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Approved for public release; distribution is unlimited / The analysis of a break in production is usually performed by a government negotiator or cost analyst. The more effectively they are able to estimate the loss of learning due to breaks in production, the more likely that the final contract will be fair and reasonable. The research of this study focused on identifying the factors which contribute to a loss of learning due to a break in production and the methods which are available to quantify these factors. The four methods identified were the George Anderlohr, the DCAA, the Pinchon and Richardson, and the Cubic Curve. These methods were then analyzed using the data from two aircraft, the Grumman C-2A and the Bell Helicopter Textron AH-1W, both of which experienced breaks in production. This study concludes that the George Anderlohr approach is the most effective method to evaluate the loss of learning due to a break in production. / http://archive.org/details/measuringlosseso00ever / Captain, United States Marine Corps

We study the effects of a deformation via the heat equation on closed, plane curves. We begin with an overview of the theory of curves in R3. In particular, we develop the Frenet-Serret equations for any curve parametrized by arc length. This chapter is followed by an examination of curves in R2, and the resultant adjustment of the Frenet-Serret equations. We then prove the rotation index for closed, plane curves is an integer and for simple, closed, plane curves is ±1. We show that a curve is convex if and only if the curvature does not change sign, and we prove the Isoperimetric Inequality, which gives a bound on the area of a closed curve with fixed length. Finally, we study the deformation of plane curves developed by M. Gage and R. S. Hamilton. We observe that convex curves under deformation remain convex, and simple curves remain simple.

Elliptic curves have played a large role in modern cryptography. Most notably, the Elliptic Curve Digital Signature Algorithm (ECDSA) and the Elliptic Curve Di e-Hellman (ECDH) key exchange algorithm are widely used in practice today for their e ciency and small key sizes. More recently, the Supersingular Isogeny-based Di e-Hellman (SIDH) algorithm provides a method of exchanging keys which is conjectured to be secure in the post-quantum setting. For ECDSA and ECDH, e cient and secure algorithms for scalar multiplication of points are necessary for modern use of these protocols. Likewise, in SIDH it is necessary to be able to compute an isogeny from a given nite subgroup of an elliptic curve in a fast and secure fashion. We therefore nd strong motivation to study and improve the algorithms used in elliptic curve cryptography, and to develop new algorithms to be deployed within these protocols. In this thesis we design and develop d-MUL, a multidimensional scalar multiplication algorithm which is uniform in its operations and generalizes the well known 1-dimensional Montgomery ladder addition chain and the 2-dimensional addition chain due to Dan J. Bernstein. We analyze the construction and derive many optimizations, implement the algorithm in software, and prove many theoretical and practical results. In the nal chapter of the thesis we analyze the operations carried out in the construction of an isogeny from a given subgroup, as performed in SIDH. We detail how to e ciently make use of parallel processing when constructing this isogeny. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2018. / FAU Electronic Theses and Dissertations Collection

by Yan Cho Hung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 122-124). / Abstract also in Chinese. / Chapter 1 --- Complex algebraic curves --- p.6 / Chapter 1.1 --- Foundations --- p.6 / Chapter 1.1.1 --- Hilbert Nullstellensatz --- p.6 / Chapter 1.1.2 --- Complex algebraic curves in C2 --- p.9 / Chapter 1.1.3 --- Complex projective curves in P2 --- p.11 / Chapter 1.1.4 --- Affine and projective curves --- p.13 / Chapter 1.2 --- Algebraic properties of complex projective curves in P2 --- p.16 / Chapter 1.2.1 --- Intersection multiplicity --- p.16 / Chapter 1.2.2 --- Bezout's theorem and its applications --- p.18 / Chapter 1.2.3 --- Cubic curves --- p.21 / Chapter 1.3 --- Topological properties of complex projective curves in P2 --- p.23 / Chapter 1.4 --- Riemann surfaces --- p.26 / Chapter 1.4.1 --- Weierstrass &-function --- p.26 / Chapter 1.4.2 --- Riemann surfaces and examples --- p.27 / Chapter 1.5 --- Differentials on Riemann surfaces --- p.28 / Chapter 1.5.1 --- Holomorphic differentials --- p.28 / Chapter 1.5.2 --- Abel's Theorem for tori --- p.31 / Chapter 1.5.3 --- The Riemann-Roch theorem --- p.32 / Chapter 1.6 --- Singular curves --- p.36 / Chapter 1.6.1 --- Resolution of singularities --- p.37 / Chapter 1.6.2 --- The topology of singular curves --- p.45 / Chapter 2 --- Coding theory --- p.48 / Chapter 2.1 --- An introduction to codes --- p.48 / Chapter 2.1.1 --- Efficient noiseless coding --- p.51 / Chapter 2.1.2 --- The main coding theory problem --- p.56 / Chapter 2.2 --- Linear codes --- p.58 / Chapter 2.2.1 --- Syndrome decoding --- p.63 / Chapter 2.2.2 --- Equivalence of codes --- p.65 / Chapter 2.2.3 --- An introduction to cyclic codes --- p.67 / Chapter 2.3 --- Special linear codes --- p.71 / Chapter 2.3.1 --- Hamming codes --- p.71 / Chapter 2.3.2 --- Simplex codes --- p.72 / Chapter 2.3.3 --- Reed-Muller codes --- p.73 / Chapter 2.3.4 --- BCH codes --- p.75 / Chapter 2.4 --- Bounds on codes --- p.77 / Chapter 2.4.1 --- Spheres in Zn --- p.77 / Chapter 2.4.2 --- Perfect codes --- p.78 / Chapter 2.4.3 --- Famous numbers Ar (n，d) and the sphere-covering and sphere packing bounds --- p.79 / Chapter 2.4.4 --- The Singleton and Plotkin bounds --- p.81 / Chapter 2.4.5 --- The Gilbert-Varshamov bound --- p.83 / Chapter 3 --- Algebraic curves over finite fields and the Goppa codes --- p.85 / Chapter 3.1 --- Algebraic curves over finite fields --- p.85 / Chapter 3.1.1 --- Affine varieties --- p.85 / Chapter 3.1.2 --- Projective varieties --- p.37 / Chapter 3.1.3 --- Morphisms --- p.89 / Chapter 3.1.4 --- Rational maps --- p.91 / Chapter 3.1.5 --- Non-singular varieties --- p.92 / Chapter 3.1.6 --- Smooth models of algebraic curves --- p.93 / Chapter 3.2 --- Goppa codes --- p.96 / Chapter 3.2.1 --- Elementary Goppa codes --- p.96 / Chapter 3.2.2 --- The affine and projective lines --- p.98 / Chapter 3.2.3 --- Goppa codes on the projective line --- p.102 / Chapter 3.2.4 --- Differentials and divisors --- p.105 / Chapter 3.2.5 --- Algebraic geometric codes --- p.112 / Chapter 3.2.6 --- Codes with better rates than the Varshamov- Gilbert bound and calculation of parameters --- p.116 / Bibliography

