Journal of the Ramanujan Mathematical Society

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Explicit Algorithm for the Arithmetic on the Hyperelliptic Jacobians of Genus 3


Affiliations
1 Kasten Chase Applied Research, Orbitor Place, 5100 Orbitor Drive, Missisauga, Ontario L4W 4Z4, Canada
2 1984 Mathematics Road, Department of Mathematics, University of British Columbia, Vancouver, Bristish Columbia V6T 1Z2, Canada
3 100 St. George Street, Department of Mathematics, University of Toronto, Toronto, Ontario M6G 2M6, Canada
     

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We investigate efficient formulae to double and add divisors on the Jacobian of a hyperelliptic 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.
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  • Explicit Algorithm for the Arithmetic on the Hyperelliptic Jacobians of Genus 3

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Authors

Cyril Guyot
Kasten Chase Applied Research, Orbitor Place, 5100 Orbitor Drive, Missisauga, Ontario L4W 4Z4, Canada
Kiumars Kaveh
1984 Mathematics Road, Department of Mathematics, University of British Columbia, Vancouver, Bristish Columbia V6T 1Z2, Canada
Vijay M. Patankar
100 St. George Street, Department of Mathematics, University of Toronto, Toronto, Ontario M6G 2M6, Canada

Abstract


We investigate efficient formulae to double and add divisors on the Jacobian of a hyperelliptic 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.