Asymptotic results for the number of Wagner's solutions to a generalised birthday problem

TL;DR

This paper analyzes the asymptotic behavior of the number of solutions to a generalized birthday problem and the solutions detected by Wagner's algorithm, providing limit theorems and bounds for their distributions as parameters grow large.

Contribution

It introduces new asymptotic results and bounds for the distributions of solutions in a generalized birthday problem and Wagner's algorithm, extending understanding of their probabilistic behavior.

Findings

Asymptotic ratio of solutions detected by Wagner's algorithm to total solutions

Chen-Stein bounds for total variation distance between distributions and Poisson

Limit theorems describing the behavior of solution counts as M approaches infinity

Abstract

We study two functionals of a random matrix $\boldsymbol A$ with independent elements uniformly distributed over the cyclic group of integers $\{0,1,\ldots, M-1\}$ modulo $M$. One of them, $V_0(\boldsymbol A)$ with mean $\mu$, gives the total number of solutions for a generalised birthday problem, and the other, $W(\boldsymbol A)$ with mean $\lambda$, gives the number of solutions detected by Wagner's tree based algorithm. We establish two limit theorems. Theorem 2.1 describes an asymptotical behaviour of the ratio $\lambda/\mu$ as $M\to\infty$. Theorem 2.2 suggests Chen-Stein bounds for the total variation distance between Poisson distribution and distributions of $V_{0}$ and $W$.