Author details

5.1.2. Lower bounds

The lower bound (10) for width(ψHS

32 Advanced Technologies of Quantum Key Distribution

The lower bound (11) for time(ψHS

6. Concluding remarks

Reed-Solomon code.

that Q produces the same quantum hashes |ψ(w)〉 and |ψ(w<sup>0</sup>

development of quantum hash functions construction.

strong extractors against quantum storage developed by Ta-Shma [32].

)〉 = |ψ〉. The last contradicts to the fact that states |ψ(w)〉 and |ψ(w<sup>0</sup>

function ψHS Theorem 4


Here, we show that the quantum branching program from Theorem 2 is optimal for

Þ ¼ Ωð Þ log log q , (43)

Þ ¼ Ωð Þ log q : (44)

� � <sup>p</sup> : (46)

)〉 for w and w<sup>0</sup>

: (45)

, that is, |ψ(w)〉 =

)〉 are ε orthogonal.

widthðψHS

timeðψHS

Proof. Let Q be a QBP for the function ψHS computation. ψHS presented by Q as follows:

<sup>ψ</sup>HS : fjψ0〉g � f g <sup>0</sup>; <sup>1</sup> log <sup>q</sup> ! <sup>ℋ</sup><sup>2</sup> � � <sup>⊗</sup> <sup>s</sup>

<sup>s</sup> <sup>≥</sup> log log <sup>q</sup> � log log 1 <sup>þ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

function. Indeed, the assumption that QBP Q for ψHS can test less than logq (that is, not all logq) variables of inputs x ∈F<sup>q</sup> means existence of (at least) two different inputs w, w<sup>0</sup> ∈F<sup>q</sup> such

To conclude, we first like to mention the results of the paper [31], which presents further

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

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

) follows immediately from Property 4

2=ð Þ 1 � ε

) follows from the fact that ψHS is collision ε-resistant

ψð Þj w ψ w<sup>0</sup> j j h i ð Þ ≤ ε: (47)

Farid Ablayev\* and Marat Ablayev

\*Address all correspondence to: fablayev@gmail.com

Kazan Federal University, Kazan, Russia
