"Security issues will play an important role in the majority of communication and computer networks of the future. As the Internet becomes more and more accessible to the public, security measures will have to be strengthened. Elliptic curve cryptosystems allow for shorter operand lengths than other public-key schemes based on the discrete logarithm in finite fields and the integer factorisation problem and are thus attractive for many applications. This thesis describes an implementation of a crypto engine based on elliptic curves. The underlying algebraic structure are composite Galois fields GF((2n)m) in a standard base representation. As a major new feature, the system is developed for a reconfigurable platform based on Field Programmable Gate Arrays (FPGAs). FPGAs combine the flexibility of software solutions with the security of traditional hardware implementations. In particular, it is possible to easily change all algorithm parameters such as curve coefficients, field order, or field representation. The thesis deals with the design and implementation of elliptic curve point multiplication architectures. The architectures are described in VHDL and mapped to Xilinx FPGA devices. Architectures over Galois fields of different order and representation were implemented and compared. Area and timing measurements are provided for all architectures. It is shown that a full point multiplication on elliptic curves or real-world size can be implemented on commercially available FPGAs."

Cheung Ka Luen. / "December 2003." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 95-101). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Leung Pak Keung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 69-70). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Introduction to Elliptic Curve Crypto-processor --- p.1 / Chapter 1.2 --- Aims --- p.2 / Chapter 1.3 --- Contributions --- p.2 / Chapter 1.4 --- Thesis Outline --- p.3 / Chapter Chapter 2 --- Cryptography --- p.5 / Chapter 2.1 --- Introduction to Cryptography --- p.5 / Chapter 2.2 --- Public-key Cryptosystems --- p.6 / Chapter 2.3 --- Secret-key Cryptosystems --- p.9 / Chapter 2.4 --- Discrete Logarithm Problem --- p.9 / Chapter 2.5 --- Comparison between ECC and RSA --- p.10 / Chapter 2.6 --- Summary --- p.13 / Chapter Chapter 3 --- Mathematical Background in Number Systems --- p.14 / Chapter 3.1 --- Introduction to Number Systems --- p.14 / Chapter 3.2 --- "Groups, Rings and Fields" --- p.14 / Chapter 3.3 --- Finite Fields --- p.15 / Chapter 3.4 --- Modular Arithmetic --- p.16 / Chapter 3.5 --- Optimal Normal Basis --- p.16 / Chapter 3.5.1 --- What is a Normal Basis? --- p.17 / Chapter 3.5.2 --- Addition --- p.17 / Chapter 3.5.3 --- Squaring --- p.18 / Chapter 3.5.4 --- Multiplication --- p.19 / Chapter 3.5.5 --- Optimal Normal Basis --- p.19 / Chapter 3.5.6 --- Generation of the Lambda Matrix --- p.20 / Chapter 3.5.7 --- Inversion --- p.22 / Chapter 3.6 --- Summary --- p.24 / Chapter Chapter 4 --- Introduction to Elliptic Curve Mathematics --- p.26 / Chapter 4.1 --- Introduction --- p.26 / Chapter 4.2 --- Mathematical Background of Elliptic Curves --- p.26 / Chapter 4.3 --- Elliptic Curve over Real Number System --- p.27 / Chapter 4.3.1 --- Order of the Elliptic Curves --- p.28 / Chapter 4.3.2 --- Negation of Point P --- p.28 / Chapter 4.3.3 --- Point at Infinity --- p.28 / Chapter 4.3.4 --- Elliptic Curve Addition --- p.29 / Chapter 4.3.5 --- Elliptic Curve Doubling --- p.30 / Chapter 4.3.6 --- Equations of Curve Addition and Curve Doubling --- p.31 / Chapter 4.4 --- Elliptic Curve over Finite Fields Number System --- p.32 / Chapter 4.4.1 --- Elliptic Curve Operations in Optimal Normal Basis Number System --- p.32 / Chapter 4.4.2 --- Elliptic Curve Operations in Projective Coordinates --- p.33 / Chapter 4.4.3 --- Elliptic Curve Equations in Projective Coordinates --- p.34 / Chapter 4.5 --- Curve Multiplication --- p.36 / Chapter 4.6 --- Elliptic Curve Discrete Logarithm Problem --- p.37 / Chapter 4.7 --- Public-key Cryptography in Elliptic Curve Cryptosystem --- p.38 / Chapter 4.8 --- Diffie-Hellman Key Exchange in Elliptic Curve Cryptosystem --- p.38 / Chapter 4.9 --- Summary --- p.39 / Chapter Chapter 5 --- Design Architecture --- p.40 / Chapter 5.1 --- Introduction --- p.40 / Chapter 5.2 --- Criteria for the Low Power System Design --- p.40 / Chapter 5.3 --- Simplification in ONB Curve Addition Equations over Projective Coordinates --- p.41 / Chapter 5.4 --- Finite Field Adder Architecture --- p.43 / Chapter 5.5 --- Finite Field Squaring Architecture --- p.43 / Chapter 5.6 --- Finite Field Multiplier Architecture --- p.44 / Chapter 5.7 --- 3-way Parallel Finite Field Multiplier --- p.46 / Chapter 5.8 --- Finite Field Arithmetic Logic Unit --- p.47 / Chapter 5.9 --- Elliptic Curve Crypto-processor Control Unit --- p.50 / Chapter 5.10 --- Register Unit --- p.52 / Chapter 5.11 --- Summary --- p.53 / Chapter Chapter 6 --- Specifications and Communication Protocol of the IC --- p.54 / Chapter 6.1 --- Introduction --- p.54 / Chapter 6.2 --- Specifications --- p.54 / Chapter 6.3 --- Communication Protocol --- p.57 / Chapter Chapter 7 --- Results --- p.59 / Chapter 7.1 --- Introduction --- p.59 / Chapter 7.2 --- Results of the Public-key Cryptography --- p.59 / Chapter 7.3 --- Results of the Session-key Cryptography --- p.62 / Chapter 7.4 --- Comparison with the Existing Crypto-processor --- p.65 / Chapter 7.5 --- Power Consumption --- p.66 / Chapter Chapter 8 --- Conclusion --- p.68 / Bibliography --- p.69 / Appendix --- p.71 / 173-bit Type II ONB Multiplication Table --- p.71 / Layout View of the Elliptic Curve Crypto-processor --- p.76 / Schematics of the Elliptic Curve Crypto-processor --- p.77 / Schematics of the System Level Design --- p.78 / Schematics of the I/O Control Interface --- p.79 / Schematics of the Curve Multiplication Module --- p.80 / Schematics of the Curve Addition Module --- p.81 / Schematics of the Curve Doubling Module --- p.82 / Schematics of the Field Inversion Module --- p.83 / Schematics of the Register Unit --- p.84 / Schematics of the Datapath --- p.85 / Schematics of the Finite Field ALU --- p.86 / Schematics of the 3-way Parallel Multiplier --- p.87 / Schematics of the Multiplier Elements --- p.88 / Schematics of the Field Adder --- p.89 / Schematics of Demultiplexer --- p.90 / Schematics of the Control of the Demultiplexer --- p.91