"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."

Zhang Jia-jin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 60-62). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.4 / Chapter 2 --- Configurations Of Points In P1 And Local Systems Of Rank One --- p.6 / Chapter 2.1 --- Configurations Of Points In P1 --- p.6 / Chapter 2.2 --- Local Systems Of Rank One --- p.7 / Chapter 2.3 --- Arithmeticity And Integral Monodromy --- p.12 / Chapter 3 --- Generalized Jacobians --- p.13 / Chapter 4 --- Stable Reductions Of Family Of Curves --- p.17 / Chapter 4.1 --- Normalization Of Cyclic Branched coverings --- p.17 / Chapter 4.2 --- Stable Reductions --- p.19 / Chapter 5 --- "Family Of n-th Cyclic Coverings Of P1, Abelian Va- rieties And CM-type" --- p.22 / Chapter 5.1 --- Family Of n-th Cyclic Coverings Of P1 --- p.22 / Chapter 5.2 --- Abelian Varieties And CM-type --- p.24 / Chapter 6 --- Families of Jacobians Coming From [6] --- p.27 / Chapter 6.1 --- Example 1. Family y3 = x(x ´ؤ l)(x ´ؤ λ)(x ´ؤμ) --- p.27 / Chapter 6.2 --- Example 2. Family y5=x(x ´ؤ l)(x ´ؤ λ)(x ´ؤμ) --- p.34 / Chapter 6.3 --- Other Families --- p.38 / Bibliography --- p.60

Huang Yong Dong. / Thesis submitted in: November 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 59-60). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Abelian Varieties And Shimura Varieties --- p.5 / Chapter 1.2 --- Jacobians of Some Families of Curves --- p.7 / Chapter 1.3 --- Endomorphism Algebras of Jacobians of Curves --- p.8 / Chapter 2 --- Families of Abelian Varieties --- p.11 / Chapter 2.1 --- Abelian Varieties --- p.11 / Chapter 2.2 --- The Endomorphism Algebra of A Simple Abelian Varieties --- p.13 / Chapter 2.3 --- Family of Abelian Varieties and Shimura Varieties --- p.15 / Chapter 2.3.1 --- Real Multiplication --- p.16 / Chapter 2.3.2 --- Totally Indefinite Quaternion Multiplication --- p.19 / Chapter 2.3.3 --- Totally Definite Quaternion Multiplication --- p.22 / Chapter 2.3.4 --- Complex Multiplication --- p.25 / Chapter 2.3.5 --- Shimura Varieties --- p.28 / Chapter 2.4 --- The Endomorphism Algebra of A General Member --- p.29 / Chapter 3 --- Jacobians of Some Families of Curves --- p.32 / Chapter 3.1 --- Some Families of Curves --- p.32 / Chapter 3.2 --- Kodaira-Spencer Map --- p.37 / Chapter 3.3 --- Infinity of CM Type Points --- p.46 / Chapter 3.3.1 --- Φ Is Dominant --- p.46 / Chapter 3.3.2 --- Infinity of CM Points --- p.50 / Chapter 4 --- Endomorphism Algebras of Jacobians of Some Fam- ilies of Curves --- p.51 / Chapter 4.1 --- Jacobians Between Finite Coverings of Curves --- p.51 / Chapter 4.2 --- Endomorphism Algebras of Families of Jacobians --- p.53 / Chapter 4.2.1 --- The Case For μ = 1 --- p.53 / Chapter 4.2.2 --- The Case For μ = 2 --- p.55 / Chapter 4.2.3 --- The Case For μ =3 --- p.58 / Bibliography

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

In this dissertation we study tropicalization curves which have a theta characteristic with large rank. This fits in the more general framework of studying the limit linear series on a curve which degenerates to a singular curve. We explore this when the singular curve is not of compact type. In particular we investigate the case when dual graph of the degenerate curve is a chain of g-loops. The fundamental object under consideration is a family of curves over a complete discrete valuation ring. In the first half of the dissertation we study geometry of such a family. In the third chapter we study metric graphs and divisors on them. This could be a thought of as the theory of limit linear series on a curve of non-compact type. In the fourth chapter we make this connection via tropicalization. We consider a family of curves with smooth generic fiber X η of genus g such that the dual graph of the special fiber is a chain of g loops. The main theorem we prove is that if X η has a theta characteristic of rank r then there are at least r linear relations on the edge lengths of the dual graph.