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.

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Elliptic curves have a rich mathematical history dating back to Diophantus (c. 250 C.E.), who used a form of these cubic equations to find right triangles of integer area with rational sides. In more recent times the deep mathematics of elliptic curves was used by Andrew Wiles et. al., to construct a proof of Fermat's last theorem, a problem which challenged mathematicians for more than 300 years. In addition, elliptic curves over finite fields find practical application in the areas of cryptography and coding theory. For such problems, knowing the order of the group of points satisfying the elliptic curve equation is important to the security of these applications. In 1985 René Schoof published a paper [5] describing a polynomial time algorithm for solving this problem. In this thesis we explain some of the key mathematical principles that provide the basis for Schoof's method. We also present an implementation of Schoof's algorithm as a collection of Mathematica functions. The operation of each algorithm is illustrated by way of numerical examples. / Master of Science

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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

This dissertation emphasizes the security aspects of real-time media. The problems of existing real-time media protections are identified in this research, and viable solutions are proposed. First, the security of real-time media depends on the Secure Real-time Transport Protocol (SRTP) mechanism. We identified drawbacks of the existing SRTP Systems, which use symmetric key encryption schemes, which can be exploited by attackers. Elliptic Curve Cryptography (ECC), an asymmetric key cryptography scheme, is proposed to resolve these problems. Second, the ECC encryption scheme is based on elliptic curves. This dissertation explores the weaknesses of a widely used elliptic curve in terms of security and describes a more secure elliptic curve suitable for real-time media protection. Eighteen elliptic curves had been tested in a real-time video transmission system, and fifteen elliptic curves had been tested in a real-time audio transmission system. Based on the performance, X9.62 standard 256-bit prime curve, NIST-recommended 256-bit prime curves, and Brainpool 256-bit prime curves were found to be suitable for real-time audio encryption. Again, X9.62 standard 256-bit prime and 272-bit binary curves, and NIST-recommended 256-bit prime curves were found to be suitable for real-time video encryption.The weaknesses of NIST-recommended elliptic curves are discussed and a more secure new elliptic curve is proposed which can be used for real-time media encryption. The proposed curve has fulfilled all relevant security criteria, but the corresponding NIST curve could not fulfill two such criteria. The research is applicable to strengthen the security of the Internet of Things (IoT) devices, especially VoIP cameras. IoT devices have resource constraints and thus need lightweight encryption schemes for security. ECC could be a better option for these devices. VoIP cameras use a similar methodology to traditional real-time video transmission, so this research could be useful to find a better security solution for these devices.

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.