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
We can view the hashing process as mapping the free abelian group generated by to a finite group 4. The Role of Group Actions
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: Traditional hash functions (like SHA-256) are designed for
In a practical setting (like the AZMATH blog might suggest), you would implement this using: Using XOR ( ⊕circled plus ) as the group operation.
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 This paper explores how provide a formal framework
Group Actions and Hashing Unordered Multisets: An Algebraic Approach to Data Integrity 1. Introduction
or a wide bit-length (e.g., 64-bit or 128-bit) minimizes the risk of two different multisets producing the same algebraic sum. The Role of Group Actions To achieve order
Note: This is often more robust against certain collision attacks but requires careful prime selection.