Document Type thesis Author Name Guajardo, Jorge Email Address guajardo at ece.wpi.edu URN etd-0328100-225237 Title Efficient Algorithms for Elliptic Curve Cryptosystems Degree MS Department Electrical & Computer Engineering Advisors Dr. Christof Paar, Advisor Dr. David Cyganski, Committee Dr. Gabor Sarkozy, Committee Dr. Robert Dulude, Committee Dr. John Orr, Department Head Keywords cryptography elliptic curves Galois Fields Date of Presentation/Defense 1997-05-03 Availability unrestricted
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.
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