Residue Number Systems: Theory and Implementation
by Omondi, Amos R.; Premkumar, BenjaminFormat: Hardcover
Pub. Date: 2007-09-10
Publisher(s): World Scientific Pub Co Inc
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Summary
Residue number systems (RNSs) and arithmetic are useful for several reasons. First, a great deal of computing now takes place in embedded processors, such as those found in mobile devices, for which high speed and low-power consumption are critical; the absence of carry propagation facilitates the realization of high-speed, low-power arithmetic. Second, computer chips are now getting to be so dense that full testing will no longer be possible; so fault tolerance and the general area of computational integrity have become more important. RNSs are extremely good for applications such as digital signal processing, communications engineering, computer security (cryptography), image processing, speech processing, and transforms, all of which are extremely important in computing today.
Table of Contents
| Preface | p. vii |
| Introduction | p. 1 |
| Conventional number systems | p. 2 |
| Redundant signed-digit number systems | p. 5 |
| Residue number systems and arithmetic | p. 6 |
| Choice of moduli | p. 9 |
| Negative numbers | p. 10 |
| Basic arithmetic | p. 11 |
| Conversion | p. 13 |
| Base extension | p. 14 |
| Alternative encodings | p. 14 |
| Using residue number systems | p. 15 |
| Summary | p. 17 |
| References | p. 18 |
| Mathematical fundamentals | p. 21 |
| Properties of congruences | p. 22 |
| Basic number representation | p. 24 |
| Algebra of residues | p. 27 |
| Chinese Remainder Theorem | p. 39 |
| Complex residue-number systems | p. 40 |
| Redundant residue number systems | p. 42 |
| The Core Function | p. 44 |
| Summary | p. 47 |
| References | p. 47 |
| Forward conversion | p. 49 |
| Special moduli-sets | p. 50 |
| {2[superscript n-1], 2[superscript n], 2[superscript n+1]} moduli-sets | p. 52 |
| Extended special moduli-sets | p. 56 |
| Arbitrary moduli-sets: look-up tables | p. 58 |
| Serial/sequential conversion | p. 59 |
| Sequential/parallel conversion: arbitrary partitioning | p. 62 |
| Sequential/parallel conversion: periodic partitioning | p. 65 |
| Arbitrary moduli-sets: combinational logic | p. 68 |
| Modular exponentiation | p. 68 |
| Modular exponentiation with periodicity | p. 78 |
| Summary | p. 80 |
| References | p. 80 |
| Addition | p. 83 |
| Conventional adders | p. 84 |
| Ripple adder | p. 85 |
| Carry-skip adder | p. 88 |
| Carry-lookahead adders | p. 91 |
| Conditional-sum adder | p. 97 |
| Parallel-prefix adders | p. 101 |
| Carry-select adder | p. 108 |
| Residue addition: arbitrary modulus | p. 111 |
| Addition modulo 2[superscript n] - 1 | p. 119 |
| Ripple adder | p. 122 |
| Carry-lookahead adder | p. 123 |
| Parallel-prefix adder | p. 127 |
| Addition modulo 2[superscript n] + 1 | p. 130 |
| Diminished-one addition | p. 130 |
| Direct addition | p. 131 |
| Summary | p. 134 |
| References | p. 134 |
| Multiplication | p. 137 |
| Conventional multiplication | p. 138 |
| Basic binary multiplication | p. 139 |
| High-radix multiplication | p. 142 |
| Conventional division | p. 151 |
| Subtractive division | p. 151 |
| Multiplicative division | p. 160 |
| Modular multiplication: arbitrary modulus | p. 162 |
| Table lookup | p. 162 |
| Modular reduction of partial products | p. 165 |
| Product partitioning | p. 169 |
| Multiplication by reciprocal of modulus | p. 173 |
| Subtractive division | p. 176 |
| Modular multiplication: modulus 2[superscript n] - 1 | p. 177 |
| Modular multiplication: modulus 2[superscript n] + 1 | p. 185 |
| Summary | p. 191 |
| References | p. 191 |
| Comparison, overflow-detection, sign-determination, scaling, and division | p. 193 |
| Comparison | p. 194 |
| Sum-of-quotients technique | p. 195 |
| Core Function and parity | p. 197 |
| Scaling | p. 198 |
| Division | p. 201 |
| Subtractive division | p. 201 |
| Multiplicative division | p. 207 |
| Summary | p. 210 |
| References | p. 210 |
| Reverse conversion | p. 213 |
| Chinese Remainder Theorem | p. 213 |
| Pseudo-SRT implementation | p. 220 |
| Base-extension implementation | p. 223 |
| Mixed-radix number systems and conversion | p. 227 |
| The Core Function | p. 234 |
| Reverse converters for [characters not reproducible]2n - 1, 2n,2n + 1[characters not reproducible] moduli-sets | p. 237 |
| High-radix conversion | p. 248 |
| Summary | p. 251 |
| References | p. 251 |
| Applications | p. 255 |
| Digital signal processing | p. 256 |
| Digital filters | p. 257 |
| Sum-of-products evaluation | p. 264 |
| Discrete Fourier Transform | p. 272 |
| RNS implementation of the DFT | p. 275 |
| Fault-tolerance | p. 278 |
| Communications | p. 286 |
| Summary | p. 288 |
| References | p. 289 |
| Index | p. 293 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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