A Novel Quantum Visual Secret Sharing Scheme

Inspired by Naor <italic>et al.</italic>’s visual secret sharing (VSS) scheme, a novel <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> out of <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> quantum visual secret sharing (QVSS) scheme is proposed, which consists of two phases: sharing process and recovering process. In the first process, the color information of each pixel from the original secret image is encoded into an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-qubit superposition state by using the strategy of quantum expansion instead of classical pixel expansion, and then these <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> qubits are distributed as shares to <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> participants, respectively. During the recovering process, all participants cooperate to collect these <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> shares of each pixel together, then perform the corresponding measurement on them, and execute the <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-qubit <italic>XOR</italic> operation to recover each pixel of the secret image. The proposed scheme has the advantage of single-pixel parallel processing that is not available in the existing analogous quantum schemes and perfectly solves the problem that in the classic VSS schemes the recovered image has the loss in resolution. Moreover, its experiment implementation with the IBM Q is conducted to demonstrate the practical feasibility.


Privacy-Preserving Quantum Two-Party Geometric Intersection
Wen-Jie Liu 1, 2, * , Yong Xu 2 , James C. N. Yang 3 , Wen-Bin Yu 1, 2 and Lian-Hua Chi 4 Abstract：Privacy-preserving computational geometry is the research area on the intersection of the domains of secure multi-party computation (SMC) and computational geometry.As an important field, the privacy-preserving geometric intersection (PGI) problem is when each of the multiple parties has a private geometric graph and seeks to determine whether their graphs intersect or not without revealing their private information.In this study, through representing Alice's (Bob's) private geometric graph A G ( B G ) as the set of numbered grids A S ( B S ), an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol is proposed.In the protocol, the oracle operation A O ( B O ) is firstly utilized to encode the private elements of 0 1 1 ( , , , ) ) into the quantum states, and then the oracle operation f O is applied to obtain a new quantum state which includes the XOR results between each element of A S and B S .Finally, the quantum counting is introduced to get the amount ( t ) of the states i j a b 

Introduction
The problem of privacy-preserving computational geometry is an important research area on the intersection of the domains of secure multi-party computation (SMC) [Oleshchuk and Zadorozhny (2007)] and computational geometry [Preparata and Shamos (2012)].It focuses on how cooperative users can use their own private geometric information as inputs in collaborative computing in the distributed systems, and they can obtain the correct results while ensuring their privacy.Since the privacy-preserving computational geometry is firstly proposed by Atallah et al. [Atallah and Du (2001)], the other researchers have drawn extensive attention on some related problems, such as point inclusion [Troncoso-Pastoriza, Katzenbeisser, Celik et al. (2007); Luo, Huang and Zhong (2007)], geometric intersection [Erlebach, Jansen and Seidel (2005); Pawlik, Kozik, Krawczyk et al. ( 2013)], nearest points or closest pair [Li and Ni (2002); Tao, Yi, Sheng et al. (2010)], and convex hull [Huang, Luo and Wang (2008); Löffler and van Kreveld (2010); Assarf, Gawrilow, Herr et al. (2017)], which have been applied to many important military and commercial fields.2 Consider the following scenario, two countries A and B intend to build a railway in an offshore area.Before the completion of the railway, the construction route is confidential.In order to prevent future collisions of trains, countries A and B hope to determine if there are any two disjoint routes without revealing their own routes, and to negotiate with the location of the intersection.The above problem is a typical application of privacy-preserving geometric intersection (PGI).Different from the protocols based on circuit evaluation schemes, recently Qin et al. [Qin (2014)] proposed the Lagrange multiplier method to solve the intersection of the two private curves, and this method is suitable for solving general geometry intersection problems.On the other way, some researchers tried to study the geometric problems in three dimensional space [Li, Wu, Wang et al. (2014)].However, most of these classical solutions are based on computational complexity assumptions, and they cannot ensure the participants' privacy under the attack of quantum computation.Fortunately, quantum cryptography can provide the unconditional security, which is guaranteed by some physical principles of quantum mechanics, to resist against such impact.In additional, quantum parallelism makes it possible to greatly speed up solving some specific computational tasks, such as large-integer factorization [Shor (1994)] and database search [Grover (1996)].With quantum mechanics utilized in the information processing, many important research findings are presented in recent decades, such as quantum key distribution (QKD) [Bennett and Brassard (1984)], quantum key agreement (QKA) [Liu, Chen, Ji et al. (2017); Liu, Xu, Yang et al. (2018)], quantum secure direct communication [Liu, Chen, Ma et al. (2009); Liu, Chen, Liu (2016), Liu and Chen (2016)], quantum private comparison [Liu, Liu, Liu et al. (2014); Liu, Liu, Chen et al. (2014); Liu, Liu, Wang et al. (2014)], and quantum sealed-bid auction (QSBA) [Naseri (2009); Liu, Wang, Ji et al. (2014); Liu, Wang, Yuan et al. (2016)], and deterministic remote state preparation [Liu, Chen, Liu et al. (2015); Qu, Wu, Wang et al. (2017)].These findings have shown the potential power in either the efficiency improvements or the security enhancements.In this study, we pay attention to the PGI problem: Alice owns a private geometric graph A G , Bob has the other geometric graph B G , and they want to determine whether these two graphs intersect without revealing any private information to each other.By utilizing some specific oracle operations and quantum counting algorithm, we propose an efficient privacy-preserving quantum two-party geometric intersection (PQGI) protocol.The rest of this paper is organized as follows, the PQGI protocol is proposed in Sect.2, and the correctness, security and efficiency analysis of PQGI protocol are discussed in Sect.3, while the conclusion is drawn in the last section.

