New Research Solves Longstanding Mystery in Superspecial Isogeny Cryptography and Humbert Surface Geometry

In a significant development for both algebraic geometry and post-quantum cryptography, researcher Tanush Shaska of Oakland University has released a comprehensive theoretical study resolving the exact probabilities associated with $(N,N)$-splitting on superspecial principally polarized abelian surfaces. Published as Paper 2026/2522 on the Cryptology ePrint Archive, the work establishes rigorous first-order asymptotics and explicit error bounds that replace decades-old empirical estimates and one-sided bounds.

The paper, titled "Superspecial points on Humbert surfaces and the probability of (N,N)-splitting," targets an intricate intersection of arithmetic geometry and cryptographic security analysis. Specifically, the research derives the proportion of superspecial principally polarized abelian surfaces over the algebraic closure of finite fields of characteristic $p$—where $p$ is a prime congruent to 3 modulo 4—that are optimally $(N,N)$-split for a given integer $N geq 2$.

For cryptographers and mathematicians alike, these findings carry direct implications for the security analysis of isogeny-based cryptosystems in dimension two. The proportion of split surfaces calculated in the study serves as the stationary target density of the splitting-detection attack on the superspecial isogeny problem in dimension two. Until now, this crucial metric was known only through experimental computations or restricted, one-sided bounds, leaving a theoretical gap in the evaluation of cryptographic hardness assumptions.

At the heart of Shaska’s mathematical framework is a careful weighting of the surfaces. The surfaces under investigation are weighted by the reciprocals of their automorphism orders, which corresponds precisely to the stationary measure of the Richelot random walk. Utilizing this weighted framework, the main theorem of the paper demonstrates that for primes $p$ not dividing $N$, the expected number of optimal $(N,N)$-splittings of a surface is given by a precise formula: $5NJ_2(N)(p-1)/(p^2+1)$, where $J_2$ denotes Jordan’s totient function.

Expanding upon the first moment, the research advances into second factorial moments to establish strict variance and proportion boundaries. The paper establishes that for primes $p$ strictly greater than $2N^2$, the second factorial moment takes the form of $c_N/(p^2+1)$, featuring a constant $c_N$ that remains entirely independent of the prime $p$. Consequently, for a fixed integer $N$, the overall proportion of split surfaces is proven to be $5NJ_2(N)/p$ with an error term of order $O_N(p^-2)$. To ground these theoretical constants in concrete data, Shaska determines the explicit values of $c_N$ for small integers ranging from $N = 2$ up to $N = 7$.

The methodology driving these proofs bridges classical modular arithmetic with advanced geometric principles. Shaska demonstrates that superspecial surfaces effectively sample the genus of their refined Humbert invariants according to Siegel’s weights. This crucial analytical connection makes Siegel’s celebrated theorem on genus averages directly applicable to the problem, unlocking a pathway to exact evaluations that were previously inaccessible through standard counting techniques.

Beyond the primary results concerning $(N,N)$-splitting, this methodological innovation yields a suite of secondary breakthroughs. The same analytical approach successfully provides the first moment of primitive representations for every discriminant coprime to $p$, calculates mixed moments spanning several distinct levels, establishes an exact probability value specifically at level $2$, and delivers a formal proof verifying that the geometric upper bound identified in previous scholarly work indeed possesses the correct leading term.

Every theoretical claim and derived formula within the paper has been rigorously validated. While the broader enumerations enter the framework strictly through the computed values of the constants $c_N$, every single formula presented in the study was additionally confirmed by exact, high-precision computations carried out across twelve distinct primes.

The availability of this paper on the Cryptology ePrint Archive provides the academic and cryptographic communities with open access to the complete proofs and formal BibTeX documentation necessary for further integration into ongoing research initiatives.