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Elliptic curves are the basis for a relative new class of public-key schemes. It is predicted that elliptic curves will replace many existing schemes in the near future. It is thus of great interest to develop algorithms which allow efficient implementations of elliptic curve crypto systems. This thesis deals with such algorithms. Efficient algorithms for elliptic curves can be classified into low-level algorithms, which deal with arithmetic in the underlying finite field and high-level algorithms, which operate with the group operation. This thesis describes three new algorithms for efficient implementations of elliptic curve cryptosystems. The first algorithm describes the application of the Karatsuba-Ofman Algorithm to multiplication in composite fields GF((2n)m). The second algorithm deals with efficient inversion in composite Galois fields of the form GF((2n)m). The third algorithm is an entirely new approach which accelerates the multiplication of points which is the core operation in elliptic curve public-key systems. The algorithm explores computational advantages by computing repeated point doublings directly through closed formulae rather than from individual point doublings. Finally we apply all three algorithms to an implementation of an elliptic curve system over GF((216)11). We provide ablolute performance measures for the field operations and for an entire point multiplication. We also show the improvements gained by the new point multiplication algorithm in conjunction with the k-ary and improved k-ary methods for exponentiation.

Thesis (M.Sc.) - Carleton University, 2008. / Includes bibliographical references (p. 212-213). Also available in electronic format on the Internet.

Honors Project--Smith College, Northampton, Mass., 2010. / Includes bibliographical references (p. 42).

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We provide an explicit algorithm for computing the trace of an endomorphism of an elliptic curve which is given by a chain of small-degree isogenies. We analyze its complexity, determining that if the length of the chain, the degree of the isogenies, and the log of the field-size are all O(n), the trace of the endomorphism can be computed in O(n⁶) bit operations. This makes explicit a theorem of Kohel which states that such a polynomial time algorithm exists. The given procedure is based on Schoof's point-counting algorithm. / Master of Science / The developing technology of quantum computers threatens to render current cryptographic systems (that is, systems for protecting stored or transmitted digital information from unauthorized third parties) ineffective. Among the systems proposed to ensure information security against attacks by quantum computers is a cryptographic scheme known as SIKE. In this thesis, we provide and analyze an algorithm that comprises one piece of a potential attack against SIKE by a classical computer. The given algorithm is also useful more generally in the field of arithmetic geometry.

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Integral solutions to y² = X³ + k, where either the x's or the y's, or both, are in arithmetic progression are studied. When both the x's and the y's are in arithmetic progression, then this situation is completely solved. One set of solutions where the y's formed an arithmetic progression of length 4 have already been constructed. In this dissertation, we construct infinitely many set of solutions where there are 4 x's in arithmetic progression and we also disprove Mohanty's Conjecture[8] by constructing infinitely many set of solutions where there are 4, 5 and 6 y's in arithmetic progression.