Multiqudit quantum hashing and its implementation based on orbital angular momentum encoding D O Akatev1 A V Vasiliev12 N M Shafeev2 F M Ablayev12

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Multiqudit quantum hashing and its implementation

based on orbital angular momentum encoding

D O Akat’ev1, A V Vasiliev1,2, N M Shafeev2, F M Ablayev1,2,

and A A Kalachev1,2

1Zavoisky Physical-Technical Institute, FRC Kazan Scientiﬁc Center of RAS, Kazan,

Russian Federation

2Kazan Federal University, Kazan, Russian Federation

E-mail: akatevdmitrijj@gmail.com, vav.kpfu@gmail.com

October 2022

Abstract. A new version of quantum hashing technique is developed wherein a

quantum hash is constructed as a sequence of single-photon high-dimensional states

(qudits). A proof-of-principle implementation of the high-dimensional quantum

hashing protocol using orbital-angular momentum encoding of single photons is

implemented. It is shown that the number of qudits decreases with increase of their

dimension for an optimal ratio between collision probability and decoding probability of

the hash. Thus, increasing dimension of information carriers makes quantum hashing

with single photons more eﬃcient.

Keywords: single-photon states, orbital angular momentum, quantum hashing, SPDC

1. Introduction

Hashing algorithms today have become essential in cybersecurity, cryptography, data-

intensive research, etc., as they can reliably inform us whether two ﬁles are identical

without opening and comparing them. As an important part of cryptography, a hash

function compresses a message of any length into a digest of ﬁxed length and it is

the key technology of veriﬁcation of message integrity, digital signatures, ﬁngerprinting

and other cryptographic applications [1, 2]. Moreover, a universal hash function is an

important part of the privacy ampliﬁcation process of the quantum key distribution [3].

For such applications, a good hashing algorithm should satisfy two main properties:

one-way property and collision resistance. The ﬁrst means that restoring an input from

its hash should be a computationally hard problem. The second property means that

the situation when two diﬀerent inputs have the same hash (such a situation is called a

collision) is hard to ﬁnd.

Recently, a promising generalization of the cryptographic hashing concept on the

quantum domain, which is called quantum hashing, has been suggested and developed

[4, 5, 6, 7]. In this case, the hash function encodes a classical input state into a quantum

arXiv:2210.10501v2 [quant-ph] 26 Feb 2023

Multiqudit quantum hashing and its implementation based on orbital angular momentum encoding2

state so that to optimize the trade-oﬀ between one-way property and collision resistance.

In particular, in [7], it was suggested to construct a quantum hash via a sequence

of single-photon qubits, and a proof-of-principle experiment using single photons with

orbital angular momentum (OAM) encoding was implemented. In the present paper,

we further develop this approach both theoretically and experimentally and construct

a quantum hash as a sequence of single-photon high-dimensional states (qudits). We

show that the use of high-dimensional states increases the collision resistance of hashing

protocols and enhances the resistance against extraction of information about the

classical input.

2. Theory

2.1. Preliminaries

In [4] we have proposed a cryptographic quantum hash function and later in [8] provided

its generalized version for arbitrary ﬁnite abelian groups based on the notion of ε-biased

sets. Here we consider its version for a cyclic group Zq. In this case, for a set S⊆Zq

we can deﬁne its bias with respect to x∈Zqas following:

bias(S, x) = 1

|S|



X

s∈S

e2πsx/q



,(2.1)

and the set Sis called ε-biased if for any x6= 0 bias(S, x)≤ε.

These sets are especially interesting when |S|  |Zq|(as S=Zqis obviously 0-

biased). In their seminal paper [9] Naor and Naor deﬁned these small-biased sets, gave

the ﬁrst explicit constructions of such sets, and demonstrated the power of small-biased

sets for several applications. Note that ε-biased sets of size O(log q/ε2) exist as proved

in [10].

In [4] we have introduced the notion of quantum hashing and its main properties.

