A Three-stage Phase Encoding Technique for Quantum Key Distribution

Transcription

1 A Three-stage Phase Encoding Technique for Quantum Key Distribution F. Zamani, S. Mandal, and P. K.Verma School of Electrical and Computer Engineering, University of Oklahoma, Tulsa, Oklahoma, USA Abstract - The only known transmission technique that provides unconditional security is based on quantum cryptography. The current workhorse of quantum cryptography, more precisely, quantum key distribution (QKD), is based on the BB 84 protocol. Most commercial QKD implementations are based on phase-coding the BB 84 protocol, where the unbalanced Mach-Zehnder Interferometer (UMZI) is used as the information coder and decoder. This paper presents a three-stage phase encoding quantum key distribution protocol which is based only on one two-way quantum channel. This protocol does not need key sifting for key establishment. The proposed technique thus obviates the use of a classical communication channel for key sifting and key distillation which are subject to eavesdropping. The proposed technique thus increases the key efficiency and security compared with the current techniques. Keywords: Quantum key distribution (QKD); sifted key; key reconciliation; Phase coding protocol; Plug and play system 1 Introduction The aim of cryptography is to prevent a cryptanalyst from deciphering information sent by one party to another party. In classical cryptography, the process of encryption and decryption of information is based on mathematical algorithms and the security of the classical cryptography is based on the difficulty of these algorithms. Quantum cryptography is a good solution for providing key distribution. Quantum key distribution (QKD) applies the quantum mechanics concept to guarantee security of a key exchange between two legitimate parties in the presence of cryptanalyst [1, 2, 3, 4]. According to the Heisenberg uncertainty principle and no-cloning theorem, measuring or copying the quantum state of the qubit alters the original information in the qubit [5]. In quantum key distribution, we are able to detect whether the eavesdropper has eavesdropped on the message or not, while the classical cryptography does not offer this feature. The BB 84 is the most commonly used quantum key distribution protocol proposed by Bennett and Brassard in 1984 [5, 6]. It utilizes one quantum channel and one two-way classical channel. The classical channel is a regular telecommunication channel, over which Alice and Bob exchange information. The aim is establishing a secret key between two authorized parties, Alice and Bob, with the possibility that Eve can be present during the key establishment process. The original proposal of BB 84 relies on the polarization encoding of a photon. Since the alignment and the stabilization of polarization axes between Alice and Bob is practically difficult and depolarization happens due to medium transmission changes, the phase-coding technique for implementing BB 84 is used. Called the Plug and Play protocol, it is one of the common implementations of BB 84. The Plug and Play system utilizes one two-way quantum channel and one two-way classical channel which is basically an Ethernet channel. This classical channel is used for key sifting and key distillation. In this paper we propose a three stage QKD protocol which uses a single two-way quantum channel, thus removing the need for depending on a classical channel. The proposed mechanism will increase the security and key efficiency of the key establishment procedure and there will be no need to have the key sifting step which reduces the size of the final key. The rest of the paper is organized as follows: the phase coding BB 84 protocol reviewed in Section II, Section III describes Plug and Play system, Section IV describes our proposed protocol, Section V illustrates the scenario in the presence of an eavesdropper and in Section VI the various advantages of the proposed protocol are described as compared with other phase coding protocols. Finally, in Section VII, we present our conclusions. 2 Phase coding BB 84 Protocol As mentioned earlier, the BB 84 protocol was first implemented using polarization encoding. At present, phase encoding is used because using the polarization states causes some alignment and stabilization difficulties in practice. The basic configuration of phase coding is shown in Fig. 1 [7, 11]. The sender (Alice) transmits a photon through an asymmetric Mach Zehnder interferometer. When the photon passes through the interferometer the phase difference a between the two paths is randomly chosen from one of four values, namely {0, π} and {π/2, 3π/2} by PMA (phase modulator of Alice). After sending the photon to the receiver, Bob receives the photon and then passes it through his interferometer which is identical to Alice s. The phase difference b on Bob s side is randomly chosen from {0, π/2} by PMB (phase modulator of Bob).The photon is then detected at one of the two interferometer s outputs, each of which has a detector. Using this method, a secret key is generated by the protocol which is as follows:

