Safe two-party quantum scalar product calculation method based on bidirectional quantum migration on circle

Abstract

The invention provides a safe two-party quantum scalar product calculation method based on bidirectional quantum migration on a circle. The scalar product of secret vectors of two communication parties is calculated under the condition that secret information is not leaked. The method provided by the invention needs the assistance of a semi-loyal third party, and the third party can act mistakenly according to own willingness but cannot collude with others. According to the method, a two-particle product state instead of a quantum entanglement state is adopted as an initial quantum resource, and only d-dimensional and two-dimensional single particle measurement instead of quantum entanglement state measurement is needed. The method provided by the invention not only has information theory security, but also can resist external attacks and participant attacks at the same time.

Description

Technical Field

[0001] The present invention relates to the field of quantum cryptography. The present invention designs a secure two-party quantum scalar product calculation method based on bidirectional quantum walk on a circle, which can calculate the scalar product of secret vectors of two communicating parties without leaking secret information. Background Art

[0002] In Secure Multi-party Computation (SMC) [1] In the problem, several mutually untrusting participants always collaborate to compute a specified function without the risk of leaking their secret inputs. Secure two-party scalar product computation is defined as computing the scalar product or dot product of two secret vectors without leaking the secret. Scalar product computation is an important basic branch of SMC and can be used for point inclusion. [2] , distance calculation [3] and k nearest neighbor calculation [4] However, the security of SMC, including scalar product calculation, is based on the computational complexity assumption, which may be affected by quantum computing. [5-6] In 1984, Bennett and Brassard [7] Combining the basic principles of quantum mechanics with classical cryptography, the first quantum cryptography method, the BB84 quantum key distribution (QKD) method, was proposed. Since then, scholars have begun to study quantum secure multi-party computation (QSMC) with theoretical unconditional security, including quantum secure multi-party summation. [8-13] , quantum secure multi-party multiplication [12-14] And quantum scalar product (QSP) calculation [15-18] .

[0003] Currently, there are only four methods for calculating QSP, namely the methods described in the literature [15-18]. In 2012, He et al.

[15] For the first time, a secure two-party quantum scalar product (TPQSP) calculation method based on four-particle quantum entanglement with the participation of a third party (TP).

[16] A secure TPQSP calculation method based on continuous variable cluster states is proposed.

[17] Using single-particle states and seven-particle quantum entangled states, a TPQSP calculation method with strong privacy protection is proposed.

[18] A secure and efficient TPQSP calculation method based on four-particle quantum entangled state is proposed. However, these QSP calculation methods have some shortcomings. Specifically, the methods in references [15, 17, 18] require multi-particle quantum entangled state as the initial quantum resource, while the method in reference

[16] requires continuous variable cluster state, but the preparation of multi-particle quantum entangled state and continuous variable cluster state is quite difficult; the methods in references [15, 16] require pre-sharing of keys between participants, which will consume additional quantum resources and classical resources; the methods in references [17, 18] require quantum Fourier transform (QFT) or inverse quantum Fourier transform (IQFT), which is quite challenging to implement.

[19] ; Moreover, reference

[16] adopts complex projection measurement.

[0004] Quantum walks (QW) are quantum simulations of classical random walks.

[20] , has been widely used in many quantum information processing tasks, such as quantum information segmentation

[21] , quantum entangled state generation

[22] , Quantum identity authentication

[23] , Quantum Public Key Cryptography

[24] , Quantum Privacy Comparison

[25] and quantum secret sharing

[26] Etc. The QW state is a two-particle product state, which is easier to prepare than the quantum entangled state.

[0005] Based on the above analysis, the present invention introduces QW into scalar product calculation and proposes a novel secure TPQSP calculation method based on two-direction quantum walks on a circle (TDQWC). The method of the present invention requires the assistance of a semi-loyal TP, who can act according to his own will but cannot collude with others.

[27] The method of the present invention uses the QW state as the initial quantum resource and only requires d-dimensional and 2-dimensional single-particle measurements rather than quantum entangled state measurements. The method of the present invention does not require QFT or IQFT. In addition, the method of the present invention not only has information-theoretic security, but also can resist external attacks and participant attacks.

[0006] References

[0007] 1Yao A C.Protocols for secure computations[C] / / 23rd Annual Symposiumon Foundations of Computer Science(SFCS1982),IEEE,1982:160-164.

[0009] 2Shen C H,Zhan J,Hsu T S,et al.Scalar-product based secure two-partycomputation[C] / / 2008IEEE International Conference on Granular Computing,IEEE,2008:556-561.

[0010] 3 Huang H,Gong T,Chen P,et al.Secure two-party distance computationprotocol based on privacy homomorphism and scalar product in wireless sensornetworks[J].Tsinghua Science and Technology,2016,21(4):385-396.

[0011] 4 Zhu Y,Takagi T.Efficient scalar product protocol and its privacy–preserving application[J].International Journal of Electronic Security andDigital Forensics,2015,7(1):1-19.

[0012] 5 Shor P W.Algorithms for quantum computation:discrete logarithms andfactoring[C] / / Proceedings 35th Annual Symposium on Foundations of ComputerScience,IEEE,1994:124-134.

