Elliptic Curves on Finite Fields
Kumar Harsha1, Anupam Saikia2
1Kumar Harsha, Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India.
2Dr. Anupam Saikia, Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, India.
Manuscript received on 09 April 2016 | Revised Manuscript received on 16 April 2016 | Manuscript Published on 30 April 2016 | PP: 35-40 | Volume-5 Issue-11, April 2016 | Retrieval Number: K22920451116/2016©BEIESP
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© The Authors. Blue Eyes Intelligence Engineering and Sciences Publication (BEIESP). This is an open access article under the CC-BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Abstract: This paper explores the algebraic properties of elliptic curves over finite fields. Elliptic curves are being widely used in modern cryptographic techniques. The rational points on an elliptic curve obey group theoretic laws. As such, computing the order of these groups forms the basis of more complex computations. The first section of this paper deals with the basic group properties of rational points on elliptic curves and an introduction to projective geometry. In the second, algorithms for computing multiplication maps are explained. The later section has point counting algorithms followed by code snippets in SAGE. Also included, is a section on some unsolved problems in the domain. Index Terms—
Keywords: Elliptic curves, SAGE
Scope of the Article: Applied Mathematics and Mechanics