The phase matching quantum key distribution protocol with 3-state systems
aa r X i v : . [ qu a n t - ph ] M a r Noname manuscript No. (will be inserted by the editor)
The phase matching quantum key distributionprotocol with 3-state systems
Han Duo · Li Zhihui B · Liu Chengji · Gao Feifei
Received: date / Accepted: date
Abstract
Quantum Key Distribution, as a branch of quantum mechanicsin cryptography, can distribute keys between legal communication parties inan unconditionally secure manner, thus can realize in transmitting confiden-tial information with unconditional security. We consider a Phase-MatchingQuantum Key Distribution protocol with 3-state systems for the first time,where the phase of the coherent state is 3,thus we propose three different waysto response to every successful detection and two parties gain their raw keysby “flip and flip”. The simulation results show that compared with Phase-Matching Quantum Key Distribution protocol where the phase equals 2, theproposed protocol breaks the limit of linear key generation rate in a shorterdistance, and the longest practical transmission distance is about 470 km ,whereas the ones of BB84 protocol is lower than 250 km . Keywords
Quantum Key Distribution · Phase · PM-QKD protocol
Han DuoSchool of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China,710119.Tel.: +18829287176E-mail: [email protected] Zhihui B School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China,710119.Tel.: +13032989886E-mail: [email protected] ChengjiState Key Laboratory of Integrated Services Networks, Xidian University, Xian, China,710071.Tel.: +15091056770E-mail: [email protected] FeifeiSchool of Mathematics and Information Science, Shaanxi Normal University, Xi’an, China,710119.E-mail: [email protected] Han Duo et al.
Quantum cryptography [1] is an interdisciplinary subject combining cryp-tography and quantum mechanics [2] .It is an important research topic.Itssecurity is based on the basic principles of quantum mechanics, such as quan-tum non-cloning theorem, uncertainty principle [3] et al., the quantum keydistribution technology in quantum cryptography provides a means of com-munication for both parties to obtain unconditional security keys. Securityand practical applications are the core of this research.The first quantum cryptography protocol,the BB84 quantum key distribu-tion protocol [1] , was proposed by Bennett and Brassard in 1984, which in-troduced quantum mechanics into practical applications. However, until 1999,Lo and Chau proved the security of the BB84 protocol by equating the BB84protocol with an entanglement and purification protocol [4] ;but the quantumcomputer was needed in the proof process. Then in 2000, Shor and Preskill pro-posed a more concise proof method for CSS quantum error correction code forentanglement and purification [5] ;which removed the dependence on quantumcomputers. Lo et al’s research further proved bit error correction and phaseerror correction can be implemented separately [6] .Although the theoretical security of the BB84 protocol has been proved,its actual implementation still has a large security risk. Since there is no idealsingle-photon source [7] in practice, a weak coherent pulse (Weak CoherentPulse) [8] is commonly used to simulate a single-photon source, which leadsto the generation of Photon Number Splitting [9] . In 2003, the problem wassolved for the first time, since a decoy-state scheme [10] was proposed to de-fend against PNS attacks. Besides the hidden dangers of the light source,the detector side channel attack [11] also greatly threatened the security ofthe password. HK proposed decoy Measurement-Device-Independent Quan-tum Key Distribution (MDI-QKD) protocol [12] in 2012, which eliminates thedetector side channel attack [13] without introducing more implementationequipment and double the transmission distance covered by the traditionalQKD scheme at the same