[0013] 6 Grover L K.A fast quantum mechanical algorithm for database search[C] / / Proceedings of the Twenty-eighth Annual ACM Symposium on Theory ofComputing,1996:212-219.

[0014] 7 Bennett C H,Brassard G.Quantum cryptography:Public key distributionand coin tossing.Proceedings of the IEEE International Conference onComputers,Systems,and Signal Processing,IEEE Press,1984,175-179.

[0015] 8 Yang H Y,Ye T Y.Secure multi-party quantum summation based onquantum Fourier transform[J].Quantum Information Processing,2018,17(6):129.

[0016] 9 Lin S,Wang N,Liu X F.An efficient secure multiparty quantumcomputation protocol[J].Scientia Sinica Physica,Mechanica&Astronomica,2023,53(4):240314.

[0017] 10 Li H J,Shi R H,Ke W Y,et al.Quantum protocol for secure multipartyXOR with application to secure communication in metropolitan area networks[J].Scientia Sinica Physica,Mechanica&Astronomica,2024,54(3):230312.

[0018] 11 Wang J T,Li X,Ye T Y.Aquantum secure multi-party summationprotocol based on one-direction quantum walks on a circle[J].Scientia SinicaPhysica,Mechanica&Astronomica,2024,54(4):240311.

[0019] 12 Shi R H,Mu Y,Zhong H,et al.Secure multiparty quantum computationfor summation and multiplication[J].Scientific Reports,2016,6(1):19655.

[0020] 13 Lv S X,Jiao X F,Zhou P.Multiparty quantum computation forsummation and multiplication with mutually unbiased bases[J].InternationalJournal of Theoretical Physics,2019,58(9):2872-2882.

[0021] 14 Sutradhar K,Om H.Secret sharing based multiparty quantumcomputation for multiplication[J].International Journal of TheoreticalPhysics,2021,60(9):3417-3425.

[0022] 15 He L B,Huang L S,Yang W,et al.A protocol for the secure two-partyquantum scalar product[J].Physics Letters A,2012,376(16):1323-1327.

[0023] 16 Wang Y,He G.Quantum secure scalar product with continuous-variableclusters[C] / / Proceedings of the 18th AQIS Conference(8-12September 2018,Nagoya,Japan)2018.

[0024] 17 Shi R,Zhang M.Strong privacy-preserving two-party scalar productquantum protocol[J].International Journal of Theoretical Physics,2019,58(12):4249-4257.

[0025] 18 Liu W J,Li Z X.Secure and efficient two-party quantum scalarproduct protocol with application to privacy-preserving matrix multiplication[J].IEEE Transactions on Circuits and Systems I:Regular Papers,2023,70(11):4456-4469.

[0026] 19 Nielsen M A,Chuang I L.Quantum computation and quantum information[M].Cambridge:Cambridge University Press,2001.

[0027] 20 Kadian K,Garhwal S,Kumar A.Quantum walk and its applicationdomains:A systematic review[J].Computer Science Review,2021,41:100419.

[0028] 21 Li H J,Li J,Xiang N,Zheng Y,Yang Y G,Naseri M.A new kind ofuniversal andflexible quantum information splitting scheme with multi-coinquantum walks[J].Quantum Information Processing,2019,18:316.

[0029] 22 Li M,Shang Y.Entangled state generation via quantum walks withmultiple coins[J].npj Quantum Information,2021,7(1):70.

[0030] 23 Lou X P,Wang S,Ren S X,Zan H R,Xu X J.Quantum identityauthentication scheme based on quantum walks on graphs with IBM quantum cloudplatform[J].International Journal of Theoretical Physics,2022,61:40.

[0031] 24 Vlachou C,Rodrigues J,Mateus P,et al.Quantum walk public-keycryptographic system[J].International Journal of Quantum Information,2015,13(07):1550050.

[0032] 25 Chen F L,Zhang H,Chen S G,et al.Novel two-party quantum privatecomparison via quantum walks on circle[J].Quantum Information Processing,2021,20(5):178.

[0033] 26 Wang Y,Lou X P,Fan Z,Wang S,Huang G.Verifiable multi-dimensional(t,n)threshold quantum secret sharing based on quantum walk[J].InternationalJournal of Theoretical Physics,2022,61:24.

[0034] 27 Yang Y G,Xia J,Jia X,Zhang H.Comment on quantum private comparisonprotocols with a semi-honest third party[J].Quantum Information Processing,2013,12(2):877.

[0035] 28 Fujiwara S,Osaki H,Buluta I M,et al.Scalable networks for discretequantum random walks[J].Physical Review A,2005,72(3):032329.

[0036] 29 Douglas B L,Wang J B.Efficient quantum circuit implementation ofquantum walks[J].Physical Review A,2009,79(5):052335.

[0037] 30 Long G L,Liu X S.Theoretically efficient high-capacity quantum-key-distribution scheme[J].Physical Review A,2002,65(3):032302.

[0038] 31 Herrero-Collantes M,Garcia-Escartin J C.Quantum random numbergenerators[J].Reviews of Modern Physics,2017,89(1):015004.

[0039] 32 Ambainis A,Mosca M,Tapp A,et al.Private quantum channels[C] / / Proceedings 41st Annual Symposium on Foundations of Computer Science,IEEE,2000:547-553.