Preliminaries
Before introducing the procedures of PQGI protocol, we firstly make some definitions of PGI problem and PQGI protocol.Without loss of generality, we suppose there are two parties, i.e., Alice and Bob, and the formal definitions are given as below.without disclosing their respective private information.

Problem 1 (Privacy-preserving point inclusion):
As a point can be viewed as a special geometric graph whose area is small enough to be one dot, Problem 1 is a typical case of Problem 2. In the study, we only consider the geometric intersection of problem 2.

The definition of PQGI
In order to solve Problem 2, the private geometric graph can be represented as the set of grids in the area of the graph (suppose these girds are divided sufficiently), then the intersection of two geometric graphs is transformed into the intersection of two sets.Without loss of generality, we suppose Alice and Bob have a private geometric graph A G and B G on the plane, and they divide and number the whole plane into R grids (Here R is a large enough integer), then Alice's and Bob's graphs can be denoted as , , ) , respectively (shown in Fig. 1)., , , ) After executing this protocol, they can obtain the result of whether the two graphs intersect without revealing their private information.To be specific, the PQGI protocol should guarantee the following privacy:  Alice's Privacy Bob cannot learn any secret information about Alice's geometric graph without risking Alice's detection. Bob's Privacy Alice cannot get any secret information about Bob's geometric graph without risking Bob's detection.

The privacy-preserving quantum two-party geometric intersection protocol
Suppose Alice and Bob's private geometric graphs A G and B G are located on a unified plane, and the plane is uniformly divided into R grids, here R is a large enough integer that the whole plane can be represented by these grids with sufficient accuracy.Thus Alice's and Bob's graphs A G , B G can be represented as the sets of grids: , where , , respectively, where 0 (shown in Fig. 3 and Fig. 4) .
Then Alice and Bob obtain the result states   (shown in Fig. 5), where f O works as follows:  as follow (shown in Fig. 6) :    , where t is the After executing the quantum algorithm, Bob obtains the result of t .Then Alice judges whether A G and B G intersect according to the value of t : if 0 t  , then it can be deduced that there exists i j a b  for any i and j, then get the conclusion that A G intersects with B G , otherwise, A G and B G are not intersect.

Security analysis
Now we discuss the security of our protocol.To realize such a secure PQGI protocol, two security requirements should be satisfied, that are Alice's privacy and Bob's privacy.

Alice's privacy
Suppose Bob wants to extract information about private graph A G (i.e., i a without affecting the final result of the protocol.If Bob performs the projective measurement on state , he can randomly obtain one element i a from A   .The state A   can also be represented by an ensemble { , ( ) , here is the probability that Bob obtains Alice's coordinates: Here, we get the upper bound of information that Bob can get from Alice's coordinates is determined by the Holevo's bound [Holevo (2011)]: where ( ) S  denotes the Von Neumann entropy of quantum state  , ( : ) H A B means the information Bob can get about Alice's secret information , we have: , he can randomly obtain one element, i.e., i j a b  .However, the state Alice received is , and Alice does not know choose which base to measure and obtain j b .On the other hand, the received information is in the form of i j a b  , which means he even does not know which i a encodes the j b , and therefore prevents his cheating on Bob's privacy.