time.However, these QKD protocols has same limitations—they never exceedthe limit of the Secret Key Capacity (SKC) [14] of the lossy optical quantumchannel. X.B.Wang et al. proposed the Twin-field Quantum Key Distribution(TF-QKD) protocol [15] in 2018, which broke the SKC bound of the previousQKD protocols under the condition of ensuring key security. The square rootdependence of the key generation rate on the channel transmittance is ob-tained. However, the security of the agreement has not been proven. X.F.Maet al. proposed the Phase-matching Quantum Key Distribution (PM-QKD)protocol in 2018 [16] which illustrated a security proof based on optical mode,and resisted all possible measuring attacks.In the PM-QKD protocol, the communication parties Alice and Bob eachgenerate coherent state pulses independently. For a d-phase PM-QKD proto-col, Alice and Bob encode their key information κ a , κ b ∈ { , , · · · · d − } intothe phase of the coherent state. Paper[16] mainly studied the the PM-QKD he phase matching quantum key distribution protocol with 3-state systems 3 in the case of phase d=2 (2-PM-QKD protocol) with phase randomization. Intheory, the protocol is immune to all possible measurement attacks, and its keyrate can even exceed the transmission probability η between two communicat-ing parties; In practice, the protocol applies phase compensation to devise apractical version of the scheme without phase locking [17] , which makes theproposed scheme feasible in current technology.Inspired by the PM-QKD protocol in [18] , this paper proposes a new PM-QKD whose phase d=3 with phase randomization(For simplicity,we use thename “3-PM-QKD protocol” in the text below).In the 2-PM-QKD protocol, each transmitted 32-bit binary bit can encodethe largest unsigned number, but if the 3-PM-QKD protocol is used, every 32-bit ternary trit can be successfully transmitted. In addition, in this protocol,the range of random phase matching is wider and the probability of successfulmatching is higher. Alice and Bob retain their key trits when their declaredrandom phases difference is 0, π , or π , which significantly improves thesuccess rate of the phase sifting phase and also results in a higher final keyrate.The paper organized as follows. Following the 2-PM-QKD protocol, wepropose the 3-PM-QKD protocol that can surpass the linear key-rate boundand make the key rate increase, whose details are given In Sec.2. Then, thesecurity of 3-PM-QKD is proved in Sec.3, and in Sec.4, we consider all practicalfactors to simulate the 3-PM-QKD key rate and compare it to the previousQKD protocol. Finally, we summarize this work , put forward the the n -PM-QKD protocol and expound its some curious features in Sec.5. This paper proposes a 3-PM-QKD protocol with phase randomization.Thatis,Alice and Bob add extra random phases on their coherent state pulses be-fore sending these pulses to Eve. After Eve’s announcement,Alice and Bobannounce the extra random phases and postselect the signals based on theirrandom phases.The specific steps and related descriptions are as follows.2.1 Specific steps
Step1
State Preparation - Alice randomly generates a key trit κ a ∈ { , , } anda random phase φ a ∈ [0 , π ), and then prepares a coherent state (cid:12)(cid:12)(cid:12) √ µ a e i ( φ a + π κ a ) (cid:12)(cid:12)(cid:12) A .Similarly, Bob generates κ b ∈ { , , } and φ b ∈ [0 , π ) then prepare (cid:12)(cid:12)(cid:12) √ µ b e i ( φ b + π κ b ) (cid:12)(cid:12)(cid:12) B . Step2
Measurements - Alice and Bob send their light pulses A and Bto an untrusted Eve,which needs to perform interferometry and record theresponse detector ( D , D , or D ). In particular,the detector response rulesare as follows: ∆ φ = (cid:12)(cid:12)(cid:12)(cid:12) φ a + 2 π κ a − (cid:18) φ b + 2 π κ b (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) π κ a − κ b ) + ( φ a − φ b ) (cid:12)(cid:12)(cid:12)(cid:12) . Han Duo et al.