Later in [6] we have considered the trade-oﬀ and balancing between two main properties

of quantum hashing, and proposed a more general deﬁnition of the quantum (δ, ε)-

resistant hash function. Here we recall it in the concise manner and refer for details to

[6].

Deﬁnition 1 Let δ∈(0,1] and ε∈[0,1). We call a function ψ:X→ HKa quantum

(δ, ε)-resistant hash function if it has two main properties:

(i) δ-one-wayness, i.e.

|X|≤δ,

(ii) ε-collision-resistance, i.e. for any pair x1, x2of diﬀerent inputs

|hψ(x1)|ψ(x2)i| ≤ ε.

In other words a quantum function ψencodes an input x∈Xinto the quantum state

|ψ(x)iof dimension K. The properties of such a function include resistance to inversion

Multiqudit quantum hashing and its implementation based on orbital angular momentum encoding3

(known as “one-way property” or “preimage resistance”), which makes it unlikely to

“extract” encoded information out of the quantum state, and resistance to quantum

collisions, which means that quantum images for diﬀerent inputs can be distinguished

with high probability.

Note that the measure of collision resistance (denoted above by ε) is not the

probability of quantum collisions. The probability of collisions follows from the

particular comparison procedure that we use. It can be the well-known SWAP-test

[11], REVERSE-test [12] or simply a result of projection of |ψ(x1)ionto |ψ(x2)i. In

the latter case the probability of collisions would be described by the ﬁdelity between

|ψ(x1)iand |ψ(x2)i, and thus bounded by ε2.

2.2. Multiqudit Quantum Hashing

Here we deﬁne a new version of the quantum hashing technique for a cyclic group, i.e.

we consider X=Zqand |X|=q. It is based on small-biased sets and high-dimensional

states (qudits). But ﬁrst we note the following equivalence between ε-biased sets.

Property 1 Let S={s1, . . . , sd}and S0={0,(s2−s1),...,(sd−s1)}. Then for any

x∈Zqbias(S, x) = bias(S0, x), i.e. the set Sis equivalent (in terms of its bias) to S0.

Proof. The proof of this statement is based on the following considerations:

d



k=1

e2πskx/q



d

e2πs1x/q





k=1

e2π(sk−s1)x/q



d



k=1

e2π(sk−s1)x/q



therefore

bias(S, x) = 1

d



k=1

e2πskx/q



d



k=1

e2π(sk−s1)x/q



=bias(S0, x).



Now let S1, S2, . . . , Sm⊂Zqbe the ε-biased subsets of Zq, and we denote Sj=

{sj,1, . . . , sj,d}for j= 1, . . . , m. By the Property 1 without loss of generality we may

consider all sj,1to be equal 0. In other words for all j= 1, . . . , m it holds that

max

x6=0

d



1 + ei2πsj,2x

q+. . . +ei2πsj,d x

q



≤ε.

Then for x∈Zqwe deﬁne a multiqudit quantum hash function in the following way:

|ψj(x)i=1

√d(|`1i+ei2πsj,2x/q|`2i+. . . +ei2πsj,dx/q|`di),(2.2)

|ψ(x)i=|ψ1(x)i⊗···⊗|ψm(x)i,(2.3)

where |`kiare the basis states (k= 1 . . . d,dis the dimension of the qudit state space), q

is the size of the input state space, x∈ {0,1, . . . , q−1}is a classical input that is encoded

by the relative phase of mqudit states, si,k are numeric parameters (elements of the ε-

biased sets) of the quantum hash function that provide its collision resistance. The main

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MultiquditquantumhashinganditsimplementationbasedonorbitalangularmomentumencodingDOAkat'ev1,AVVasiliev1;2,NMShafeev2,FMAblayev1;2,andAAKalachev1;21ZavoiskyPhysical-TechnicalInstitute,FRCKazanScienticCenterofRAS,Kazan,RussianFederation2KazanFederalUniversity,Kazan,RussianFederationE-mail:akatevdmitr...

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Multiqudit quantum hashing and its implementation based on orbital angular momentum encoding D O Akatev1 A V Vasiliev12 N M Shafeev2 F M Ablayev12

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