Detecting split surfaces, RM surfaces, and minimum walks on isogeny graphs
Abstract
We study the detectability of split surfaces in isogeny graphs of principally polarized superspecial abelian surfaces, a question relevant to the security analysis of dimension-$2$ isogeny-based cryptography. Our approach uses refined Humbert invariants to replace explicit isogeny computations with primitive representation problems for positive definite quadratic forms in five variables. We develop algorithms to detect (N,N)-splittings and to compute the minimum (N,N)-splitting level of a superspecial Jacobian without constructing the corresponding isogeny path. The same framework detects embeddings of real multiplication (RM) orders through primitive representations of discriminants of real quadratic orders. For RM, we use exhaustive small-prime data together with 100,000 random polarizations per prime for 227≤ p≤1619, testing primitive representations of square-free discriminants D≤100; the first nontrivial primitively represented discriminant is typically small. We apply these methods experimentally in two regimes. For 11≤ p≤251, where the irreducible principal polarizations are known exhaustively, the automorphism data recovered from the refined Humbert invariant reproduces the counts of Ibukiyama--Katsura--Oort for the irreducible polarizations. For large parameters, we reach primes of $1000$ bits, where the splitting degrees produced satisfy log2N≈log2p with a distribution whose shape is stable across the whole range, so the detected (N,N)-isogeny has degree about p2; the smallest observed largest prime-power divisor of N remains small up to $250$ bits and increases sharply from $300$ bits onward.
Related
- A uniform bound for the minimal degree of an isogeny between principally polarized superspecial abelian surfaces
- Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields
- On supersingular isogeny graphs of Drinfeld modules
- Generalized Spectral Closedness and the Nonexistence of Walk-Realizable Supporters for Split Graphs
- Minimum Weakly Saturated Graphs and Bootstrap Percolation in General Host Graphs
BibTeX
@misc{kırımlı2026detecting,
title = {{Detecting split surfaces, RM surfaces, and minimum walks on isogeny graphs}},
author = {Eda Kırımlı and Gaurish Korpal},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23941},
eprint = {2609.23941},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23941},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Detecting split surfaces, RM surfaces, and minimum walks on isogeny graphs
AU - Eda Kırımlı
AU - Gaurish Korpal
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23941
UR - https://arxiv.org/abs/2609.23941
AB - We study the detectability of split surfaces in isogeny graphs of principally polarized superspecial abelian surfaces, a question relevant to the security analysis of dimension-$2$ isogeny-based cryptography. Our approach uses refined Humbert invariants to replace explicit isogeny computations with primitive representation problems for positive definite quadratic forms in five variables. We develop algorithms to detect (N,N)-splittings and to compute the minimum (N,N)-splitting level of a superspecial Jacobian without constructing the corresponding isogeny path. The same framework detects embeddings of real multiplication (RM) orders through primitive representations of discriminants of real quadratic orders. For RM, we use exhaustive small-prime data together with 100,000 random polarizations per prime for 227≤ p≤1619, testing primitive representations of square-free discriminants D≤100; the first nontrivial primitively represented discriminant is typically small. We apply these methods experimentally in two regimes. For 11≤ p≤251, where the irreducible principal polarizations are known exhaustively, the automorphism data recovered from the refined Humbert invariant reproduces the counts of Ibukiyama--Katsura--Oort for the irreducible polarizations. For large parameters, we reach primes of $1000$ bits, where the splitting degrees produced satisfy log_2N≈log_2p with a distribution whose shape is stable across the whole range, so the detected (N,N)-isogeny has degree about p^2; the smallest observed largest prime-power divisor of N remains small up to $250$ bits and increases sharply from $300$ bits onward.
ER -