### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 053, 21 pages      arXiv:1912.13277      https://doi.org/10.3842/SIGMA.2020.053

### Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials

Julia Bernatska a and Yaacov Kopeliovich b
a)  National University of Kyiv-Mohyla Academy, 2 Skovorody Str., Kyiv, 04655, Ukraine
b)  University of Connecticut, 2100 Hillside Rd, Storrs Mansfield, 06269, USA

Received February 05, 2020, in final form May 29, 2020; Published online June 14, 2020

Abstract
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.

Key words: reduced divisor; inverse divisor; non-special divisor; generalised Jacobi inversion problem.

pdf (378 kb)   tex (21 kb)

References

1. Baker H.F., Abelian functions. Abel's theorem and the allied theory of theta functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995.
2. Bukhshtaber V.M., Leykin D.V., Heat equations in a nonholomic frame, Funct. Anal. Appl. 38 (2004), 88-101.
3. Bukhshtaber V.M., Leykin D.V., Addition laws on Jacobians of plane algebraic curves, Proc. Steklov Inst. Math 251 (2005), 49-120.
4. Cantor D.G., Computing in the Jacobian of a hyperelliptic curve, Math. Comp. 48 (1987), 95-101.
5. de Jong R., Müller J.S., Canonical heights and division polynomials, Math. Proc. Cambridge Philos. Soc. 157 (2014), 357-373, arXiv:1306.4030.
6. Gaudry P., Fast genus 2 arithmetic based on theta functions, J. Math. Cryptol. 1 (2007), 243-265.
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.
9. Sutherland A.V., Fast Jacobian arithmetic for hyperelliptic curves of genus 3, in Proceedings of the Thirteenth Algorithmic Number Theory Symposium, Open Book Ser., Vol. 2, Math. Sci. Publ., Berkeley, CA, 2019, 425-442, arXiv:1607.08602.
10. Uchida Y., Division polynomials and canonical local heights on hyperelliptic Jacobians, Manuscripta Math. 134 (2011), 273-308.