[0040] 33 Boykin P O,Roychowdhury V.Optimal encryption of quantum bits[J].Physical review A,2003,67(4):042317.

[0041] 34 Li C Y,Zhou H Y,Wang Y,and Deng F G.Secure quantum keydistribution network with Bell states and local unitary operations[J].ChinesePhysics Letters,2005,22(5):1049-1052.

[0042] 35 Li C Y,Li X H,Deng F G,Zhou P,Liang Y J,and Zhou H Y.Efficientquantum cryptography network without entanglement and quantum memory[J].Chinese Physics Letters,2006,23(11):2896-2899.

[0043] 36 Shor P W,Preskill J.Simple proof of security of the BB84 quantumkey distribution protocol[J].Physical Review Letters,2000,85(2):441.

[0044] 37 Ye TY,Jiang L Z.Improvement of controlled bidirectional quantumdirect communication using a GHZ state[J]. Chinese Physics Letters, 2013,30(4):040305.

[0045] 38 Gao F, Qin SJ, Wen QY, et al. A simple participant attack on the protocol[J].Quantum Information&Computation,2007,7(4):329-334.

[0046] 39 Cabello A.Quantum key distribution in the Holevo limit[J].PhysicalReview Letters,2000,85(26):5635.

[0047] 40 Geng MJ,Xu TJ,Chen Y,Ye T Y.Semiquantum private comparison of size relationship based on d-level single-particle states[J].Scientia SinicaPhysica,Mechanica&Astronomica,2022,52(9):290311. Summary of the invention

[0048] The purpose of the present invention is to design a secure two-party quantum scalar product calculation method based on bidirectional quantum walk on a circle, so as to calculate the scalar product of secret vectors of two communicating parties without leaking secret information.

[0049] A secure two-way quantum scalar product calculation method based on bidirectional quantum walk on a circle includes the following nine processes:

[0050] S1) Alice’s secret vector is x=(x 1 ,x 2 ,…,x L ), Bob’s secret vector is y=(y 1 ,y 2 ,…,y L ),in And i=1,2,…,L. Alice sets And transmit ∈ to the semi-loyal TP through quantum secure direct communication, where ∈∈{0,1,…,d-1},e i ∈{0,1,…,d-1}, i=1,2,…,L, and Alice generates r=(r 1 ,r 2 ,…,r L ),in and i = 1, 2, ..., L. Afterwards, Alice prepares the initial QW state where p a ∈{0,1,…,d-1} and i=1,2,…,L. Then, Alice Apply in random order and get in symbol represents modulo d addition, i = 1, 2, ..., L. Then, Alice generates λ random c decoy particles of the basis or {|+>,|->} basis, where M c ={|0>,|1>} and and will Finally, Alice sends this particle sequence to Bob through the quantum channel, and sends r to Bob and p to Bob through the authenticated classical channel. a Give TP.

[0051] S2) Alice informs Bob of the specific locations of the λ decoy particles and their preparation bases. Bob performs quantum measurements on the λ decoy particles using the corresponding preparation bases and informs Alice of the measurement results. Alice calculates the error rate by determining whether the initial preparation states of the λ decoy particles are consistent with Bob's corresponding measurement results. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue.

[0052] S3) Bob Imposition get Afterwards, Bob uses M p Base measurement The wandering particles get Using M c Base measurement The coin particles get Among them, M p={|0>,|1>,…,|d-1>} and i=1,2,…,L. Bob calculates in Then, Alice and Bob announce and , and calculate the error rate The symbol “∨” represents a bitwise XOR operation. If θ≠0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0053] S4) TP preparation where p b ∈{0,1,…,d-1} and c b ∈{0,1}. Then, TP generates λ random c decoy particles of the basis or {|+>,|->} basis, and The λ decoy particles are randomly mixed with the above to form a new particle sequence. Finally, TP sends this particle sequence to Bob through the quantum channel.

[0054] S5) TP and Bob perform eavesdropping detection similar to step S2. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0055] S6) Bob generates a quantum random number generator Afterwards, Bob Apply in random order and get And calculate and Then, Bob generates λ random c decoy particles of the basis or {|+>,|->} basis, and The particles are randomly mixed with the above λ decoy particles to form a new particle sequence. Finally, Bob sends this particle sequence to TP through the quantum channel and authenticates the classical channel. b Transmit to TP, and transmit s via quantum secure direct communication b and c Give TP.

[0056] S7) Bob and TP perform eavesdropping detection similar to step S2. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0057] S8)TP calculation Then Apply in random order and G ρ get Afterwards, TP uses M c Base measurement The coin particles get And calculate the error rate If δ≠0, the communication will be stopped immediately; otherwise it will continue to be executed.

