Hash Collision Probability Calculator

Estimate the probability of at least one hash collision (birthday problem) given the number of possible hash values and the number of items hashed.

Hash Size (bits) Common values: 32, 64, 128, 160, 256, 512

Number of Items (n) How many distinct items are hashed

Preset Hash Functions

Fill in the fields and click Calculate.

Birthday Problem (Exact):

P(collision) = 1 − ∏_k=0ⁿ⁻¹ (1 − k / H)

where H = 2^b is the number of possible hash values for a b-bit hash.

Approximation (accurate when n ≪ H):

P(collision) ≈ 1 − e^{−n(n−1) / (2H)}

Items needed for 50% collision probability:

n_50% ≈ √(2H · ln 2) ≈ 1.1774 · √H

Expected number of colliding pairs:

E[collisions] ≈ n(n−1) / (2H)

Hash outputs are assumed to be uniformly distributed across all 2^b possible values.
The birthday approximation P ≈ 1 − e^−n²/(2H) is used for numerical stability with large hash spaces; it is accurate to within 1% when n ≤ 0.1·√H.
Calculations are performed in log-space to handle hash sizes up to 512 bits without floating-point overflow.
A "collision" means at least two of the n items share the same hash value.
This does not account for cryptographic weaknesses or non-uniform distributions in real hash functions.
References: Flajolet & Odlyzko (1990), "Random Mapping Statistics"; NIST FIPS 180-4 (SHA family); Knuth, The Art of Computer Programming, Vol. 2, §6.4.

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