Bernoulli numbers and the probability of a birthday surprise

TL;DR

The paper investigates the birthday-surprise probability $\beta^k_n$, the chance that a collision occurs when drawing $k$ items from an $n$-element space. It derives arbitrarily precise bounds by expressing the collision-free probability $\pi^k_n$ through $-\ln(\pi^k_n)=\sum_{m\ge1} \frac{1}{m n^m} \mathfrak p(k-1,m)$ and bounding the tail with $\epsilon^k_n(N)$; here $\mathfrak p(k-1,m)$ are power sums obtained via Bernoulli numbers in Faulhaber’s formula. The main contributions are provable, adjustable bounds that can be computed to any desired accuracy, enabling exact calculations in arbitrary-precision calculators and tight security assessments in cryptography. Practically, the results link classical Bernoulli-number techniques to modern applications in pseudorandom number generation and security, providing precise, non-asymptotic bounds for collision probabilities.

Abstract

A birthday surprise is the event that, given k uniformly random samples from a sample space of size n, at least two of them are identical. We show that Bernoulli numbers can be used to derive arbitrarily exact bounds on the probability of a birthday surprise. This result can be used in arbitrary precision calculators, and it can be applied to better understand some questions in communication security and pseudorandom number generation.