[0058] S9)TP uses M p Base measurement The wandering particles get Then calculate The symbol Represents modulo d subtraction. TP announces the value of SP(x,y) to Alice and Bob respectively. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 (a) and 1(b) are T 1 and T -1 quantum circuits; Figure 2 (a), 2(b) and 2(c) are U, U -1 and G k quantum circuits; Figure 3 (a) and 3(b) are U and U respectively. -1 The simulation circuit of Figure 4 (a) and 4(b) are G 1 and G -1 The simulation circuit of Figure 5 It is used to generate and measure U 5 |8>|0> simulation circuit; Figure 6 It is the U k |8>|0>Measurement results of wandering particles, where k=0,1,…,5; Figure 7 It is the simulation circuit of TPQSP calculation method; Figure 8 yes Figure 7 The measurement results. DETAILED DESCRIPTION

[0060] The technical solution of the present invention is further described below in conjunction with embodiments.

[0061] 1 Method Description

[0062] 1.1d-dimensional shift operator

[0063] The d-dimensional shift operator is described as

[0064]

[0065] in symbol represents modulo d addition. According to the literature

[28] , T 1 and T-1 The quantum circuit is Figure 1 Here, T -1 It is T 1 Obviously, T -1 The quantum circuit is T 1 A version of a quantum circuit with the gate order reversed.

[0066] 1.2 Bidirectional quantum walk on a circle

[0067] For discrete QW

[20] , ground state M c ={|0>,|1>} to form a 2D coin space describing the direction of the coin particles Ground state M p = {|0>,|1>,…,|d-1>} to describe the d-dimensional wandering particle space of the wandering particle position The QW state can be described as

[0068]

[0069] in In order to make the wandering particles in the QW state evolve in two directions, the QW evolution operator is defined as

[0070]

[0071] Here, I d is a unit operator of size d×d; It is the Hadamard gate, whose function is to transform the quantum bit |0> or |1> into a superposition state

[19] ,Right now

[0072]

[0073] where c∈{0,1}; S is a bidirectional controlled shift operator, defined as

[0074]

[0075] When U is applied to the QW state, if the coin particle is |0>, its wandering particle moves forward one step on the circle; if the coin particle is |1>, its wandering particle moves back one step on the circle. This QW system is called TDQWC. The inverse operator of U is described as

[0076]

[0077] Where S -1 is the inverse of S and is described as

[0078]

[0079] The QW state after applying U k times can be described as

[0080] The wandering particles are p After the basis is measured, it may collapse to some positions with a certain probability. Assume n = 4, d = 2 n =16, and the initial quantum wandering state is After a series of mathematical derivations, U k |8>|0>The positions of the wandering particles are shown in Table 1, where k = 0, 1, …, 5.

[0081] Table 1 U in TDQWC system k |8>|0>Probability distribution of the position of wandering particles, where k=0,1,…,5

[0082]

[0083] In addition, the global shift operator G k is defined as

[0084]

[0085] According to the literature

[29] , U,U -1 and G The schematic diagrams of the quantum circuits of k are as follows Figure 2 Here, the first qubit represents the coin particle |c>, and the remaining n qubits represent the wandering particles |p>.

[0086] 1.3 Description of the method of the present invention

[0087] Assume Alice's secret vector is x=(x 1 ,x 2 ,…,x L ), Bob’s secret vector is y=(y 1 ,y 2 ,…,y L ),in and i = 1, 2, ..., L. With the assistance of the semi-loyal TP, Alice and Bob hope to calculate the scalar product function SP(x, y) by the following TPQSP calculation method. Here, TP can act wrongly according to its own will but cannot collude with others.

[27] ; Define SP(x,y) as

[0088]

[0089] S1) Alice sets up

[0090]

[0091] And through Quantum Secure Direct Communication (QSDC)

[30] Send ∈ to TP, where ∈∈{0,1,…,d-1},e i ∈{0,1,…,d-1}, i=1,2,…,L, and Alice through the quantum random number generator

[31] (Quantum Random Number Generator, QRNG) generates r = (r 1 ,r 2 ,…,r L ),in and i = 1, 2, ..., L. Afterwards, Alice prepares the initial QW state where p a ∈{0,1,…,d-1} and i=1,2,…,L. Then, Alice Apply in random order and get Where i=1,2,…,L.

[0092] Then, Alice generates λ random c decoy particles of the basis or {|+>,|->} basis, where and will Finally, Alice sends this particle sequence to Bob through the quantum channel, and sends r to Bob and p to Bob through the authenticated classical channel. a Give TP.

[0093] S2) Alice informs Bob of the specific locations of the λ decoy particles and their preparation bases. Bob performs quantum measurements on the λ decoy particles using the corresponding preparation bases and informs Alice of the measurement results. Alice calculates the error rate by determining whether the initial preparation states of the λ decoy particles are consistent with Bob's corresponding measurement results. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue.

[0094] S3) Bob Imposition get Afterwards, Bob uses M p Base measurement The wandering particles get Using M c Base measurement The coin particles get Where i = 1, 2, ..., L. Bob calculates

[0095]

[0096] in Then, Alice and Bob announce and , and calculate the error rate

[0097]

[0098] The symbol “∨” represents a bitwise XOR operation. If θ≠0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0099] S4) TP preparation where p b ∈{0,1,…,d-1} and c b ∈{0,1}. Then, TP generates λ random c decoy particles of the basis or {|+>,|->} basis, and The λ decoy particles are randomly mixed with the above to form a new particle sequence. Finally, TP sends this particle sequence to Bob through the quantum channel.

