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PreprintCC BY-SA 4.0arXiv

Supersingularity and Superspeciality Verification of Abelian Surfaces

  • Maria Corte-Real Santos
  • Gioella Lorenzon
  • Krijn Reijnders
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Abstract

Supersingular abelian surfaces are essential in isogeny-based cryptography. Despite this, we have no efficient algorithm to verify if a given abelian surface is supersingular. In this work, we initiate this research topic by giving an efficient Monte Carlo algorithm to verify if an abelian surface over 𝔽p is supersingular in O(log p) with negligible failure probability, and an efficient conclusive algorithm if the order is smooth. We derive this algorithm by a careful analysis on the structure of supersingular Jacobians over 𝔽p. Furthermore, we derive efficient algorithms to verify if an abelian variety of any dimension is minimal or maximal, and to verify if a Jacobian of any dimension is superspecial.

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Cite

BibTeXRIS
BibTeX
@misc{santos2026supersingularity,
  title = {{Supersingularity and Superspeciality Verification of Abelian Surfaces}},
  author = {Maria Corte-Real Santos and Gioella Lorenzon and Krijn Reijnders},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.01924},
  eprint = {2610.01924},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.01924},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Supersingularity and Superspeciality Verification of Abelian Surfaces
AU  - Maria Corte-Real Santos
AU  - Gioella Lorenzon
AU  - Krijn Reijnders
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.01924
UR  - https://arxiv.org/abs/2610.01924
AB  - Supersingular abelian surfaces are essential in isogeny-based cryptography. Despite this, we have no efficient algorithm to verify if a given abelian surface is supersingular. In this work, we initiate this research topic by giving an efficient Monte Carlo algorithm to verify if an abelian surface over 𝔽_p is supersingular in O(log p) with negligible failure probability, and an efficient conclusive algorithm if the order is smooth. We derive this algorithm by a careful analysis on the structure of supersingular Jacobians over 𝔽_p. Furthermore, we derive efficient algorithms to verify if an abelian variety of any dimension is minimal or maximal, and to verify if a Jacobian of any dimension is superspecial.
ER  -