Explicit algorithm for the arithmetic on the hyperelliptic Jacobians of genus 3 Cyril Guyot * Kasten Chase Applied Research Orbitor Place, 5100 Orbitor Drive Missisauga, Ontario L4W 4Z4, Canada cyril@zoy.org Kiumars Kaveh * 1984 Mathematics Road Department of Mathematics University of British Columbia Vancouver, Bristish Columbia V6T 1Z2, Canada kaveh@math.ubc.ca Vijay M. Patankar * 100 St. George Street Department of Mathematics University of Toronto Toronto, Ontario, Canada M6G 2M6 vijay@math.utoronto.ca Abstract We investigate efficient formulae to double and add divisors on the Jacobian of a hyper- elliptic curve of genus 3. The main contributions of this paper are as follows: (1) Overall improvements in the complexity of the addition and doubling algorithms for both even and odd characteristics, (2) Algorithms applicable to almost all hyperelliptic curves of genus 3, and (3) Efficient computation of the resultant of two polynomials and of the inverse of one polynomial modulo another. This paper is specifically written in an implementation-ready format. * Most of the work in this paper was carried out while the first and the last authors were employed by Kas- ten Chase Applied Research. We wish to thank the said organisation for providing us with a stimulating and encouraging work environment. All the authors wish to thank Karthika Technologies for providing the research atmosphere where the research on this topic had begun. 1