[0100] S5) TP and Bob perform eavesdropping detection similar to step S2. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0101] S6) Bob uses QRNG

[31] generate Afterwards, Bob Apply in random order and get And calculate

[0102]

[0103] and

[0104]

[0105] Then, Bob generates λ random c decoy particles of the basis or {|+>,|->} basis, and The particles are randomly mixed with the above λ decoy particles to form a new particle sequence. Finally, Bob sends this particle sequence to TP through the quantum channel and authenticates the classical channel. b Sent to TP and through QSDC

[30] Teleports b andc Give TP.

[0106] S7) Bob and TP perform eavesdropping detection similar to step 2. If the error rate is not 0, the communication will be stopped immediately; otherwise, it will continue to be executed.

[0107] S8)TP calculation Then Apply in random order and G ρ get Afterwards, TP uses M c Base measurement The coin particles get And calculate the error rate If δ≠0, the communication will be stopped immediately; otherwise it will continue to be executed.

[0108] S9)TP uses M p Base measurement The wandering particles get Then calculate

[0109]

[0110] The symbol Represents modulo d subtraction. TP announces the value of SP(x,y) to Alice and Bob respectively.

[0111] 2 Correctness Analysis

[0112] Property 1 describes the relationship between U and G k The proof of property 1 is as follows. Reference

[24] only proves that “U and G k The "commutative" between them is a sufficient condition for "Formula (12) equals Formula (13)". Here we prove more strictly that U and G k There is an commutative relationship between them.

[0113] Property 1: In the TDQWC system, when and When [U,G k ]≡UG k -G k U=0.

[0114] Proof: Insert equation (5) into equation (3) to obtain

[0115]

[0116] Reference

[26] has proved that T m T n =T m+n ,in According to formula (9) and formula (17), we have

[0117]

[0118] as well as

[0119]

[0120] Therefore, there is

[0121] [U,G k ]≡UG k -G k U=0, (20)

[0122] This formula shows that U and G k There is commutativity between them. This completes the proof of Property 1.

[0123] Property 2 describes U -1 With G k The proof of property 2 is as follows.

[0124] Property 2: In the TDQWC system, when and When [U -1 ,G k ]≡U -1 G k -G k U -1 =0.

[0125] Proof: Insert equation (7) into equation (6) to obtain

[0126] According to formula (9) and formula (21), we have

[0127]

[0128] as well as

[0129]

[0130] Therefore, there is

[0131] [U -1 ,G k ]≡U -1 G k -G k U -1 =0, (24)

[0132] This formula shows that U -1 With G k There is commutativity between them. This completes the proof of Property 2.

[0133] Next, Conclusion 1 is used to verify in detail the output correctness of the TPQSP calculation method of the present invention.

[0134] Conclusion 1: The output of the TPQSP calculation method of the present invention is correct.

[0135] Proof: According to properties 1 and 2, Alice obtains in step S1

[0136]

[0137] In step S3, Bob obtains

[0138]

[0139] According to formula (26), in step S3, if the quantum channel is secure, when Bob measures The coin particles get When and θ=0; when Bob measures The wandering particles get When

[0140]

[0141] According to formula (11), formula (14), formula (15), formula (27), property 1 and property 2, we have

[0142]

[0143] In step S8, when TP Apply in random order and G ρ , according to properties 1 and 2, she can obtain

[0144]

[0145] Therefore, if the quantum channel is secure, when TP measures The coin particles get When and δ=0; in step S9, when TP is measured The wandering particles get When This completes the proof of Conclusion 1.

[0146] 3 Safety Analysis

[0147] 3.1 Information Theoretical Security

[0148] The TPQSP calculation method of the present invention has two quantum one-time pad (QOTP) processes. The first QOTP is: in step S1, Alice prepares the initial QW state where p a ∈{0,1,…,d-1}, and i = 1, 2, ..., L, then use random order to and Encrypt it to get In step S3, Bob uses right Decrypt to get The second QOTP is: In step S4, TP is prepared where px∈{0,1,…,d-1} and c b ∈{0,1}; In step S6, Bob uses random order to and Encrypt it to get In step S8, TP uses random order to and G ρ right Decrypt to get Next, Conclusion 2 is used to verify the information-theoretic security of the method of the present invention.

[0149] Conclusion 2: The TPQSP calculation method of the present invention has information-theoretic security.

[0150] Proof: References [32,33] have proved the information-theoretic security of QOTP. According to reference

[32] , for an n-qubit system, if for all possible input states ρ in , its output state ρ out is a completely mixed state, that is,

[0151]

[0152] It has information-theoretic security. in is the density matrix of any possible input state, Uk 是 With p k The probabilities are imposed on the input states for the corresponding unitary matrix used to encrypt, It's U k The Hermitian conjugation of Serves 2 n ×2 b The identity matrix of .

[0153] In the method of the present invention, all possible input states of |p> constitute Mp All possible input states of |c> form M c Therefore, the density matrix of the input QW state is

[0154]

[0155]

[0156] Here, d = 2 n ; and the QW state is composed of n+1 quantum bits.

