On the Goppa morphism
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916593230938112 |
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| author | Castañeda, Ángel Luis Muñoz |
| author_facet | Castañeda, Ángel Luis Muñoz |
| contents | We investigate the geometric foundations of the space of geometric Goppa codes using the Tsfasman-Vladut H-construction. These codes are constructed from level structures, which extend the classical Goppa framework by incorporating invertible sheaves and their trivializations over rational points. A key contribution is the definition of the Goppa morphism, a map from the universal moduli space of level structures, denoted $LS_{g,n,d}$, to certain Grassmannian $\mathrm{Gr}(k,n)$. This morphism allows problems related to distinguishing attacks and key recovery in the context of Goppa Code-based Cryptography to be translated into a geometric language, addressing questions about the equations defining the image of the Goppa morphism and its fibers. Furthermore, we identify the ranges of the degree parameter $d$ that should be avoided to maintain security against distinguishers. Our results, valid over arbitrary base fields, also apply to convolutional Goppa codes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_19088 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Goppa morphism Castañeda, Ángel Luis Muñoz Algebraic Geometry Information Theory 14L24, 13A50, 13A02 We investigate the geometric foundations of the space of geometric Goppa codes using the Tsfasman-Vladut H-construction. These codes are constructed from level structures, which extend the classical Goppa framework by incorporating invertible sheaves and their trivializations over rational points. A key contribution is the definition of the Goppa morphism, a map from the universal moduli space of level structures, denoted $LS_{g,n,d}$, to certain Grassmannian $\mathrm{Gr}(k,n)$. This morphism allows problems related to distinguishing attacks and key recovery in the context of Goppa Code-based Cryptography to be translated into a geometric language, addressing questions about the equations defining the image of the Goppa morphism and its fibers. Furthermore, we identify the ranges of the degree parameter $d$ that should be avoided to maintain security against distinguishers. Our results, valid over arbitrary base fields, also apply to convolutional Goppa codes. |
| title | On the Goppa morphism |
| topic | Algebraic Geometry Information Theory 14L24, 13A50, 13A02 |
| url | https://arxiv.org/abs/2411.19088 |