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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.19268 |
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| _version_ | 1866911267259678720 |
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| author | Shaska, Elira Mello, Jorge Salami, Sajad Shaska, Tony |
| author_facet | Shaska, Elira Mello, Jorge Salami, Sajad Shaska, Tony |
| contents | This paper examines the arithmetic of the loci \(\cL_n\), parameterizing genus 2 curves with \((n, n)\)-split Jacobians over finite fields \(\F_q\). We compute rational points \(|\cL_n(\F_q)|\) over \(\F_3\), \(\F_9\), \(\F_{27}\), \(\F_{81}\), and \(\F_5\), \(\F_{25}\), \(\F_{125}\), derive zeta functions \(Z(\cL_n, t)\) for \(n = 2, 3\). Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_19268 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant Shaska, Elira Mello, Jorge Salami, Sajad Shaska, Tony Number Theory This paper examines the arithmetic of the loci \(\cL_n\), parameterizing genus 2 curves with \((n, n)\)-split Jacobians over finite fields \(\F_q\). We compute rational points \(|\cL_n(\F_q)|\) over \(\F_3\), \(\F_9\), \(\F_{27}\), \(\F_{81}\), and \(\F_5\), \(\F_{25}\), \(\F_{125}\), derive zeta functions \(Z(\cL_n, t)\) for \(n = 2, 3\). Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics. |
| title | Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.19268 |