[0157] Assume a random integer r ≥ 0, from U r After encryption, the input state ρ in becomes

[24]

[0158]

[0159] When r<0, the situation is similar. Assume that the secret information By G k After encryption, the input state ρ′ in =ρ out becomes

[24]

[0160]

[0161] According to equations (32) and (33), when U is applied r and G k After that, the input state ρ in The output state after evolution is the maximum mixed state, which means that the TPQSP calculation method of the present invention has information-theoretic security.

[0162] 3.2 External Attacks

[0163] Obviously, in the process of Alice sending to Bob, TP sending to Bob, and Bob sending to TP particle sequence, the method of the present invention adopts the decoy photon technology

[34] ,

[35] Detect eavesdropping from external attacker Eve. This eavesdropping detection technology is actually the BB84 QKD method [7] A variant of the eavesdropping detection technique that has been proven to be unconditionally secure

[36] In addition, the literature

[37] verified in detail the effectiveness of using decoy photon technology to resist measurement-retransmission attack, interception-retransmission attack and entanglement-measurement attack from Eve.

[0164] 3.3 Participant Attack

[0165] Gao et al. pointed out in the literature

[38] that in quantum communication methods, attacks by dishonest participants are always more dangerous than attacks by external eavesdroppers and should be taken seriously. Next, we analyze the participant attacks initiated by different participants.

[0166] 3.3.1 Attack from dishonest Alice

[0167] In order to obtain Bob’s secret information y=(y 1 ,y 2 ,…,y L ), Alice may launch the following attack: In step S1, Alice uses QSDC

[30] Send ∈ to TP and generate r through QRNG i , prepare pseudo-initial QW state right Imposition get Then With r i Send to Bob, and send p a Sent to TP, where p a ∈{0,1,…,d-1}, and i=1,2,…,L; in step S3, Bob Imposition get Using M p Base measurement The wandering particles get Using M c Base measurement The coin particles get And calculate m p y=(y 1 ,y 2 ,…,y L ), where i = 1, 2, ..., L; in step S4, TP is prepared and will Sent to Bob, where p b ∈{0,1,…,d-1} and c b ∈{0,1}; In step S6, Bob uses QRNG

[31] Generate r b ,right Apply in random order and get Then Send to TP, through the authenticated classic channel to send rb 传Send to TP, and pass through QSDC

[30] Teleports b and c Give TP, where In step S8, TP is calculated right Apply in random order and G ρ get Using M c Measurement The coin particles get In step S9, TP uses M p Base measurement The wandering particles get Then calculate And announce the calculation results to Alice and Bob respectively. When Alice learns from TP in step S9 Without knowing ρ, she cannot obtain sb, let alone the specific value of y.

[0168] 3.3.2 Participant Attack from Dishonest Bob

[0169] In step S3, Bob Imposition get Using M p Base measurement The wandering particles get Using M c Base measurement The coin particles get Where i = 1, 2, ..., L. Obviously, xx is p a and e i Encryption, where i = 1, 2, ..., L. Alice uses QSDC

[30] Send ∈ to TP so that Bob has no chance to know e i , although Bob can get Where i = 1, 2, ..., L. Therefore, Bob cannot get x i , where i=1,2,…,L.

[0170] In step S6, Bob Apply in random order and get where s a =SUM(x 1 ,x 2 ,…,x L ), and then and c =0 is sent to TP, where c b ∈{0,1}; In step S8, TP calculates ρ=0, Apply in random order and G ρ =G 0 get Using M c Measurement The coin particles get In step S9, TP uses M p Base measurement The wandering particles get Then calculate And announce the calculation results to Alice and Bob respectively. When Bob learns from TP in step S9 When, because we do not know e i , he couldn't get s a , not to mention x i Specific value of , where i=1,2,…,L.

[0171] 3.3.3 Attacks from Participants of Semi-loyal TPs

[0172] The method of the present invention allows TP to act erroneously according to their own will but cannot collude with others

[27] In step S1, Alice uses QSDC

[30] Send ∈ to TP, and pass p through the authenticated classical channel a Send to TP and r i Send it to Bob, and include TP may attack the particle sequence sent by Alice to Bob, where i = 1, 2, …, L. However, since TP does not know the preparation basis and specific location of the decoy particles, she will inevitably be detected as an external attacker by Alice and Bob in step S2. Even if TP is not detected by the eavesdropping detection and successfully steals r from Alice, i , then Imposition After using M p Base measurement of wandering particles, she still can't get x i This is because TP does not know the specific value of e i Specific value of , where i=1,2,…,L.

[0173] In step S4, TP preparation and sends it to Bob; in step S6, Bob Apply in random order and get In step S8, TP Apply in random order and G ρ get In step S9, TP is calculated Since Bob will m p y is encoded in After calculation, TP can only know SP(x,y) but cannot get the specific value of y. Moreover, in this process, TP itself prepares the initial QW state in Therefore, TP cannot obtain the specific value of y by preparing a pseudo-initial QW state.

[0174] In addition, in step S6, Bob uses QSDC

[30] Teleports b and c Give TP. Obviously, if Established, that is holds, where i,k∈{1,2,…,L}, then there will be s c =y k ;if Established, that is holds, where i,k∈{1,2,…,L}, then we have However, since TP does not know The specific value of , and step S1 requires So y will not appear k The situation was leaked to TP.

