First-hit analysis of algorithms for computing quadratic irregularity
Joshua Holden
TL;DR
The paper extends the theory of regular and irregular primes to real quadratic fields, introducing a first-hit analysis to search for large primes $p$ and discriminants $D$ with $p$ dividing $\zeta_D(1-2m)$ or related $L$-values. It relies on Siegel-type equidistribution heuristics, positing a product distribution for numerators of $\zeta_k(1-2m)$ across characters and providing two algorithmic strategies with predicted subexponential times in $D$ and $m$, augmented by a guard against small numerator pitfalls. The work presents both theoretical predictions and empirical data (chi-squared tests) for quadratic and quasi-quadratic settings, offering partial support for the hypothesized distributions and outlining extensions to general totally real fields. It also discusses cryptographic applications via class groups and the practical constraints (storage and time) involved in realizing such groups, motivating further large-scale data collection and implementation work. Overall, the study advances understanding of quadratic irregularity, offers concrete first-hit timing predictions, and lays groundwork for cryptographic use of number-field class groups pending further validation.
Abstract
The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are evenly distributed modulo p for every p. This is the basis of a well-known heuristic given by Siegel for estimating the frequency of irregular primes. So far, analyses have shown that if Q(\sqrt{D}) is a real quadratic field, then the values of the zeta function ζ_{D}(1-2m)=ζ_{Q(\sqrt{D})}(1-2m) at negative odd integers are also distributed as expected modulo p for any p. We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.