The way the detector responds in this protocol depends on the phase dif-ference between Alice and Bob,The detector response mechanism is set to: D response, when∆φ = 0 (mod2 π ) ,D response, when∆φ = π (mod2 π ) ,D response, when∆φ = π (mod2 π ) . Step3
Statement - Eve announces his detection result. Then Alice andBob announce random phase φ a and φ b ,respectively. Step4
Sifting - Alice and Bob repeat the above steps multiple times. WhenEve announces a successful response (just one detector response),Alice andBob make the κ a and κ b the raw key trits.According to Eve’s statement, Bob flips his key trits κ b accordingly.Theflipped key trits are recorded as κ ′ b . The flip rule is as follows: κ ′ b = κ b , if D responce,κ ′ b = κ b + 1(mod2 π ) , if D responce,κ ′ b = κ b + 2(mod2 π ) , if D responce. When Alice and Bob respectively announce random phases, Bob flips hiskey trits κ ′ b again according to their random phase difference | φ a − φ b | .Theflipped key trits are recorded as κ ′′ b .Theflipping rules are as follows: κ ′′ b = κ ′ b (mod2 π ) , if | φ a − φ b | = 0 (mod2 π ) ,κ ′′ b = κ ′ b − π ) , if | φ a − φ b | = π (mod2 π ) ,κ ′′ b = κ ′ b − π ) , if | φ a − φ b | = π (mod2 π ) . Finally, Bob’s key trits are κ ′′ b . Step5
Parameter Estimation - Alice and Bob derive the gain Q µ andquantum trit error rate E Zµ from all the retained raw data and then estimate E Xµ . Step6
Key Distillation - Alice and Bob perform error correction and pri-vacy amplification on the sifted key trits to generate a private key (note thatthe error correction and privacy amplification of this protocol are the sameas in all QKD protocols except that we must use trits Not bits, so the paritybecomes a ternary test, ie the modulus is 3 [18] ).In the actual implementation of this protocol, Alice and Bob retain theirsignals only when their declared random phases difference is 0, π , or π .However, due to the announcement of phase continuity, the probability ofsuccessful sifting is 0. In addition, the phase locking technique required inactual implementation is very difficult . Therefore, we use the phase postcompensation method [19] like paper.The post-phase compensation method used here is similar to 2-PM-QKDprotocol. Alice and Bob divide the phase interval [0 , π ) into M slices first.When a random phase is declared, Alice and Bob only compare the slice indi-cators, not the exact phase. This makes the step of phase sifting practical, butintroduces inherent bias errors. This bias error can compensate for the inher-ent bias error by sacrificing a portion of the data, minimizing the quiz error he phase matching quantum key distribution protocol with 3-state systems 5 rate QBER based on random sampling, and calculating the appropriate phaseoffset. In addition, Alice and Bob do not perform phase sifting immediately ineach round,but perform this phase sifting in data post-processing. This makesthe 3-PM-QKD protocol practical.More importantly,in the 2-PM-QKD protocol, each successfully transmit-ted 32-bit binary bit can encode the largest unsigned number to 2 − − π , or π , which significantlyimproves the success rate of the phase screening phase and also results in ahigher final key rate.To illustrate the feasibility of this protocol, this paper presents a simple keycorrespondence table to illustrate how Alice and Bob match key informationby “flip and flip” successfully.2.2 Key-correspondence table of this protocolIn the 3-PM-QKD protocol, the key information κ a ( b ) ∈ { , , } , so a success-ful key is generated if and only if the random phase difference is an integermultiple π (less than anon-negative integer multiple of 3).There are 27 cases,and we list the situation when φ a and φ b satisfy with | φ a − φ b | = π (mod2 π )in this section, and other cases are equally available. Table 1
Key-correspondence table with | φ a − φ b | = π (mod2 π ) κ a κ b | φ a − φ b | ∆ φ Response κ ′ b κ ′′ b π/ π/ D π/ π/ D π/ D π/ D π/ π/ D π/ π/ D π/ π/ D π/ D π/ π/ D The Table I shows that accurate detection response and key sifting canguarantee a successful key match with a probability close to 1, and the prob-ability of successful match is higher than that of 2-PM-QKD protocol.
Han Duo et al.
Unlike the general QKD protocol security proof, the commonly used pho-ton number channel model [20] and the “tagging” method used in the securityproof by Gottsman et al.(GLLP security proofs) [21] are no longer applicablehere. This is because the random phases of Alice and Bob are annonced inthis protocol, and the quantum source can no longer be regarded as a mixtureof photon number states. But as mentioned in [16] , we can directly analyzethe optical mode by applying Lo-Chau entangled distillation theory to demon-strate the security of PM-QKD with coherent pluses.The proof of the security of this protocol is followed by the analysis ofdistillable entanglement based on the equivalent entanglement protocol, andtransforms it into 3-PM-QKD protocol gradually. The security performanceis proved in each operation. The proof process is similar to the 2-PM-QKDprotocol, except we must use trits instead of bits, so we will not repeat themhere.
