Acknowledgements
Abstract
Contents
Acronyms
List ofΒ Figures
List ofΒ Tables
1 Introduction
1.1 Problem Statement and Motivation
1.2 Structure of the Thesis
2 Elliptic Curve Cryptography
2.1 Cryptography for Flash Memory Controllers
2.2 Applications of Elliptic Curve Cryptographic (ECC) Systems
2.2.1 Elliptic Curves
2.2.2 Key Exchange
2.2.3 Digital Signatures
2.3 Elliptic Curve Point Multiplication
2.4 Elliptic Curve Geometry and Group Laws
2.4.1 Elliptic Curve Geometry
2.4.2 Group Laws for Prime Curves
2.5 Reducing the Number of Field Inversions for Elliptic Curves over Prime Fields
2.6 Discussion
3 Elliptic Curve Cryptography over Gaussian Integers
3.1 Gaussian Integer Rings and Fields
3.2 Point Multiplication over Gaussian Integers
3.2.1 Determining the Ο-adic Expansions
3.2.2 Elliptic Curve Point Multiplication for Complex Expansions
3.3 Resistance Against side Channel Attacks using Gaussian Integers
3.3.1 Improved Ο-adic Expansion Algorithm
3.3.2 Comparison with Existing Non-binary Expansions
3.4 Discussion
4 Montgomery Arithmetic over Gaussian Integers
4.1 Montgomery Arithmetic
4.2 Reduction over Gaussian Integers using the Absolute Value
4.3 Reduction over Gaussian integers using the Manhattan Weight
4.3.1 Montgomery reduction algorithm using the Manhattan weight
4.3.2 Reduction after addition (or subtraction)
4.4 Simplifying the Reduction based on the Manhattan Weight
4.5 Discussion
5 Architecture of the ECC Coprocessor for Gaussian Integers
5.1 Coprocessor Architecture for Gaussian Integers
5.1.1 Basic Concepts of the Proposed Design
5.1.2 Hardware Architecture
5.1.3 Instruction Set Architecture
5.1.4 Data Memory
5.1.5 Arithmetic Unit for Gaussian Integer Fields
5.2 ECC Coprocessor Architecture for Prime Fields
5.2.1 Preliminary Considerations and Modulo Reduction
5.2.2 Architecture
5.2.3 Instruction Set
5.2.4 Arithmetic Unit
5.3 Implementation Results
5.4 Discussion
6 Compact Architecture of the ECC Coprocessor for Binary Extension Fields
6.1 Group Laws and Projective Coordinates for Binary Extension Fields
6.1.1 Group Law for Binary Extension Curves
6.1.2 Projective Coordinates for Elliptic Curves over Binary Extension Fields
6.2 ECC Coprocessor Architecture
6.2.1 Instruction Set Architecture
6.2.2 Data Memory
6.2.3 Arithmetic Unit
6.3 Results and Discussion
7 The Parallel Dictionary LZW Algorithm for Flash Memory Controllers
7.1 Data Compression for Flash Memory Devices
7.1.1 Reducing the Write Amplification
7.1.2 Combining Data Compression and Error-correcting Codes
7.1.3 Suitable Data Compression Scheme
7.2 Parallel Dictionary LZW (PDLZW) Algorithm
7.3 Address Space Partitioning for the PDLZW
7.3.1 Data Model
7.3.2 Partitioning the PDLZW Address Space
7.4 Reducing the Memory Requirements of the PDLZW
7.4.1 Recursive PDLZW Algorithm
7.4.2 Basic Concept of the Word Partitioning Technique
7.4.3 Dimensioning the Layers
7.4.4 Dictionary Architecture
7.4.5 Implementation
7.5 Compression and Implementation Results
7.6 Discussion and Comparison with Other Data Compression Schemes
8 Conclusion
Bibliography