[0175] Example:

[0176] 1 Application examples of the method of the present invention

[0177] First, the IBM Qiskit platform is used to simulate the TDQWC system, and then the correctness and feasibility of the method of the present invention are verified through simulation of specific examples.

[0178] 1.1 Simulation of TDQWC

[0179] In the TDQWC system, assuming d = 2 n =16, a total of n+1=5 qubits are needed to describe the QW state |p 0 >|c 0 >, where c 0 ∈{0,1} and p 0 ∈{0,1,…,15}, that is, |c 0> By quantum bit q 0 Indicates,|p 0 > By quantum bit q 4 q 3 q 2 q 1 Based on equations (3), (5)-(7), (9) and Figure 1 and Figure 2 ,U and U -1 The simulation circuits are as follows Figure 3 (a) and Figure 3 (b) shows that G 1 and G -1 The simulation circuits are as follows Figure 4 (a) and Figure 4 (b) is shown. Figure 3 and Figure 4 In each subgraph of , the first X gate is used to activate the flip-flop controlled by zero, and the second X gate is used to flip it. Figure 3 In (a), the second dashed line is used to separate the T controlled by |0> 1 and T controlled by |1> -1 ;exist Figure 3 In (b), the first dashed line is used to separate the T controlled by |0> -1 and T controlled by |1> 1 ; and in Figure 4 In (a), the dashed line is used to separate the T controlled by |0> 1 and T controlled by |1> 1 ;exist Figure 4 In (b), the dashed line is used to separate the T controlled by |0> -1 and T controlled by |1> -1 .according to Figure 3 and Figure 4 You can directly get U k With G k The simulation circuit.

[0180] Assume that the initial QW state is Obviously, The measurement result of the wandering particle in the basis {|0>,|1>,…,|15>} must be |8>. 5 |8>|0>'s simulation circuit is as follows Figure 5 As shown. The left side of the first dotted line is the initial QW state |8>|0>; the left side of the second dotted line is 5 times U; and the right side of the second dotted line is U 5 |8>|0> performs {|0>,|1>,…,|15>} basis measurements. After 10,000 simulations, U k |8>|0>The measurement results of wandering particles are as follows Figure 6 (a)-(f), where k = 0, 1, ..., 5. For clarity, Table 2 lists the U obtained by simulation. k |8>|0>The position of the wandering particle. Obviously, the simulation results in Table 2 are almost consistent with the results of mathematical derivation in Table 1, which verifies the correctness of the above simulation circuit.

[0181] Table 2 U after TDQWC system simulation k |8>|0>Probability distribution of the position of wandering particles, where k=0,1,…,5

[0182]

[0183] 1.2 Simulation of TPQSP calculation method

[0184] Assume d = 16, L = 4, p a =5,p b =13,r b =19 and c b =1. Alice's secret vector is x=(1,4,-5,-8), and Bob's secret vector is y=(2,3,-6,-7). So theoretically we have

[0185] In step S1, assuming ∈ = 9, Therefore, e=(e 1 ,e 2 ,e 3 ,e 4 )=(8,9,8,8). In step S3, Bob uses M p Base measurement The wandering particles obtain Where i = 1, 2, 3, 4, so Bob gets m p =(14,2,8,5), furthermore, m p y=(28,6,-48,-35). In step S4, TP generates In step S6, Bob generates r through QRNG b =19, apply U 19 and G 28 ,G 6 ,G -48 ,G -35 get calculate and Then send to TP r b ,s b and cIn step S8, TP is calculated And apply U -19 and G 5 get In step S9, TP uses M p Base measurement The wandering particles get

[0186] Figure 7 The simulation circuit of the TPQSP calculation method of the present invention is shown. The three dotted lines Figure 7 Divided into four parts: (1) The simulation line on the left side of the first dotted line represents (2) The simulation line on the left side of the second dashed line represents the simulation of Bob applying U 19 and G 28 ,G 6 ,G -48 ,G -35 ; (3) The simulation line on the left side of the third dotted line represents the TP application U -19 and G 5 ; (4) The simulation line on the right side of the third dotted line represents the TP pair Quantum measurement of wandering particles. Figure 8 Shown Figure 7 The measurement results after 10,000 simulations consistently show Because p b =13, so we have This verifies the correctness of the TPQSP calculation method of the present invention.

[0187] 2 Discussions

[0188] 2.1 Efficiency

[0189] Reference

[39] proposed the concept of quantum bit efficiency to evaluate the communication performance of quantum methods in 2D quantum systems. Later, reference

[40] generalized the quantum bit efficiency of reference

[39] to quantum bit efficiency, which was used to calculate the communication efficiency of quantum methods in dD quantum systems. Quantum bit efficiency is defined as

[40]

[0190]

[0191] in, is the length of the secret input, is the quantum bottom number used in the quantum channel, is the number of classical bits consumed in the classical channel. It should be noted that this efficiency ignores QRNG, eavesdropping detection, and the transmission of ∈,s through QSDC b and cThe quantum and classical resources consumed.

