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Memetic Search for Supersingular Elliptic Curves over $\mathbb{F}_p$ — arXiv:2609.03249

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.03249
License (as archived): CC0-1.0
Archived: 2026-10-17T02:03:58.200Z
SHA-256 of the archived text: 54b5cd7e8c2c030b63ed0f715d14e3e280dbd3c245407af9128c68cad7bfc83e
Stable URL of this page: https://metraipolis.com/documento/dc_e62fed387ebf

How to cite

Memetic Search for Supersingular Elliptic Curves over $\mathbb{F}_p$ — arXiv:2609.03249 — arXiv (en), https://arxiv.org/abs/2609.03249 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_e62fed387ebf (Metraipolis archive dc_e62fed387ebf, sha256 54b5cd7e8c2c030b63ed0f715d14e3e280dbd3c245407af9128c68cad7bfc83e)

BibTeX

@misc{dce62fed387ebf,
  title = {Memetic Search for Supersingular Elliptic Curves over $mathbbF_p$ — arXiv:2609.03249},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.03249},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_e62fed387ebf; sha256 54b5cd7e8c2c030b63ed0f715d14e3e280dbd3c245407af9128c68cad7bfc83e},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.03249. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The search for supersingular elliptic curves is a fundamental computational problem in isogeny-based cryptography. A recent metaheuristic formulation over $\mathbb{F}_{p^2}$ introduced the NonMultiplicity Distance (NMD) objective, measuring the deviation of the Frobenius trace from a multiple of $p$, and showed that uninformed random search fails beyond $\approx 10^{13}$ candidates. This work investigates metaheuristic search over the prime field $\mathbb{F}_p$, the setting for oriented isogeny protocols such as CSIDH, OSIDH, and SQISign. Although the candidate space decreases from $p^2$ to $p$, the supersingular locus is asymptotically sparse ($O(\sqrt{p}\log p)$ curves), keeping the search exponentially difficult. We formulate a memetic algorithm tailored to $\mathbb{F}_p$ using a one-dimensional $j$-invariant chromosome, bit-level recombination, adaptive mutation, and periodic local search under the NMD objective. Benchmarks across 30 independent seeds at 40-bit, 46-bit, and 51-bit prime sizes ($p \approx 1.13\times 10^{15}$) show that the algorithm discovers an exact supersingular curve at 46 bits and consistently converges to ``near-supersingular'' ordinary curves with Frobenius traces remarkably close to zero: best NMD values of 19 at 40 bits and 3 at 51 bits, corresponding to relative trace deviations of $1.3\times 10^{-5}$ and $4.5\times 10^{-8}$ across the Hasse interval. These results demonstrate that NMD-driven memetic search effectively navigates the sparse $\mathbb{F}_p$ landscape and systematically locates near-supersingular structures.

Tema degli esploratori: fiducia digitale e crittografia (retro-catalogo arXiv: annunci del 2026-09-04 in cs.CR).
Categorie arXiv: Cryptography and Security (cs.CR).
Pagina del lavoro: https://arxiv.org/abs/2609.03249 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T01:28:02.796Z dalla pagina https://arxiv.org/catchup/cs.CR/2026-09-04?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).