What Is the “Birthday Paradox” in the Context of Cryptographic Hash Collisions?

The birthday paradox demonstrates that the probability of a collision (two people sharing a birthday) becomes surprisingly high with a relatively small group size. In cryptography, this means the number of attempts needed to find a hash collision is much lower than expected.

For an N-bit hash, a collision is likely after approximately 2^(N/2) attempts, not 2^N. This principle dictates that hash functions must use a sufficiently long output (like 256 bits) to keep collision finding computationally infeasible.