In a significant development for cryptographic research and algebraic geometry, mathematician Tanush Shaska of Oakland University has released a comprehensive new study addressing longstanding theoretical questions regarding superspecial points on Humbert surfaces. The paper, designated as Paper 2026/2522 on the Cryptology ePrint Archive, establishes precise first-order determinations and explicit error bounds for the proportion of superspecial principally polarized abelian surfaces over algebraic closures of finite fields that exhibit optimal $(N,N)$-splitting.
For years, cryptographers and number theorists have sought exact formulations for these mathematical structures, which play a critical role in evaluating the security foundations of specific post-quantum cryptographic schemes. Previously, the proportion of such split surfaces was known only through targeted empirical experiments or from one-sided theoretical bounds. Shaska’s latest work bridges this gap by providing rigorous mathematical proofs that confirm the behavior of these geometric objects, offering precise insights into the stationary target density utilized in advanced cryptographic cryptanalysis.
The research focuses on the properties of abelian surfaces defined over the algebraic closure of a finite field of characteristic $p$, where $p$ is a prime congruent to $3$ modulo $4$, alongside an integer $N$ greater than or equal to $2$. Within this framework, the surfaces are systematically weighted by the reciprocals of their respective automorphism orders. This specific weighting scheme corresponds directly to the stationary measure of the Richelot random walk, a classical stochastic process frequently analyzed in the context of modular curves and isogeny graphs.
According to the paper’s main theorem, for primes $p$ that do not divide $N$, the expected number of optimal $(N,N)$-splittings of a given surface can be expressed through a precise formula involving Jordan’s totient function. Specifically, the expectation is calculated as five times $N$, multiplied by Jordan’s totient $J_2(N)$, multiplied by $p-1$, all divided by the sum of $p^2$ and $1$. Furthermore, the research demonstrates that when $p$ is strictly greater than $2N^2$, the second factorial moment is governed by a constant divided by $p^2+1$, with this constant remaining entirely independent of the prime $p$ itself.
Consequently, for any fixed integer $N$, the overall proportion of split surfaces scales inversely with the prime, yielding a leading term dependent on $N$ and $J_2(N)$ divided by $p$, alongside a controlled error term of second order. To ground these theoretical derivations in concrete values, the specific constants governing the second moments have been determined explicitly for all integer values of $N$ ranging from $2$ through $7$.
These mathematical formulations carry direct implications for modern cryptographic security evaluations. In particular, the proportion identified in Shaska’s work serves precisely as the stationary target density for the splitting-detection attack strategy targeted at the superspecial isogeny problem operating in dimension two. Understanding the exact statistical distribution of these split surfaces allows cryptographers to model the complexity and success probability of such attacks with unprecedented theoretical accuracy, moving beyond mere empirical observation.
The methodological backbone of the new research relies on establishing a profound connection between the arithmetic of superspecial surfaces and classical analytic number theory. Shaska proves that superspecial surfaces effectively sample the genus of their refined Humbert invariants in alignment with Siegel’s classical weights. This critical link successfully brings Siegel’s powerful theorem concerning genus averages into the domain of modern superspecial isogeny graphs.
By making Siegel’s theorem available for these geometric structures, the methodology yields several secondary breakthroughs. Alongside the primary results concerning $(N,N)$-splitting, the same overarching technique successfully delivers the first moment of primitive representations for every discriminant coprime to $p$, provides mixed moments spanning multiple levels, establishes an exact probability evaluation specifically at level $2$, and offers a definitive proof that the geometric upper bounds suggested in earlier literature indeed possess the correct leading asymptotic term.
Rigorous verification accompanied the theoretical developments. All analytical results presented in the paper are fully proved, with enumerations entering the framework solely through the explicit determination of the constants. To ensure absolute reliability, every mathematical formula derived in the study was independently verified and confirmed through exact computational evaluations carried out across twelve distinct primes.
Researchers interested in examining the full mathematical proofs, technical derivations, and complete bibliographic references can access the complete manuscript, titled "Superspecial points on Humbert surfaces and the probability of (N,N)-splitting," directly through the Cryptology ePrint Archive under paper identifier 2026/2522.
