Hash Collision Probability: From the Birthday Paradox to Approximations

This article delves into the probability of hash collisions. Hash functions map arbitrarily complex inputs to single numbers, but there's a risk of hash collisions (different inputs mapping to the same number). Starting with the Birthday Paradox, the article explains the exact formula for calculating hash collision probability and three approximation methods: exponential approximation, simplified approximation, and a further simplified approximation. Through comparison, the exponential approximation performs best in most cases, while the other two are more suitable for quick estimations. The article also provides mathematical proofs supporting the approximation methods.