Geodesic
Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials
Symmetry, integrability and geometry: methods and applications, Tome 16 (2020) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered as the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of solution of the Jacobi inversion problem which provides the way of computing Abelian functions on arbitrary non-special divisors. An effective reduction algorithm is proposed, which has the advantage that it involves only arithmetic operations on polynomials. The proposed addition algorithm contains more details comparing with the known in cryptography, and is extended to divisors of arbitrary degrees comparing with the known in the theory of hyperelliptic functions.
Keywords: reduced divisor, inverse divisor, non-special divisor, generalised Jacobi inversion problem. 
@article{SIGMA_2020_16_a52,
     author = {Julia Bernatska and Yaacov Kopeliovich},
     title = {Addition of {Divisors} on {Hyperelliptic} {Curves} via {Interpolation} {Polynomials}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2020},
     volume = {16},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a52/}
}
TY  - JOUR
AU  - Julia Bernatska
AU  - Yaacov Kopeliovich
TI  - Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2020
VL  - 16
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a52/
LA  - en
ID  - SIGMA_2020_16_a52
ER  - 
%0 Journal Article
%A Julia Bernatska
%A Yaacov Kopeliovich
%T Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials
%J Symmetry, integrability and geometry: methods and applications
%D 2020
%V 16
%U http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a52/
%G en
%F SIGMA_2020_16_a52
Julia Bernatska; Yaacov Kopeliovich. Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials. Symmetry, integrability and geometry: methods and applications, Tome 16 (2020). http://geodesic.mathdoc.fr/item/SIGMA_2020_16_a52/

[1] Baker H. F., Abelian functions. Abel's theorem and the allied theory of theta functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[2] Bukhshtaber V. M., Leykin D. V., “Heat equations in a nonholomic frame”, Funct. Anal. Appl., 38 (2004), 88–101 | DOI | MR | Zbl

[3] Bukhshtaber V. M., Leykin D. V., “Addition laws on Jacobians of plane algebraic curves”, Proc. Steklov Inst. Math., 251 (2005), 49–120 | MR | Zbl

[4] Cantor D. G., “Computing in the Jacobian of a hyperelliptic curve”, Math. Comp., 48 (1987), 95–101 | DOI | MR | Zbl

[5] de Jong R., Müller J. S., “Canonical heights and division polynomials”, Math. Proc. Cambridge Philos. Soc., 157 (2014), 357–373, arXiv: 1306.4030 | DOI | MR | Zbl

[6] Gaudry P., “Fast genus 2 arithmetic based on theta functions”, J. Math. Cryptol., 1 (2007), 243–265 | DOI | MR | Zbl

[7] Shaska T., Kopeliovich Y., Additiona laws on Jacobians from a geometric point of view, arXiv: 1907.11070

[8] Shoup V., A computational introduction to number theory and algebra, 2nd ed., Cambridge University Press, Cambridge, 2009 | MR | Zbl

[9] Sutherland A. V., “Fast Jacobian arithmetic for hyperelliptic curves of genus 3”, Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Ser., 2, Math. Sci. Publ., Berkeley, CA, 2019, 425–442, arXiv: 1607.08602 | DOI | MR

[10] Uchida Y., “Division polynomials and canonical local heights on hyperelliptic Jacobians”, Manuscripta Math., 134 (2011), 273–308 | DOI | MR | Zbl