**A Simple Method for Obtaining Relations Among Factor Basis Elements for Special Hyperelliptic Curves**

*Palash Sarkar and Shashank Singh*

**Abstract: **Nagao had proposed a decomposition method for divisors of hyperelliptic curves defined over a field $\rF_{q^n}$ with $n\geq 2$.
Joux and Vitse had later proposed a variant which provided relations among the factor basis elements. Both Nagao's and the
Joux-Vitse methods require solving a multi-variate system of non-linear equations. In this work, we revisit Nagao's approach
with the idea of avoiding the requirement of solving a multi-variate system. While this cannot be done in general, we are
able to identify special cases for which this is indeed possible. Our main result is for curves $C:y^2=f(x)$ of genus $g$ defined
over $\rF_{q^2}$ having characteristic greater than two. If $f(x)$ has at most $g$ consecutive coefficients which are
in $\rF_{q^2}$ while the rest are in $\rF_q$, then we show that it is possible to obtain a single relation in about
$(2g+3)!$ trials. The method combines well with a sieving method proposed by Joux and Vitse. Our implementation of the
resulting algorithm provides examples of factor basis relations for $g=5$ and $g=6$. We believe that none of the other methods
known in the literature can provide such relations faster than our method. Other than obtaining such decompositions, we
also explore the applicability of our approach for $n>2$ and also for binary characteristic fields.

**Category / Keywords: **hyperelliptic curves, index calculus algorithm, Nagao's decomposition, Joux-Vitse sieving

**Date: **received 1 Mar 2015

**Contact author: **sha2nk singh at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20150302:080016 (All versions of this report)

**Short URL: **ia.cr/2015/179

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