2 4) The linear polarization is rotated by 90 degrees in the shorter arm of the interferometer, not shown in Fig. 2. 5) Both the beams then recombine and come out one after another from Bob s side. (The first pulse passed the short and the second pulse passed the long arm of the interferometer). 6) The pulses are then transmitted to Alice through the optical fiber. Figure 1. Schematic diagram of phase coding protocol 1) A number of photons are sequentially transmitted from Alice to Bob. 2) After the completion of the transmission through the quantum channel, Bob tells Alice which phase was chosen by him to detect the photon. 3) Alice lets Bob know using the classical channel, whether she chose a from {0, π} or from {π/2, 3π/2) for the detected photon. This phase information enables Bob to know if the detection event was deterministic or probabilistic. 4) In case of deterministic detection events, Alice considers a zero (X basis) axis or π/2 (Y basis) as bit 0 and b equals π (X basis) or 3π/2 (Y basis) as bit 1, whereas, Bob considers the detector 1 click as bit 0 and the detector 2 click as bit 1. In case of probabilistic detection events, they ignore them. A bit string created in this step is the sifted key. 5) Finally, after obtaining the sifted bit string, the two parties should communicate with each other through the classical channel to apply error correction and privacy amplification to obtain the final secret key. 2.1 Plug and Play System In this section, we describe a phase coding autocompensating system which is part of the Plug and Play implementation [8, 9, 10, 11]. The structure of the phase coding auto-compensating system is as shown in Fig. 2. The steps under which the system operates are as follows: Bob s Side: 1) Laser (L) produces strongly linearly polarized pulses of photons on Bob side. 2) The beam is then separated into two equal parts at the 50/50 beam splitter (BS). 3) The long arm of the interferometer contains a delay line (DL) and the Bob s phase modulator (PMB) is not used at this step of transmission. Alice s side: 7) Pulses that reach Alice, passing the BS 10/90 (90% of the intensity will be registered in the detector DA). 8) The other output of the beam splitter which is 10% is attenuated by the variable attenuator (VA) and reflected by a Faraday Mirror (FM), where the polarization states are reversed. 9) Alice applies a phase of 0 or π (bit 0 and 1 in the X-basis) and π/2 or 3π/2 (bit 0 and 1 in the Y-basis) on the second pulse for implementing the BB 84 protocol with her phase modulator (PMA). 10) The two pulses coming out as the output from Alice s side are orthogonal to each other but they have their polarizations interchanged because they have been reflected by the Faraday Mirror (FM). Thus, a compensation of all accumulated polarizations changes can take place on the way back from Alice to Bob. Bob side: 11) Two pulses arrive at Bob s interferometer and the first pulse now enters the long arm because of the changed polarization states. 12) Bob randomly chooses the measurement basis by applying a 0 or a π/2 phase shift on the first pulse by using his phase modulator (PMB). 13) The second pulse passes the short path in the interferometer. 14) Both pulses arrive at the same time at beam splitter (BS) and interfere with each other. 15) They are detected either in the detector 1 (D1) or after passing through the circulator (C) in the detector (D2). 16) The system is a usual QKD system which is using phase encoding between coherent pulses for transmitting a key from Alice to Bob. If Bob s phase (PMB) = 0 and Alice s phase (PMA) = 0 or π, then measuring it in the X basis, one of Bob s detectors obtains a conclusive result, which thereby determines the bit to be a 0 or 1. On the other hand, when the phase of PMB is 0 and that of PMA is π/2 or 3π/2, either of the two detectors of Bob clicks with equal probability. This is because Alice chooses the Y-basis and Bob chooses the X-basis, which are different bases. A complementary process happens for PMB=π/2.