Efficiency analysis
The communication cost is one of the key indicators of the efficiency for communication protocols.In order to analyze the efficiency of our PQGI protocol, we choose the classical PGI protocols [Atallah and Du (2001);Qin, Duan, Zhao et al. (2014)] as comparative references.In the Atallah et al.'s protocol, the participants send total 2 4M messages to Bob, here M is the number of divided edges of the geometric graph, and each message requires R bits.So the transmitted messages of their protocol are , thus the total transmitted messages of our protocol are Through the above calculations, we can get the results of the three protocols' communication complexity (see Tab. 1).Obviously, our protocol achieves a great reduction in the communication complexity aspect.

Conclusion and discussion
In this paper, we present a novel quantum solution to two-party geometric intersection based on oracle and the quantum counting algorithm.The security of them is based on the quantum Copyright © 2018Tech Science Press TSP, vol.1, no.1, pp.1-5, 2018 10 cryptography instead of difficulty assumptions of mathematical problem.Compared with the classical related protocols, our solution has the advantage of higher security and lower communication complexity.In addition, our proposed protocol can also be extended to some other complicated privacy-preserving computation problems, such as privacy-preserving database queries over cloud data [Cao, Wang, Li et al. (2014);Shen, Li, Li et al. (2017)], privacy-preserving set operations in cloud computing [Cao, Li, Dang et al. (2017);Zhuo, Jia, Guo et al. (2017)], and privacy-preserving reversible data hiding over encrypted image [Cao, Du, Wei et al. (2016)].
Furthermore, the method of the oracle operation applied in the presented protocols is general and can be employed to solve other similar privacy-preserving computation geometry protocols, which have the property that geometric graphs can be divided into small enough grids, such as privacy-preserving convex hull.However, how to extend our two party scenarios to the multiparty scenario, and the more complex situations such as geometric union is another problem.We would like to investigate the applications of quantum technologies in more kinds of privacypreserving computational geometric protocols in the future.
There are two parties, Alice has a point A p , and Bob has a geometric graph B G .They want to decide whether A B p G  without revealing to each other anything more than what can be inferred from that answer.Problem 2 (Privacy-preserving two-party geometric intersection): Two parties Alice, Bob own the private geometric graphs A G , B G , respectively, and decide whether A

Figure 1 :
Figure 1: The illustration of partitioning and numbering the plane with R=400.The green (blue) part is Alice's graph A G (Bob' graph B G ), respectively, and the yellow part is the intersection area.Through representing Alice's and Bob's private geometric graphs A G , B G as the grid sets A S , B S ), the PQGI protocol is defined as follows.Definition 1 (the PQGI protocol): Alice and Bob encode their serial numbers of graph grids, i.e., A S = 0 1 . The detailed protocol is described in detail as follows (shown in Fig.2).

Figure 2 :
Figure 2: The procedure of the proposed PQGI protocol.The dotted (solid) line denotes the quantum (classic) channel.
a A and b A denote Alice's (m-qubit) and Bob's (n-qubit) address qubits, while a D and b D represent Alice's and Bob's (r-qubit) data qubits,

Figure 3 :
Figure 3: Schematic circuit of the oracle operation A O .

Figure 4 :
Figure 4: Schematic circuit of the oracle operation B O .2. After receiving A   from Alice, Bob applies oracle operation f O on A

Figure 5 :
Figure 5: Schematic circuit of the oracle operation f O .3.After receiving AB  4)Copyright © 2018 Tech Science Press   TSP, vol.1, no.1, pp.1-5, 2018  6   The result state is named AB   , Alice then performs measurement on the data qubits a D in AB   , if the measurement outcome turns to be 0 , she can conclude that Bob has not cheated.

Figure 6 :
Figure 6: Schematic circuit of the oracle operation A O .4.After the cheating check, Alice executes the quantum counting algorithm[Brassard, HØyer and Tapp (1998)] on AB  

Figure 7 :
Figure 7: The example of the two intersecting geometric graphs A G and B G .
each message requires R bits.Here M and N are the number of curves from the edges of the geometry, and its communication complexity is  protocol, Alice sends a (m+r)-qubit states A   to Bob in Step 1, and then Bob sends a (m+n+2r)-qubit state AB  and then performs measurement on them.Since the measured data qubits a  does not equal to 0 , she can conclude Bob has cheated and aborts the protocol.3.2.2Bob's privacy Suppose Alice wants to extract any information about private graph B G (i.e., i b without affecting the final result of the protocol.If Bob performs the projective measurement on state

Table 1 :
Comparison among our protocol and the other PGI protocols