[0192] In the method of the present invention, Alice's secret vector is x=(x 1 ,x 2 ,…,x L ), Bob’s secret vector is y=(y 1 ,y 2 ,…,y L ),in And i=1,2,…,L, so Alice prepares L initial QW states in step S1. TP prepares a QW state in step S4 Therefore Alice sets r=(r 1 ,r 2 ,…,r L ) to Bob and p a Send to TP, Bob in step S6 b Sent to TP, so Therefore, the quantum efficiency of the method of the present invention is When L is large enough, η converges to

[0193] 2.2 Comparison

[0194] Next, the method of the present invention is compared in detail with the four QSP calculation methods in references [15-18]. The comparison results are shown in Table 3. Table 3 can be summarized as follows: (1) Compared with the QSP calculation methods in references [15-18], the method of the present invention has advantages in the initial quantum resources used, because the QW state is a two-particle product state, which is much easier to prepare; (2) In terms of the use of QFT or IQFT, the method of the present invention is superior to the methods in references [17-18]; (3) In terms of quantum measurement, compared with the method in reference

[16] , the method of the present invention has advantages.

[0195] Table 3 Comparison of different QSP calculation methods

[0196]

[0197] 3 Conclusion

[0198] Based on the TDQWC system, the present invention designs a novel secure TPQSP calculation method, which can calculate SP(x, y) while keeping x and y secret. The method of the present invention only requires d-dimensional single-particle measurement and 2-dimensional single-particle measurement. The correctness of the method of the present invention has been deeply verified through theoretical analysis and IBM Qiskit simulation experiments. Security analysis shows that the method of the present invention has information-theoretic security and is immune to external attacks and participant attacks.

Claims

1. A secure two-party quantum scalar product calculation method based on bidirectional quantum walk on a circle, which realizes the calculation of the scalar product of the secret vectors of two communicating parties without leaking secret information; using the two-particle product state instead of the quantum entangled state as the initial quantum resource, only d-dimensional and 2-dimensional single particle measurements are required instead of quantum entangled state measurements; including the following nine processes: S1) The secret vector of A1ice is x = (x1, x2, ..., x L ), Bob’s secret vector is y=(y1, y2, …, y L ),in and i=1,2,…,L; Alice sets And transmit ∈ to the semi-loyal TP through quantum secure direct communication, where ∈∈{0,1,…,d-1}, e i ∈{0,1,…,d-1}, and Alice generates r=(r1, r2, …, r L ),in and i = 1, 2, ..., L; then, Alice prepares the initial QW state where p a ∈{0, 1, ..., d-1} and i = 1, 2, ..., L; then, Alice Apply in random order and get in symbol represents modulo d addition, i = 1, 2, ..., L; then Alice generates λ random c decoy particles of the base or {|+>,|-)} base, where M c = {|0>, |1)} and and will The particles are randomly mixed with the above λ decoy particles to form a new particle sequence. Finally, Alice sends this particle sequence to Bob through the quantum channel, and sends r to Bob and p to Bob through the authenticated classical channel. a Give TP; S2) Alice informs Bob of the specific locations of the two decoy particles and their preparation bases; Bob performs quantum measurement on the λ decoy particles with the corresponding preparation bases and informs Alice of the measurement results; Alice calculates the error rate by judging whether the initial preparation states of the λ decoy particles are consistent with the corresponding measurement results of Bob; if the error rate is not 0, the communication will be stopped immediately, otherwise it will continue; S3) Bob Imposition get Afterwards, Bob uses M p Base measurement The wandering particles get Using M c Base measurement The coin particles get where Mp = {|0>, |1>, ..., |d-1>} and i = 1, 2, ..., L; Bob calculates in Then, Alice and Bob announce and , and calculate the error rate The symbol "∨" represents a bitwise XOR operation; if θ≠0, the communication will be stopped immediately, otherwise it will continue to be executed; S4) TP preparation where p b ∈{0, 1, ..., d-1} and c b ∈{0, 1}; then, TP generates λ random c decoy particles of the basis or {|+>,|->} basis, and Randomly mix with the above λ decoy particles to form a new particle sequence; finally, TP sends this particle sequence to Bob through the quantum channel; S5) TP and Bob perform eavesdropping detection similar to step S2; if the error rate is not 0, the communication will be stopped immediately, otherwise it will continue to be executed; S6) Bob generates a quantum random number generator Afterwards, Bob Apply in random order and get And calculate and Then, Bob generates λ random nodes in M c decoy particles of the basis or {|+>,|->} basis, and The particles are randomly mixed with the above-mentioned λ decoy particles to form a new particle sequence; finally, Bob sends this particle sequence to TP through the quantum channel, transmits rb to TP through the authenticated classical channel, and transmits s through quantum secure direct communication. b and c Give TP; S7) Bob and TP perform eavesdropping detection similar to step S2; if the error rate is not 0, the communication will be stopped immediately, otherwise it will continue to be executed; S8)TP calculation Then Apply in random order and G ρ get Afterwards, TP uses M c Base measurement The coin particles get And calculate the error rate If δ≠0, the communication will be stopped immediately, otherwise it will continue to be executed; S9)TP uses M p Base measurement The wandering particles get Then calculate The symbol Represents modulo d subtraction; TP announces the value of SP(x,y) to Alice and Bob respectively.

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