3 Figure 2. Schematic diagram of Plug and Play system, (FM: Faraday Mirror, BS: beam splitter, PBS: polarization beam splitter, PMA: Phase modulator of Alice, PMB: Phase modulator of Bob, DL: delay line, VA: variable attenuator, C: circulator, D1&D2&DA: detectors, L: laser, SL: storage line) Alice and Bob after exchanging raw keys through the quantum channel need to communicate with each other through the classical channel to sift the raw key and discard Bob s random clicks. Also, after obtaining the sifted bit string, the two parties should communicate with each other through the classical channel to apply key distillation and Privacy Amplification to obtain the final secret key. In this paper, we propose a new quantum key distribution protocol, which utilizes a two way quantum channel and does not need the two parties to communicate to get the sifted key. This three stage quantum key distribution protocol will be discussed in the next section in detail. 2.2 The Proposed Protocol As we discussed in the previous section, the Plug and Play system is based on using a classical channel for key sifting and key reconciliation. Our proposed protocol is based on encoding the qubits in a similar manner to the Plug and Play system, but we propose to use a two way quantum channel for the quantum key distribution. The idea for omitting the classical channel to sift the key comes from Kak s three stage polarization based protocol which is the only QKD protocol using three stages for key establishment [14,15]. The key establishment technique of our proposed protocol is described in Fig. 3. In the proposed protocol Alice has a bit string that she wants to share with Bob as the secret key. The key distribution procedure is achieved in the following way: 1- Strong linearly polarized pulses of photons are produced by a laser from Alice s side. The qubits are encoded in the relative phase between two subsequent pulses. 2- Alice applies an arbitrary phase of α1 to the first pulse and β1 to the second pulse and sends these two pulses through the quantum channel to Bob. 3- Bob receives these pulses and applies another arbitrary phase α2 to the first pulse and β2 to the second pulse and sends them back to Alice. It should be mentioned here that these arbitrary phases are known only to Alice and Bob respectively. 4- Alice again receives the two pulses from Bob and applies α1 to the first pulse and β1 to the second pulse for encoding bit 0. On the other hand, for encoding bit 1, Alice applies ( α1+π) and β1 to the first and second pulses respectively and sends them back to Bob. 5- Bob receives the two pulses from Alice and applies α2 and β2 to the first and second pulses. These pulses after passing through the unbalanced interferometer are finally detected either in detector 1 (bit 0 ) or detector 2 (bit 1 ) according to their phase difference (0 or π). The system structure of the proposed protocol is sketched in Fig. 4. In the proposed system, at Alice s side, a laser produces strong linearly polarized pulses of photons. The beam is then separated into two parts at the 50/50 beam splitter (BS) and enters into the unbalanced interferometer with a delay line DL in the long arm, which produces two pulses. Then with the help of the phase modulator (PMA1), Alice applies an arbitrary phase of α1 to the first pulse and β1 to the second pulse and sends these two pulses through the quantum channel to Bob. Now Bob uses his phase modulator (PMB1) to apply another arbitrary phase α2 to the first pulse and β2 to the second pulse and sends them back to Alice. Next, Alice encodes bit 0 and 1, according to the key establishment procedure described above using the phase modulator (PMA2) and sends these encoded qubits back through the quantum channel (optical fiber) to Bob. Bob receives the two pulses from Alice and applies ( α2) and ( β2) to the first and second pulses using his phase modulator (PMB2). Bob then inverts the polarizations of the pulses and sends them to the interferometer. Due to this reversal of the polarizations of the pulses, the pulses which had travelled through the shorter arms of the interferometer at Alice s side will pass through the long arm of the interferometer at Bob s side and vice versa. As a result, the two pulses will reach the beam splitter where they interfere. Then they are detected

