<BIRTHDAY PARADOX> shows that in a group of just 23 people, there is about a 50% chance that two people share the same birthday. This happens because the number of possible pairs of comparisons grows rapidly as the group size increases, making collisions (shared birthdays) much more likely than intuitively expected. Application in Cryptography: Birthday Attack In cryptography, hash functions are used to generate fixed-size outputs (hash values) from variable-length inputs. A key property of hash functions is collision resistance, meaning it should be hard to find two different inputs that produce the same hash value (a collision). The birthday paradox helps attackers exploit this property. Instead of trying every possible input (as in brute force attacks), attackers use the statistical principle behind the paradox to reduce the effort needed to find collisions. For example: If a hash function generates a 128-bit hash, brute force would require attempts to guarantee finding a collision. Why It Matters Birthday attacks can be used to: Forge digital signatures by creating two different messages with the same hash value. Break systems relying on weak hash functions, compromising data integrity and security. To prevent birthday attacks, cryptographic systems use strong hash functions with large output sizes (e.g., SHA-3) and other techniques to make collisions computationally infeasible.