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Elliptic Curves CryptographyIdrees, Zunera January 2012 (has links)
In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

52 
Elliptic Curves CryptographyIdrees, Zunera January 2012 (has links)
In the thesis we study the elliptic curves and its use in cryptography. Elliptic curvesencompasses a vast area of mathematics. Elliptic curves have basics in group theory andnumber theory. The points on elliptic curve forms a group under the operation of addition.We study the structure of this group. We describe Hasse’s theorem to estimate the numberof points on the curve. We also discuss that the elliptic curve group may or may not becyclic over finite fields. Elliptic curves have applications in cryptography, we describe theapplication of elliptic curves for discrete logarithm problem and ElGamal cryptosystem.

53 
FPGA implementations of elliptic curve cryptography and Tate pairing over binary fieldHuang, Jian. Li, Hao, January 2007 (has links)
Thesis (M.S.)University of North Texas, Aug., 2007. / Title from title page display. Includes bibliographical references.

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Hardware/software optimizations for elliptic curve scalar multiplication on hybrid FPGAs /Ramsey, Glenn. January 2008 (has links)
Thesis (M.S.)Rochester Institute of Technology, 2008. / Typescript. Includes bibliographical references (p. 9597).

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FPGA Implementations of Elliptic Curve Cryptography and Tate Pairing over Binary FieldHuang, Jian 08 1900 (has links)
Elliptic curve cryptography (ECC) is an alternative to traditional techniques for public key cryptography. It offers smaller key size without sacrificing security level. Tate pairing is a bilinear map used in identity based cryptography schemes. In a typical elliptic curve cryptosystem, elliptic curve point multiplication is the most computationally expensive component. Similarly, Tate pairing is also quite computationally expensive. Therefore, it is more attractive to implement the ECC and Tate pairing using hardware than using software. The bases of both ECC and Tate pairing are Galois field arithmetic units. In this thesis, I propose the FPGA implementations of the elliptic curve point multiplication in GF (2283) as well as Tate pairing computation on supersingular elliptic curve in GF (2283). I have designed and synthesized the elliptic curve point multiplication and Tate pairing module using Xilinx's FPGA, as well as synthesized all the Galois arithmetic units used in the designs. Experimental results demonstrate that the FPGA implementation can speedup the elliptic curve point multiplication by 31.6 times compared to software based implementation. The results also demonstrate that the FPGA implementation can speedup the Tate pairing computation by 152 times compared to software based implementation.

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Toroidal algebra representations and equivariant elliptic surfacesDeHority, Samuel Patrick January 2024 (has links)
We study the equivariant cohomology of moduli spaces of objects in the derived category of elliptic surfaces in order to find new examples of infinite dimensional quantum integrable systems and their geometric representation theoretic interpretation in enumerative geometry. This problem is related to a program to understand the cohomological and Ktheoretic Hall algebras of holomorphic symplectic surfaces and to understand how it related to the DonaldsonThomas theory of threefolds fibered in those surfaces.
We use the theory of noncommutative deformations of Poisson surfaces and especially van den Berg’s noncommutative P1 bundles as well as Rains’s analysis of moduli theory for quasiruled noncommutative surfaces as well as the theory of Bridgeland stability conditions and their relative versions to understand equivariant deformations and birational transformations of Hilbert schemes of points on equivariant elliptic surfaces. The moduli spaces are closely related to elliptic versions of classical integrable systems. We also use these moduli spaces to construct vertex algebra representations of extensions of toroidal extended affine algebras on their equivariant cohomology, building on work of EswaraRao–Moody–Yokonuma, of Billig, and of Chen–Li–Tan on vertex representations of toroidal algebras, full toroidal algebras, and toroidal extended affine algebras.
Using FourierMukai transforms and their relative action on families of dgcategories we study the relationship between automorphisms of toroidal extended affine algebras and families of derived equivalences on dg categories, in particular finding a relativistic (difference) generalization of the LaumonRothstein deformation of the FourierMukai duality. Finally, using the above analysis we extend the construction of Maulik–Okounkov’s stable envelopes to moduli of framed torsionfree sheaves on noncommutative surfaces in some cases and use this to study coproducts on associated algebras assigned to elliptic surfaces with applications to understanding new representation theoretic structures in the DonaldsonThomas theory of local curves.

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Análise de desempenho de algoritmos criptográficos assimétricos em uma rede veicular (Vanet)Matos, Leila Buarque Couto de 31 January 2013 (has links)
This dissertation describes the impact of using asymmetric encryption algorithms, with emphasis on algorithms RSA, ECC and MQQ in scenarios VANET (Vehicular Ad hoc Network). In the research were investigated some simulators as GrooveNet, VANET / DSRC, VANET / CRL Epidemic, NS2, trans, NCTUns / EstiNET, SUMO, VanetMobiSim and ns3, suitable for VANET. The algorithms have been implemented in C and inserted into the ns3, where the simple scenarios created a network VANET. The results showed that it is possible to add protocollayer security services of vehicular networks (1609.2), these asymmetric algorithms and obtain secure communication between nodes in the VANET. / Esta dissertação de mestrado descreve o impacto de usar algoritmos assimétricos de criptografia, dando ênfase aos algoritmos RSA, ECC e MQQ em cenários de VANET (Vehicular Ad hoc Network). Na pesquisa foram investigados alguns simuladores como GrooveNet, VANET/DSRC, VANET/Epidemic CRL, NS2, TraNS, NCTUns/EstiNET, SUMO, VanetMobiSim e ns3, próprio para VANET. Os algoritmos foram implementados em C e inseridos no ns3, onde se criam cenários simples de uma rede VANET. Os resultados obtidos permitem concluir que é possível agregar ao protocolo, na camada de serviços de segurança das redes veiculares (1609.2), esses algoritmos assimétricos e obter comunicação segura entre os nós da VANET.

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Smart card fault attacks on public key and elliptic curve cryptographyLing, Jie January 2014 (has links)
Indiana UniversityPurdue University Indianapolis (IUPUI) / Blömmer, Otto, and Seifert presented a fault attack on elliptic curve scalar multiplication called the Sign Change Attack, which causes a fault that changes the sign of the accumulation point. As the use of a sign bit for an extended integer is highly unlikely, this appears to be a highly selective manipulation of the key stream. In this thesis we describe two plausible fault attacks on a smart card implementation of elliptic curve cryptography. King and Wang designed a new attack called counter fault attack by attacking the scalar multiple of discretelog cryptosystem. They then successfully generalize this approach to a family of attacks. By implementing King and Wang's scheme on RSA, we successfully attacked RSA keys for a variety of sizes. Further, we generalized the attack model to an attack on any implementation that uses NAF and wNAF key.

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