Since our solution is generalized from 2-PM-QKD, the simulation in thissection is mainly compared with the 2-PM-QKD protocol. According to [16],when l > km , the key rate of 2-PM-QKD exceeded the key rate of tra-ditional BB84 protocol; when transmitting distance l > km , 2-PM-QKDcould exceed the limit of linear key rate; Compared with MDI-QKD, 2-PM-QKD can achieve a longer transmission distance of l = 450km, and at thetime l > km , the key rate increased by about 4-6 orders of magnitude.This section will prove that the 3-PM-QKD protocol is better than the 2-PM-QKD protocol, and thus better than the traditional BB84 protocol and theMDI-QKD protocol.Applying the key rate formula in Shor-Preskill’s security proof [5] r = 1 − H (cid:0) E Z (cid:1) − H (cid:0) E X (cid:1) . (1)And the key rate formula in [16] R − P M ≥ M Q µ (cid:2) − f H (cid:0) E zµ (cid:1) − H (cid:0) E Xµ (cid:1)(cid:3) . (2)Where E Z and E X are the Z error rate and the X error rate, respectively. H ( x ) = − x log x − (1 − x ) log (1 − x ) are the binary Shannon entropy function, Q µ is the phase error rate, and f is the error correction efficiency.Because we have only expanded the value range of the key bits in thispaper, other parts of the key rate formula are still similar to the 2-PM-QKDprotocol, but the phase sifting factor in this article is 3 / M, Shannon’s entropyfunction becomes H ( x ) = − x log x − (1 − x ) log (1 − x ) for the three-statesystem.Finally, our key rate formula is he phase matching quantum key distribution protocol with 3-state systems 7 R − P M ≥ M Q µ (cid:2) − f H (cid:0) E zµ (cid:1) − H (cid:0) E Xµ (cid:1)(cid:3) . (3)We use the parameters given in the Table 2 below to simulate the per-formance of 3-PM-QKD. Assuming that the lossy channels of Alice and Bobare symmetrical; the dark count rate is from Ref. [22] , and other parametersare set to classic values. (Note that in order to identify the effect of the keyrate is on the protocol itself rather than other parameters , the actual settingparameters used in this article are completely consistent with the 2-PM-QKDprotocol). Table 2
Parameters used for simulation in 3-PM-QKD protocolParameters ValuesDark count rate p d × − Error correction efficiency f η d M e d The simulation results are shown in the following figure.
Communication Distance(Km) K e y R a t e l g ( R ) -14 -12 -10 -8 -6 -4 -2 Fig. 1
Simulation of our protocol. For the considered simulation parameters, the key rateis similar to 2-PM-QKD protocol, but it breaks the SKC bound in a shorter distance andthe effective transmission distance has increased by about 20 kilometers.
As can be seen, our protocol has a small increase on key rate compared withthe 2-PM-QKD protocol, but the effective transmission distance has increased
Han Duo et al. by about 20 kilometers, and it has broken through SKC bound in a shorterdistance.
This paper proposes the 3-PM-QKD protocol and proves its security. The3-PM-QKD protocol not only breaks the boundaries of SKC, but also reducesthe distance to break the boundaries of SKC,meanwhile,the proposed protocolincreases the key rate and effective transmission distance.Also,the higher-dimensional promotion of the 3-PM-QKD protocol in thispaper will be an interesting direction, such as the n -PM-QKD protocol withphase randomization. This is a very difficult problem, because when space isextended to any dimension, it is difficult to express all its properties strictly,but this attempt is very interesting. At the time κ a ( b ) ∈ { , , , · · · , n } , al-though the selection interval of the random phase was still the same, theprobability of successful screening could approach 100%.This is an interest-ing change, which is because there is always a exact detector response for thespecific value of any phase difference, but this requires extremely accurate de-tector standards. In any case, this will be one of the efforts of QKD protocolin practice.The specific steps of n -PM-QKD is similar to 2-PM-QKD, whichwill not be described in detail here.However, the n -PM-QKD protocol is currently only possible theoretically,but if it can be successfully implemented, that will greatly increase the keyrate of the QKD protocol and guarantee a 100% probability of successful phasematching, in which the corresponding parameter estimation and security proofwill be the largest challenge . Once the phase of the coherent state is extendedto infinite dimensions, it may have some distinctive properties, which is asubject worthy of study. Acknowledgements
We would like to thank anonymous reviewer for valuable comments.Thiswork is supported by the National Natural Science Foundation of China under GrantNo.11671244.