraitis, “A new approach to elliptic
curve cryptography: An rns architecture,” in Proceedings of the Mediterranean
Electro-technical Conference - MELECON, vol. 2006, 2006, pp. 1241–1245, cited By
(since 1996): 1. [Online]. Available: www.scopus.com 3
[13] A. S. Madhukumar and F. Chin, “Residue number system-based multicarrier
cdma system for high-speed broadband wireless access,” IEEE Transactions on
Broadcasting, vol. 48, no. 1, pp. 46–52, 2002, cited By (since 1996): 5. [Online].
Available: www.scopus.com 3
[14] ——, “Design and performance of residue number system based multicarrier
cdma in frequency-selective rayleigh fading channels,” in Conference Record of the
Asilomar Conference on Signals, Systems and Computers, vol. 2, 2000, pp. 884–888.
[Online]. Available: www.scopus.com 3
[15] L. Yang and L. Hanzo, “Performance of residue number system based ds-cdma over
multipath fading channels using orthogonal sequences,” European Transactions on
56
BIBLIOGRAPHY
Telecommunications, vol. 9, no. 6, pp. 525–535, 1998, cited By (since 1996): 14.
[Online]. Available: www.scopus.com 3
[16] L.-L. Yang and L. Hanzo, “Residue number system based multiple code ds-cdma
systems,” IEEE VTS 50th Vehicular Technology Conference, VTC 1999-Fall, vol. 2,
pp. 1450–1454, 1999, cited By (since 1996): 3. [Online]. Available: www.scopus.com
3
[17] D. Zhu and B. Natarajan, “Residue number system arithmetic assisted coded frequencyhopped ofdma,” EURASIP Journal on Wireless Communications and Networking, vol. Article ID 263695, 2009. 3
[18] ——, “Residue number system arithmetic inspired hopping pilot pattern design for
cellular downlink ofdma,” in 2010 Wireless Telecommunications Symposium, WTS
2010, 2010. [Online]. Available: www.scopus.com 3
[19] C. T. Clarke and T. Srikanthan, “Residue arithmetic techniques for hardware
reduction in pseudo-random sequence correlators,” in Conference Record - Asilomar
Conference on Signals, Systems and Computers, vol. 2, 2004, pp. 1864–1867.
[Online]. Available: www.scopus.com 3
[20] M. F. Griffin and F. J. Taylor, “Residue number system reduced instruction set
computer (risc) concept,” in ICASSP, IEEE International Conference on Acoustics,
Speech and Signal Processing - Proceedings, vol. 4, 1989, pp. 2581–2584, cited By
(since 1996): 1. [Online]. Available: www.scopus.com 3
[21] H.T.Vergos, “A 200-mhz rns core,” European Conference on Circuit Theory and Design, pp. II/249–252, August 2001. 3
[22] C. H. Huang, D. G. Peterson, H. E. Rauch, J. W. Teague, and D. F. Fraser,
“Implementation of a fast digital processor using the residue number system.”
IEEE transactions on circuits and systems, vol. CAS-28, no. 1, pp. 32–38, 1981,
cited By (since 1996): 1. [Online]. Available: www.scopus.com 4
57
BIBLIOGRAPHY
[23] R. Chaves and L. Sousa, “Rdsp: a risc dsp based on residue number system,”
IEEE Proceedings. Euromicro Symposium on Digital Object Identifier, pp. 128 – 135,
September 2003. 4
[24] A. Skavantzos, “Efficient residue to weighted converter for a new residue number
system,” in Proceedings of the IEEE Great Lakes Symposium on VLSI, 1998, pp.
185–191, cited By (since 1996): 10. [Online]. Available: www.scopus.com 4
[25] J. Ramírez, A. García, S. López-Buedo, and A. Lloris, “Rns-enabled digital signal
processor design,” Electronics Letters, vol. 38, no. 6, pp. 266–268, 2002, cited By
(since 1996): 30. [Online]. Available: www.scopus.com 4
[26] A. Omondi and B. Premkumar, Residue Number System: Theory and Implementation. Imperial College Press, 2007, vol. 2. 7, 9, 10, 11, 15, 24
[27] N. Stamenkovit’c, “Digital fir filter architecture based on the residue number system,” ELEC. ENERG. vol. 22, no. 1, April , 125-140, vol. 22, no. 1, pp. 125–140, April
2009. 13, 18
[28] W. Wang, M. N. S. Swamy, and M. O. Ahmad, “Moduli selection in rns for efficient
vlsi implementation,” in Proceedings - IEEE International Symposium on Circuits
and Systems, vol. 4, 2003, pp. IV512–IV515, cited By (since 1996): 6. [Online].
Available: www.scopus.com 18, 24
[29] R. Conway and J. Nelson, “Improved rns fir filter architectures,” IEEE Transactions
on Circuits and Systems II: Express Briefs, vol. 51, no. 1, pp. 26–28, 2004, cited By
(since 1996): 24. [Online]. Available: www.scopus.com 18, 24
[30] K. A. Fuatai, Concepts of Mapping.
Springer, 2009, vol. DOI 10.1007/978-0-387-
89194-1. 19, 24
[31] A. M. Vikram Pasham and K. Chapman, Transposed for FIR Filter, v1.2 ed., Xilinx,
October 2001. 21
58
BIBLIOGRAPHY
[32] M. Bahrami and B. Sadeghiyan, “Efficient modulo 2n+1 multiplication schemes for
idea,” in Proceedings - IEEE International Symposium on Circuits and Systems,
vol. 4, 2000, pp. IV–653–IV–656, cited By (since 1996): 2. [Online]. Available:
www.scopus.com 24
[33] W. K. Jenkins and B. J. Leon, “Use of residue number systems in the design of finite
impulse response digital filters.” IEEE Trans Circuits Syst, vol. CAS-24, no. 4, pp.