4 either in detector 1(D1) or detector 2 (D2) according to their phase difference. Now Bob considers the detector 1 clicks as bit 0 and the detector 2 clicks as bit 1.Thus, in this way, Alice shares her identical key with Bob without the process of key sifting and hence obviating the need for a classical channel in this step. We will consider the existence of Eve in our protocol in the next section. 2.3 Impact of an Intruder One of the primary advantages of the proposed protocol is that Alice and Bob use arbitrary phases each time for the key establishment and thus an intruder Eve cannot guess the actual guess the actual phase which is applied to the pulses in each transmission. So, in this case, Eve cannot get any information from avesdropping on the channel and she can only apply some random phases and disturb the information which Bob receives. Alice α1 β 1 Bob α1 β 1 For Encoding bit 0 α1+α2- α1 β 1+β 2- β 1 α1+α2- α1 β 1+β 2- β 1 α1+α2- α1-α2 β 1+β 2- β 1-β 2 D1 For Encoding bit 1 α1+α2- α1+π β 1+β 2- β 1 Δθ = 0 α1+α2- α1+π β 1+β 2- β 1 α1+α2- α1-α2 +π β 1+β 2- β 1-β 2 Δθ = π D2 Fig.3. Schematic diagram of key establishment procedure of the proposed protocol Alice Bob LASER BS DL H V (α1,β1) PMA1 1 2 PMA2 Optical fiber (-α2,-β2) PMB2 2 1 PMB1 V H PBS DL D2 D1 For bit 0 :(-α1,-β1) For bit 1 :(-α1+π,-β1) Encoding Box (α2,β2) Figure 4. Schematic diagram of proposed protocol system, (BS: beam splitter, PBS: polarization beam splitter,pma1&pma2: Phase modulator of Alice, PMB1&PMB2 : Phase modulator of Bob, DL: delay line, C: circulator, D1&D2: detectors)

5 In the case where Eve applies some random phase to the pulses in order to disturb the information, the phase difference of the two pulses will be detected randomly by the two detectors. To overcome this vulnerability, Alice and Bob put some portion of the previous established key as a test key and send it at the first of the transmission in order to check the presence of any intruder on the channel. If any of the two parties get some error in the test key, they know that there is an intruder present and they abort the transmission. Moreover, for avoiding the man in the middle attack, Alice and Bob use this test key part for authentication and thus in this case there is no chance for Eve to attack the protocol by disguising to be the man-in-the-middle. 2.4 The Advantages of Proposed Protocol in Comparison with the Other Phase-Coding Protocols 1. Raw key efficiency is defined as the length of the raw key shared by Alice and Bob divided by the length of the random bits generated by Alice [10]. In the proposed protocol due to the usage of two-way quantum channel and three stages of transmission, the raw key efficiency of the protocol can reach 100 percent, which is twice as much as that of the other phase coding protocols [12,13], which is only 50 per cent. 2. The other phase coding protocols need both quantum channel and classical channel for raw key generation, which is basically a weak channel in the sense of security but the proposed protocol only utilizes a twoway quantum channel for raw key generation and this causes Eve s information reduction dramatically. 3. The proposed protocol is more secure according to other protocol because there is no need to transmit the phase information through the classical channel and Eve cannot get any information about the key so the privacy amplification step which is necessary for the other protocols is obviated. As a result the key establishment security and efficiency in our protocol is more than others. 4. In the proposed protocol, Alice can send the encoded key stream to Bob and because there is no key sifting in this method, an effective coding scheme based on forward error correction will reduce the probability of error in the final key to an arbitrary small value so there is no need to have classical channel for error correction. Proposed method uses a three-stage protocol. Fiberbased polarization changes are addressed through the use of phase encoding and an auto compensation technique used in the plug and play system. 5. The proposed protocol is robust against photon number splitting attack and it does not need to have single photon generator because if Eve intercept the photons cannot get any information without having the phase of pulses. 3 Conclusions This paper has presented a novel means for exchanging quantum key between two entities, Alice and Bob. The technique is based on the use of a two-way quantum channel instead of one-way quantum channel as in BB 84. The two way quantum channel is used both for exchanging the quantum key as well as for replacing the conventional key sifting and key distillation on a classical channel. The proposed method thus obviates the need for a classical channel which is an additional security threat. The two-way quantum channel uses a three-stage protocol described in this paper. 4 References [1] Lo HK, Chau HF. Unconditional security of quantum key distribution over arbitrary long distances. Science 1999; 283(5410): [2] V.L. Kurochkin, I.G. 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