In a significant development for both algebraic geometry and post-quantum cryptography, researcher Tanush Shaska of Oakland University has released a comprehensive paper resolving the exact probability of $(N,N)$-splitting for superspecial principally polarized abelian surfaces. The paper, titled "Superspecial points on Humbert surfaces and the probability of (N,N)-splitting," was published as Paper 2026/2522 on the Cryptology ePrint Archive. The findings provide rigorous mathematical proofs for phenomena previously understood only through experimental observations or constrained theoretical bounds, with direct implications for the security analysis of cryptographic protocols reliant on higher-dimensional isogenies.
For cryptographers studying the superspecial isogeny problem in dimension two, understanding the distribution and behavior of superspecial points on Humbert surfaces is vital. These geometric structures arise naturally when examining the moduli of abelian surfaces over finite fields. Specifically, the research focuses on prime characteristics $p$ where $p equiv 3 pmod 4$ and an integer $N geq 2$. By examining the proportion of superspecial principally polarized abelian surfaces over the algebraic closure of the finite field with $p$ elements that are optimally $(N,N)$-split, Shaska has successfully moved the field past decades of reliance on heuristics and empirical data.
In the framework of the research, these surfaces are weighted by the reciprocals of their automorphism orders. This specific weighting serves a crucial role because it represents the stationary measure of the Richelot random walk, a fundamental stochastic process used to navigate through sets of abelian surfaces in algorithmic implementations and cryptographic attacks.
The core of the new paper centers on the main theorem established by Shaska. The theorem demonstrates that for a prime $p$ that does not divide $N$, the expected number of optimal $(N,N)$-splittings of a given surface is precisely calculated as $5NJ_2(N)(p-1)/(p^2+1)$, where $J_2$ denotes Jordan’s totient function. Furthermore, the work establishes that for primes $p$ greater than $2N^2$, the second factorial moment takes the form of $c_N/(p^2+1)$, where the constant $c_N$ is entirely independent of the prime $p$. Consequently, for a fixed integer $N$, the overall proportion of split surfaces resolves to $5NJ_2(N)/p$ plus an error term of order $O_N(p^-2)$.
To make these formulas operationally useful for researchers, the specific constants $c_N$ have been explicitly determined for all integers $N$ ranging from $2$ through $7$. This proportion is far from being a purely theoretical abstraction; it directly corresponds to the stationary target density of the splitting-detection attack, a primary analytical tool utilized against the superspecial isogeny problem in dimension two. Prior to this work, the exact value of this target density was known only from empirical laboratory experiments or from a one-sided mathematical bound, leaving a persistent gap in the formal security analysis of these systems.
The methodology deployed by Shaska to achieve these breakthroughs relies on connecting the geometric distribution of superspecial surfaces to classical number theory. The paper shows that superspecial surfaces effectively sample the genus of their refined Humbert invariants, and that they do so in accordance with Siegel’s weights. By establishing this bridge, Shaska makes Siegel’s celebrated theorem on genus averages directly available for the study of superspecial abelian surfaces.
Beyond resolving the primary question of $(N,N)$-splitting probabilities, this innovative methodological approach unlocks several broader mathematical results. The same analytical framework yields the first moment of primitive representations of every discriminant that is prime to $p$, as well as mixed moments spanning several distinct levels. Furthermore, the research delivers an exact probability calculation at level two and provides a formal mathematical proof that the geometric upper bound established in earlier foundational work indeed possesses the correct leading term.
Every result presented in the comprehensive paper is rigorously proved. The enumerations required for the analysis enter the equations strictly through the computed values of the constants $c_N$, and every resulting formula was thoroughly cross-verified and confirmed through exact computational implementations carried out across twelve distinct primes.
The availability of this paper on the Cryptology ePrint Archive provides the academic and cryptographic communities with the definitive proofs needed to solidify the security foundations of isogeny-based cryptographic schemes operating in higher dimensions. As post-quantum cryptography continues to demand rigorous mathematical guarantees for emerging mathematical primitives, Shaska’s resolution of Humbert surface splitting probabilities marks a major milestone for the field.
