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For nearly three centuries mathematicians have been interested in polygons which simultaneously circumscribe and inscribe quadrics. They have shown in many contexts (real, complex, non-euclidean, higher dimensional, etc.) that such polygons may be ``rotated'' while maintaining their circum-inscribed quality. Of particular interest has been conditions on the quadrics which guarantee the existence of such polygons. In 1854 Arthur Cayley provided conditions for closure general to polygons of any size in the complex projective plane. We show that under suitable circumstances the curve, defined by Cayley's conditions, on a fibration of Jacobians over the space of families of quadrics is a reducible curve, particularly in genus two. We may infer additional information about points of finite order on the Jacobians based on the component of the reducible curve in which they lie. Using this information we are able to accomplish two tasks. First we provide sufficient closure conditions for Poncelet's Great Theorem in which each vertex of the polygon lies on a distinct quadric. Next, for a polygon circum-inscribed in quadrics in ℙ^3, we provide additional sufficient conditions for closure beyond what mathematicians had previously believed to be necessary and sufficient.

The most efficient algorithm for solving the elliptic curve discrete logarithm problem can only be done in exponential time. Hence, we can use it in many cryptographic applications. Weil pairing is a mapping which maps a pair of points on elliptic curves to a multiplicative group of a finite field with nondegeneracy and bilinearity. Pairing was found to reduce the elliptic curve discrete logarithm problem into the discrete logarithm problem of a finite field, and became an important issue since then. In 1986, Miller proposed an efficient algorithm for computing Weil pairings. Many researchers focus on the improvement of this algorithm. In 2006, Blake et al. proposed the reduction of total number of lines based on the conjugate of a line. Liu et al. expanded their concept and proposed two improved methods. In this paper, we use both NAF and segmentation algorithm to implement the Weil pairing and analyse its complexity.

abstract: Diophantine arithmetic is one of the oldest branches of mathematics, the search for integer or rational solutions of algebraic equations. Pythagorean triangles are an early instance. Diophantus of Alexandria wrote the first related treatise in the fourth century; it was an area extensively studied by the great mathematicians of the seventeenth century, including Euler and Fermat. The modern approach is to treat the equations as defining geometric objects, curves, surfaces, etc. The theory of elliptic curves (or curves of genus 1, which are much used in modern cryptography) was developed extensively in the twentieth century, and has had great application to Diophantine equations. This theory is used in application to the problems studied in this thesis. This thesis studies some curves of high genus, and possible solutions in both rationals and in algebraic number fields, generalizes some old results and gives answers to some open problems in the literature. The methods involve known techniques together with some ingenious tricks. For example, the equations $y^2=x^6+k$, $k=-39,\,-47$, the two previously unsolved cases for $|k|<50$, are solved using algebraic number theory and the ‘elliptic Chabauty’ method. The thesis also studies the genus three quartic curves $F(x^2,y^2,z^2)=0$ where F is a homogeneous quadratic form, and extend old results of Cassels, and Bremner. It is a very delicate matter to find such curves that have no rational points, yet which do have points in odd-degree extension fields of the rationals. The principal results of the thesis are related to surfaces where the theory is much less well known. In particular, the thesis studies some specific families of surfaces, and give a negative answer to a question in the literature regarding representation of integers n in the form $n=(x+y+z+w)(1/x+1/y+1/z+1/w).$ Further, an example, the first such known, of a quartic surface $x^4+7y^4=14z^4+18w^4$ is given with remarkable properties: it is everywhere locally solvable, yet has no non-zero rational point, despite having a point in (non-trivial) odd-degree extension fields of the rationals. The ideas here involve manipulation of the Hilbert symbol, together with the theory of elliptic curves. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / SSL protocol is one industrial standard to protect data transferred securely on Internet. Firstly SSL is analyzed, according to its characteristics, one solution plan on mobile bank based on SSL is proposed and presented, in which GPRS technology is adopted and elliptic curve algorithm is used for the session key, finally several functional modules of mobile bank are designed in details and its security is analyzed.

This thesis is devoted to investigating some properties of the sequence (Wn) of the denominators. This is a divisibility sequence; that is, Wm | Wn whenever m | n. Our task here is to examine a conjecture on the number of prime terms in (Wn), well known as the Primality conjecture. We will prove that there is a uniform lower bound on n beyond such that all terms Wn have at least two distinct prime factors. In some cases, the bound is as low as n = 2.

It was recently discovered by Joux [30] and Sakai, Ohgishi and Kasahara [47] that bilinear maps could be used to construct cryptographic schemes. Since then, bilinear maps have been used in applications as varied as identity-based encryption, short signatures and one-round tripartite key agreement. This thesis explains the notion of bilinear maps and surveys the applications of bilinear maps in the three main fields of cryptography: encryption, signature and key agreement. We also show how these maps can be constructed using the Weil and Tate pairings in elliptic curves.

Thesis advisor: Benjamin Howard / In the last twenty years, Elliptic Curve Cryptography has become a standard for the transmission of secure data. The purpose of my thesis is to develop the necessary theory for the implementation of elliptic curve cryptosystems, using elementary number theory, abstract algebra, and geometry. This theory is based on developing formulas for adding rational points on an elliptic curve. The set of rational points on an elliptic curve form a group over the addition law as it is defined. Using the group law, my study continues into computing the torsion subgroup of an elliptic curve and considering elliptic curves over finite fields. With a brief introduction to cryptography and the theory developed in the early chapters, my thesis culminates in the explanation and implementation of three elliptic curve cryptosystems in the Java programming language. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

Elliptic curve cryptosystem is one type of public-key system, and it can guarantee the same security level with Rivest, Shamir and Adleman (RSA) with a smaller key size. Therefore, the key of elliptic curve cryptography (ECC) can be more compact, and it brings many advantages such as circuit area, memory requirement, power consumption, performance and bandwidth. However, compared to private key system, like Advanced Encryption Standard (AES), ECC is still much more complicated and computationally intensive. In some real applications, people usually combine private-key system with public-key system to achieve high performance. The ultimate goal of this research is to architect a high performance ECC processor for high performance applications such as network server and cellular sites.<p> In this thesis, a high performance processor for ECC over Galois field (GF)(2^163) by using polynomial presentation is proposed for high-performance applications. It has three finite field (FF) reduced instruction set computer (RISC) cores and a main controller to achieve instruction-level parallelism (ILP) with pipeline so that the largely parallelized algorithm for elliptic curve point multiplication (PM) can be well suited on this platform. Instructions for combined FF operation are proposed to decrease clock cycles in the instruction set. The interconnection among three FF cores and the main controller is obtained by analyzing the data dependency in the parallelized algorithm. Five-stage pipeline is employed in this architecture. Finally, the u-code executed on these three FF cores is manually optimized to save clock cycles. The proposed design can reach 185 MHz with 20; 807 slices when implemented on Xilinx XC4VLX80 FPGA device and 263 MHz with 217,904 gates when synthesized with TSMC .18um CMOS technology. The implementation of the proposed architecture can complete one ECC PM in 1428 cycles, and is 1.3 times faster than the current fastest implementation over GF(2^163) reported in literature while consumes only 14:6% less area on the same FPGA device.

It was recently discovered by Joux [30] and Sakai, Ohgishi and Kasahara [47] that bilinear maps could be used to construct cryptographic schemes. Since then, bilinear maps have been used in applications as varied as identity-based encryption, short signatures and one-round tripartite key agreement. This thesis explains the notion of bilinear maps and surveys the applications of bilinear maps in the three main fields of cryptography: encryption, signature and key agreement. We also show how these maps can be constructed using the Weil and Tate pairings in elliptic curves.