Aspects relate to systems and methods for implementing a hash function using a stochastic and recurrent process, and performing arithmetic operations during the recurrence on portions of a message being hashed. In an example method, the stochastic process is a Galton-Watson process, the message is decomposed into blocks, and the method involves looping for a number of blocks in the message. In each loop, a current hash value is determined based on arithmetic performed on a previous hash value and some aspect of a current block. The arithmetic performed can involve modular arithmetic, such as modular addition and exponentiation. The algorithm can be adjusted to achieve qualities including a variable length output, or to perform fewer or more computations for a given hash. Also, randomizing elements can be introduced into the arithmetic, avoiding a modular reduction until final hash output production.