This chapter reviews the characteristics of the main digest algorithms, and presents a new derivation of the leftover hash lemma, using the collision probability to derive an upper bound on the statistical distance between the key and seed joint probability, and the hash bit sequence distribution. The ideal cryptographic hash function has some desired properties. It is a deterministic function, that is, identical messages result in the same hash. The hash function creates a unidirectional process, that makes it impossible to guess the original contents of a file based only on the message digest. The chapter discusses the use of the hash function in cryptography and presented a new derivation of the upper bound on the statistical distance between the joint distribution of the key and the seed and the distribution of the hash bit distribution based on the collision probability.