191–201, 1977, cited By (since 1996): 37. [Online]. Available: www.scopus.com 24,
27
[34] F. Pourbigharaz and H. M. Yassine, “A signed-digit architecture for residue to
binary transformation,” IEEE Transactions on Computers, vol. 46, no. 10, pp.
1146–1150, 1997, cited By (since 1996): 16. [Online]. Available: www.scopus.com
24, 27
[35] T. K. Shahana, R. K. James, B. R. Jose, K. Poulose Jacob, and S. Sasi, “Performance
analysis of fir digital filter design: Rns versus traditional,” in ISCIT 2007 2007 International Symposium on Communications and Information Technologies
Proceedings, 2007, pp. 1–5. [Online]. Available: www.scopus.com 27
[36] M. Askarzadeh, M. Hosseinzadeh, and K. Navi, “A new approach to overflow
detection in moduli set 2n-3, 2 n-1, 2n+1, 2n+3,” in 2009 International Conference
on Computer and Electrical Engineering, ICCEE 2009, vol. 1, 2009, pp. 439–442.
[Online]. Available: www.scopus.com 27
[37] R. C. Debnath, D. A. Pucknell, and D. A. Pucknell, “On multiplicative overflow
detection in residue number system.” Electronics Letters, vol. 14, no. 5, pp. 129–130,
1978. [Online]. Available: www.scopus.com 27
[38] B. A. Forouzan, Cryptography and Network Security, special indian ed.
Tata
McGraw-Hill, 2008. 29, 30
[39] N. Bourbakis and C. Alexopoulos, “Picture data encryption using scan patterns,”
59
BIBLIOGRAPHY
Pattern Recognition, vol. 25, no. 6, pp. 567–581, 1992, cited By (since 1996): 109.
[Online]. Available: www.scopus.com 30
[40] H. K. Chang and J. Liu, “A linear quadtree compression scheme for image
encryption,” Signal Processing: Image Communication, vol. 10, no. 4, pp. 279–290,
1997, cited By (since 1996): 42. [Online]. Available: www.scopus.com 30
[41] C. Chang, M. Hwang, and T. Chen, “A new encryption algorithm for image
cryptosystems,” Journal of Systems and Software, vol. 58, no. 2, pp. 83–91, 2001,
cited By (since 1996): 70. [Online]. Available: www.scopus.com 30
[42] H. Cheng, “Partial encryption of compressed images and videos,” IEEE Transactions
on Signal Processing, vol. 48, no. 8, pp. 2439–2451, 2000, cited By (since 1996): 179.
[Online]. Available: www.scopus.com 30
[43] H. M. A. K. D. S. A. Elminaam and M. M. Hadhoud, “Evaluating the performance
of symmetric encryption algorithms,” International Journal of Network Security,
vol. 10, no. 3, pp. 213–219, 2010. 30
[44] N. E. Fishawy and O. M. A. Zaid, “Quality of encryption measurement of bitmap
images with rc6, mrc6, and rijndael block cipher algorithms,” International Journal
of Network Security, vol. 5, no. 3, pp. 241–251, 2007. 30
[45] S. Moriai and Y. L. Yin, “Cryptanalysis of twofish(ii),” Technical Report of IEICE,
2000. 30, 34
[46] B. Schneier, “Description of a new variable-length key, 64-bit block cipher (blowfish),” Springer LNCS, vol. 809, pp. 191–204, 1994. 34
[47] N. Ferguson, “Impossible differentials in twofish,” 1999. 34
[48] Q. Huynh-Thu and M. Ghanbari, “Scope of validity of psnr in image/video quality
assessment,” Electronics Letters, vol. 44, no. 13, pp. 800–801, 2008, cited By (since
1996): 31. [Online]. Available: www.scopus.com 35
60
BIBLIOGRAPHY
[49] B. Girod, What’s wrong with mean-squared error? in Digital images and human
vision. MIT Press Cambridge, 1993. 35
[50] D. Salomon, Data Compression The Complete Reference, 4th ed. Springer, 2007. 35
[51] R. W. Gonzalez, R.C., Digital Image Processing Using MATLAB, 2nd ed.,
A. Dworkin, Ed.
New Jersey, Prentice Hall, 2003. 38
[52] R. C. Dixon, Spread Spectrum System with Commercial Application, 3rd ed. WileyInterscience, 1994. 43
[53] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” IEEE Transactions on Information Theory, vol. 13, pp. 619–621, 1967. 45
[54] S. Haykin, Communication Systems, 4th ed.
61
John Wiley and Sons, 2004. 45
Publications
7.1 Journal
1. Pallab Maji and GS Rath," Design of Low Pass FIR Filter based on Residue Arithmetic", International Journal of Science and Technology (IJSAT),2011.(In Press)
7.2 Conference
1. Pallab Maji and G. S. Rath," A Novel Design Approach for Low Pass Finite Impulse
Response Filter Based on Residue Number System", IEEE 3rd ICECT, Kanyakumari, India, 2011.
2. Pallab Maji and G. S. Rath," Bit Efficient Finite Impulse Response Filter based on
Residue Arithmetic", ICSSA-11, Gujarat, India, 2011.
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