Selmer-Inspired Elliptic Curve Generation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908573797187584 |
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| author | Bhowmik, Awnon |
| author_facet | Bhowmik, Awnon |
| contents | Elliptic curve cryptography (ECC) is foundational to modern secure communication, yet existing standard curves have faced scrutiny for opaque parameter-generation practices. This work introduces a Selmer-inspired framework for constructing elliptic curves that is both transparent and auditable. Drawing from $2$- and $3$-descent methods, we derive binary quartics and ternary cubics whose classical invariants deterministically yield candidate $(c_4,c_6)$ parameters. Local solubility checks, modeled on Selmer admissibility, filter candidates prior to reconciliation into short-Weierstrass form over prime fields. We then apply established cryptographic validations, including group-order factorization, cofactor bounds, twist security, and embedding-degree heuristics. A proof-of-concept implementation demonstrates that the pipeline functions as a retry-until-success Las Vegas algorithm, with complete transcripts enabling independent verification. Unlike seed-based or purely efficiency-driven designs, our approach embeds arithmetic structure into parameter selection while remaining compatible with constant-time, side-channel resistant implementations. This work broadens the design space for elliptic curves, showing that descent techniques from arithmetic geometry can underpin trust-enhancing, standardization-ready constructions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_02383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Selmer-Inspired Elliptic Curve Generation Bhowmik, Awnon Cryptography and Security Number Theory 94A60 Elliptic curve cryptography (ECC) is foundational to modern secure communication, yet existing standard curves have faced scrutiny for opaque parameter-generation practices. This work introduces a Selmer-inspired framework for constructing elliptic curves that is both transparent and auditable. Drawing from $2$- and $3$-descent methods, we derive binary quartics and ternary cubics whose classical invariants deterministically yield candidate $(c_4,c_6)$ parameters. Local solubility checks, modeled on Selmer admissibility, filter candidates prior to reconciliation into short-Weierstrass form over prime fields. We then apply established cryptographic validations, including group-order factorization, cofactor bounds, twist security, and embedding-degree heuristics. A proof-of-concept implementation demonstrates that the pipeline functions as a retry-until-success Las Vegas algorithm, with complete transcripts enabling independent verification. Unlike seed-based or purely efficiency-driven designs, our approach embeds arithmetic structure into parameter selection while remaining compatible with constant-time, side-channel resistant implementations. This work broadens the design space for elliptic curves, showing that descent techniques from arithmetic geometry can underpin trust-enhancing, standardization-ready constructions. |
| title | Selmer-Inspired Elliptic Curve Generation |
| topic | Cryptography and Security Number Theory 94A60 |
| url | https://arxiv.org/abs/2510.02383 |