What is the birthday problem in hashing?

The birthday problem asks: how many randomly chosen values do you need before two share the same hash output? With a hash space of H = 2^b values, the probability of at least one collision among n items is approximately P ≈ 1 − e^(−n² / 2H). It grows surprisingly fast — you only need √H items to reach ~50% probability.

What formula does this tool use?

P ≈ 1 − e^(−n² / 2H), where H = 2^bits is the total hash space and n is the number of items. This is the standard birthday-problem approximation, accurate when n ≪ H. For n close to H the true probability approaches 1 and the approximation still holds well.

How many items does it take to reach 50% collision probability?

Solving for P = 0.5 gives n₅₀ ≈ √(2H × ln 2) ≈ 1.177 × √H. For a 256-bit hash, that is about 1.19 × 10^38 items — effectively impossible. For a 64-bit hash, only about 5.1 billion items are needed.

Why does SHA-1 (160 bits) worry cryptographers?

A birthday attack against SHA-1 requires roughly 2^80 hash evaluations to find a collision, which is about 1.2 × 10^24 operations. While not trivial, dedicated hardware and algorithmic improvements have made chosen-prefix collisions practical (demonstrated in 2020). SHA-256 raises the bar to 2^128, far beyond current capabilities.

Is this calculator accurate for large bit counts?

The tool uses log-space arithmetic to avoid floating-point overflow, so results are numerically stable even for 512-bit hash spaces. When probability rounds to 0.00% it means the collision risk is negligible (below 0.005%), not exactly zero.

Hash Collision Probability

Name: Hash Collision Probability
Availability: InStock
Author: Nham Vu

Enter the hash size in bits and item count to get the collision probability via the birthday-problem approximation.

Parameters

Hash size (bits)

Common: MD5 = 128, SHA-1 = 160, SHA-256 = 256, SHA-512 = 512

Number of items (n)

How many distinct hashes exist in your dataset

Collision Probability

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0%25%50%75%100%

Enter values above

Hash space

—

Items (n)

—

n / √H ratio

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Details

50% collision threshold (≈ 1.177 × √H) —

1% collision threshold —

Expected collisions at n items —

Birthday attack cost (ops to 50%) —

Summary

Enter the hash size in bits and item count to get the collision probability via the birthday-problem approximation.

How it works

Enter the hash output size in bits (e.g. 256 for SHA-256, 128 for MD5).
Enter the number of items (hashes) that will be in your dataset.
The tool computes the hash space H = 2^bits and applies the approximation P ≈ 1 − e^(−n² / 2H).
Review the collision probability, expected first-collision threshold, and safety margin.
Adjust either value to explore different scenarios interactively.

Use cases

Assess collision risk when choosing a hash length for a database or cache key scheme.
Understand why SHA-1 (160 bits) is considered weak at scale vs SHA-256 (256 bits).
Estimate when birthday attacks become practical against a given hash function.
Teach the birthday paradox in cryptography or probability courses.
Verify that a UUID or random token space is large enough for your expected volume.
Compare 32-bit, 64-bit, and 128-bit hash spaces for non-cryptographic hashing.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu