6. Concluding remarks

To conclude, we first like to mention the results of the paper [31], which presents further development of quantum hash functions construction.

Recall that any ε-biased set gives rise to a Cayley expander graph [28]. We show how such graphs generate balanced quantum hash functions. Every expander graph can be converted to a bipartite expander graph. The generalization of these bipartite expander graphs is the notion of extractor graphs. Such point of view gives a method for constructing quantum hash functions based on extractors. This construction of quantum hash functions is applied to define the notion of keyed quantum hash functions. The latter is used for constructing quantum hashbased message authentication codes (QMAC). The security proof of QMAC is based on using strong extractors against quantum storage developed by Ta-Shma [32].

Secondly, in [24], we offered a design that allows to build a large amount of different quantum hash functions. The construction is based on composition of classical δ-universal hash family and a given family H<sup>δ</sup> , <sup>q</sup>, a quantum hash generator. A resulting family of functions is a new quantum hash generator. In particular, we present a quantum hash generator GRS based on Reed-Solomon code.
