Zobrist, A. L. (1970). "A New Hashing Method with Applications for Game Playing."
To achieve order invariance, we typically use algebraic operations that are and associative . Additive Hashing: Assign a hash to each element. The multiset hash is: Multiplicative Hashing: Zobrist, A
Useful for incremental updates. If you add an element to the multiset, you simply update the hash with the new element’s hash using the group operation ( 6. Security and Collisions "A New Hashing Method with Applications for Game Playing
Group Actions and Hashing Unordered Multisets: An Algebraic Approach to Data Integrity 1. Introduction If you add an element to the multiset,
This topic explores a fascinating intersection: how to use group theory to create hash functions for multisets where the order of elements doesn't matter, but their frequency does.
Unlike sets, multisets allow for multiple instances of the same element. A multiset over a universe is defined by a multiplicity function Group Actions: Let be the symmetric group Sncap S sub n acting on a sequence of elements. A hash function is "unordered" if it is invariant under the action of 3. Construction Methods
Traditional hash functions (like SHA-256) are designed for sequences. If you change the order of items in a list, the hash changes. However, in many applications—such as database query optimization, chemical informatics, or distributed state verification—we need to treat {A, A, B} the same as {B, A, A} . This paper explores how provide a formal framework for designing such "order-invariant" hash functions. 2. Mathematical Preliminaries