Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials
Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
| ISSN: | 1815-0659 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/210697 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Addition of Divisors on Hyperelliptic Curves via Interpolation Polynomials. Julia Bernatska and Yaacov Kopeliovich. SIGMA 16 (2020), 053, 21 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | Two problems are addressed: reduction of an arbitrary degree non-special divisor to the equivalent divisor of the degree equal to the genus of a curve, and addition of divisors of arbitrary degrees. The hyperelliptic case is considered the simplest model. Explicit formulas defining reduced divisors for some particular cases are found. The reduced divisors are obtained in the form of a solution of the Jacobi inversion problem, which provides a 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 compared with the known in cryptography, and is extended to divisors of arbitrary degrees compared with the known in the theory of hyperelliptic functions.
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| ISSN: | 1815-0659 |