Group law on affine conics and applications to cryptography
| Autor: | Nadir Murru, Michele Elia, Emanuele Bellini, Antonio J. Di Scala |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
0209 industrial biotechnology
Computer science Group (mathematics) business.industry Applied Mathematics 020206 networking & telecommunications Rational functions 02 engineering and technology Rational function Encryption Algebra Computational Mathematics Elliptic curve 020901 industrial engineering & automation Finite field Quadratic equation Conic section Groups over curves Algorithms Finite fields Public key cryptography 0202 electrical engineering electronic engineering information engineering Affine transformation business |
| Zdroj: | Applied Mathematics and Computation. 409:125537 |
| ISSN: | 0096-3003 |
| DOI: | 10.1016/j.amc.2020.125537 |
| Popis: | In this paper, we highlight that the point group structure of elliptic curves, over finite or infinite fields, may be also observed on reducible cubics with an irreducible quadratic component. Starting from this, we introduce in a very general way a group’s structure over any kind of conic. In the case of conics over finite fields, we see that the point group is cyclic and lies on the quadratic component. Thanks to this, some applications to cryptography are described, considering convenient parametrizations of the conics. We perform an evaluation of the complexity of the operations involved in the parametric groups and consequently in the cryptographic applications. In the case of the hyperbolas, the Redei rational functions can be used for performing the operations of encryption and decryption, and the More’s algorithm can be exploited for improving the time costs of computation. Finally, we provide also an improvement of the More’s algorithm. |
| Databáze: | OpenAIRE |
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