Quantum Hashing via Classical $\epsilon$-universal Hashing Constructions

We define the concept of a quantum hash generator and offer a design, which allows one to build a large number of different quantum hash functions. The construction is based on composition of a classical ǫ-universal hash family and a given family of functions – quantum hash generators. The relationship between ǫ-universal hash families and error-correcting codes give possibilities to build a large amount of different quantum hash functions. In particular, we present quantum hash function based on Reed-Solomon code, and we proved, that this construction is optimal in the sense of number of qubits needed. Using the relationship between ǫ-universal hash families and Freivalds’ fingerprinting schemas we present explicit quantum hash function and prove that this construction is optimal with respect to the number of qubits needed for the construction.