Efficient reduction of large divisors on hyperelliptic curves

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May 2010, 4(2): 261-279. doi: 10.3934/amc.2010.4.261

Efficient reduction of large divisors on hyperelliptic curves

Roberto Avanzi ^1, , Michael J. Jacobson, Jr. ^2, and Renate Scheidler ^3,

1.	Fakultät für Mathematik, Ruhr-Universität Bochum and Horst Gösrtz Institut für IT-Sicherheit, Universitätsstraße 150, D-44780 Bochum
2.	Department of Computer Science, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada
3.	Department of Mathematics & Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4, Canada

Received June 2009 Revised March 2010 Published May 2010

Abstract
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We present an algorithm for reducing a divisor on a hyperelliptic curve of arbitrary genus over any finite field. Our method is an adaptation of a procedure for reducing ideals in quadratic number fields due to Jacobson, Sawilla and Williams, and shares common elements with both the Cantor and the NUCOMP algorithms for divisor arithmetic. Our technique is especially suitable for the rapid reduction of a divisor with very large Mumford coefficients, obtained for example through an efficient tupling technique. Results of numerical experiments are presented, showing that our algorithm is superior to the standard reduction algorithm in many cases.

Keywords: Hyperelliptic curve, reduction, scalar multiplication., continued fraction expansion, divisor.

Mathematics Subject Classification: Primary: 11R58, 14H45; Secondary: 14G5.

Citation: Roberto Avanzi, Michael J. Jacobson, Jr., Renate Scheidler. Efficient reduction of large divisors on hyperelliptic curves. Advances in Mathematics of Communications, 2010, 4 (2) : 261-279. doi: 10.3934/amc.2010.4.261

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