Yao, Wei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 52-58). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iv / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Orbital stability without linear optical lattice --- p.12 / Chapter 2.1 --- Properties of single-spike bound states --- p.12 / Chapter 2.2 --- Proof of theorem 1.0.1 --- p.20 / Chapter 3 --- Orbital stability with linear optical lattice --- p.30 / Chapter 3.1 --- Properties of single-spike bound states --- p.30 / Chapter 3.2 --- Proof of theorem 1.0.2 --- p.33 / Chapter 3.3 --- Proof of theorem 1.0.3 --- p.34 / Chapter 3.4 --- Proof of theorem 1.0.4 --- p.37 / Chapter 4 --- Uniqueness for semilinear elliptic systems --- p.42 / Chapter 4.1 --- Proof of theorem 1.0.5 --- p.42 / Chapter 5 --- Appendix --- p.44 / Chapter 5.1 --- Appendix A --- p.44 / Chapter 5.2 --- Appendix B --- p.45 / Chapter 5.3 --- Appendix C --- p.47

In this thesis, we shall give a concise account for the classification of the structure of positive radial solutions of the semilinear elliptic equation$$Delta u+K(|x|)u^{p}=0 .$$ It is known that a radial solution $u$ is crossing if $u$ has a zero in $(0, infty)$; $u$ is slowly decaying if $u$ is positive but $displaystylelim_{r ightarrow{infty}}r^{n-2}u=infty$; u is rapidly decaying if $u$ is positive, $displaystylelim_{r ightarrow{infty}}r^{n-2}u$ exists and is positive. Using some Pohozaev identities, we show that under certain condition on $K$, by comparing some parameters $r_{G}$ and $r_{H}$, the structure of positive radial solutions for various initial conditions can be classified as Type Z ($u(r; alpha)$ is crossing for all $r>0$ ), Type S ($u(r; alpha)$ is slowly decaying for all $r>0$), and Type M (there is some $alpha_{f}$ such that $u(r; alpha)$ is crossing for $alphain(alpha_{f}, infty)$, $u(r; alpha)$ is slowly decaying for $alpha=alpha_{f}$, and $u(r; alpha)$ is rapidly decaying for $alphain(0, alpha_{f})$). The above work is due to Yanagida and Yotsutani.

Elliptic curve cryptosystems were proposed in 1985 by Victor Miller and by Neal Koblitz independently. Since elliptic curve discrete logarithm problem is harder to solve than discrete logarithm problem in finite fields. If is believed that the key length of elliptic curve cryptosystems can be shorter then that of RSA with the same security strength. The most important work of using elliptic curve cryptosystem is constructing a group from a proper elliptic curve. The major work of constructing an elliptic curve is counting points on elliptic curves over finite fields. In 1985, Schoof published a deterministic polynomial time algorithm for computing the number of points on the elliptic curves over finite fields. We consult IEEE P1363 to implement pseudo random elliptic curve.

Thesis (doctoral)--Friedrich-Wilhelms-Universität Berlin, 1870. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1882. / Vita. Includes bibliographical references.

Thesis (doctoral)--Friedrich-Wilhelms-Universität zu Berlin, 1882. / Vita.

Thesis (master's)--Kaiserliche Universität Dorpat, 1885.