New Research Resolves Longstanding Probabilities in Superspecial Isogeny Cryptography and Humbert Surfaces

In a significant development for algebraic geometry and cryptographic research, Tanush Shaska of Oakland University has released a new paper addressing the theoretical foundations of superspecial principally polarized abelian surfaces. Published in the Cryptology ePrint Archive as Paper 2026/2522, the work, titled "Superspecial points on Humbert surfaces and the probability of (N,N)-splitting," provides rigorous mathematical proofs for phenomena previously understood only through empirical observation, heuristic arguments, or one-sided bounds.

The research focuses on the arithmetic of abelian surfaces defined over algebraic closures of finite fields, specifically targeting characteristics where a prime $p$ satisfies the congruence $p equiv 3 pmod 4$. For an integer $N geq 2$, the paper determines—to the first order in $1/p$ and accompanied by an explicit error bound—the exact proportion of superspecial principally polarized abelian surfaces that are optimally $(N,N)$-split.

Within this framework, surfaces are weighted by the reciprocals of their automorphism orders, a formulation that corresponds directly to the stationary measure of the Richelot random walk. This measure plays a critical role in evaluating algorithmic processes and security assumptions in dimension-two isogeny-based cryptography, where random walks on modular curves and related surfaces form the backbone of several advanced cryptographic protocols.

At the core of the new paper is a powerful main theorem establishing the expected behavior of these geometric structures. According to the research, for primes where $p$ does not divide $N$, the expected number of optimal $(N,N)$-splittings of a given surface is exactly $5NJ_2(N)(p-1)/(p^2+1)$, where $J_2$ denotes Jordan’s totient function. Building upon this expectation, the paper demonstrates that for primes strictly greater than twice the square of $N$ ($p > 2N^2$), the second factorial moment evaluates to $c_N/(p^2+1)$, with the constant $c_N$ remaining entirely independent of the characteristic $p$.

Consequently, for a fixed integer $N$, the overall proportion of split surfaces is shown to be $5NJ_2(N)/p + O_N(p^-2)$. To ground these theoretical derivations in concrete values, the constants $c_N$ are explicitly determined for all integers $N$ from 2 through 7. This proportion is far from a purely academic abstraction; it represents the stationary target density of the splitting-detection attack directed against the superspecial isogeny problem in dimension two. Until now, cryptographers and mathematicians relied primarily on experimental data or one-sided bounds to estimate these densities when analyzing the security parameters of isogeny-based cryptographic schemes.

The methodological breakthrough driving these results relies on a deep connection to classical arithmetic geometry. Shaska demonstrates that superspecial surfaces effectively sample the genus of their refined Humbert invariants in accordance with Siegel’s weights. By establishing this distribution, the author makes Siegel’s celebrated theorem on genus averages directly applicable to the problem, opening new analytical avenues for studying modular surfaces in positive characteristic.

Beyond the primary findings regarding $(N,N)$-splittings, the exact same methodological framework yields several important collateral results. The paper successfully derives the first moment of primitive representations of every discriminant coprime to $p$, calculates mixed moments spanning multiple levels, establishes an exact probability value specifically at level 2, and provides a formal mathematical proof that the geometric upper bound established in previous literature indeed possesses the correct leading term.

To ensure the utmost rigor and reliability of the theoretical framework, all propositions and theorems in the paper are fully proved. The enumerations required for the analysis enter the equations strictly through the computed values of the constants $c_N$. Furthermore, every analytical formula derived in the study was rigorously cross-verified and confirmed through exact computations performed at twelve distinct prime numbers, bridging abstract algebraic geometry with concrete computational validation.