QUANTUM SECURE DIRECT COMMUNICATION USING ENTANGLEMENT AND SUPER DENSE CODING

Transcription

1 QANTM SECRE DIRECT COMMNICATION SING ENTANGLEMENT AND SPER DENSE CODING Ola M. Hegazy, Ayman M. Bahaa Eldin and Yasser H. Dakrury Cmputer and Systems Eng. Department, Ain Shams niversity, Cair, Egypt Keywrds: Abstract: Entanglement, Quantum secure direct cmmunicatin (QSDC), Super dense cding. This paper intrduces a new quantum prtcl fr secure direct cmmunicatin. This prtcl is based n Entanglement and Super-Dense cding. In this paper we present sme basic definitins f entanglement in quantum mechanics, present hw t use the maximally entangled states knwn as Bell States, and super dense cding technique t achieve secure direct message cmmunicatin. Finally, we will apply sme errr mdels that culd affect the transmissin f the quantum data n the quantum channels, and hw t treat these errrs and acquire a safe transmissin f the data. INTRODCTION The aim f cryptgraphy is t ensure that a secret message is transmitted between tw users in a way that any eavesdrpper cannt read it. Since classical cryptgraphy relies n difficulty and infeasibility f cmputatin t find the plain text, it is lsing security mre and mre as cmputatinal pwer is increasing by technical innvatins. In classical cryptgraphy, it is generally accepted that ne-time pad, which utilizes a previusly shared secret key t encrypt the message transmitted in the public channel, is the nly cryptsystem with prved security. Frtunately, quantum key distributin (QKD) (Bennett, 984), the apprach using quantum mechanics principles fr distributin f secret key, can vercme this bstacle skillfully. Since bth (QKD) and ne-time pad have been prved secure (Lee, 005), the cryptsystem f QKD & ne-time pad is a perfect ne when the security is cncerned. Previusly prpsed QKDPs are the theretical design (Bennett, 984), security prf (Massey, 988), and physical implementatin (Bennett, 99). Quantum secure direct cmmunicatin (QSDC) (Bström, 00, Deng, 008) is anther branch f quantum cryptgraphy. Different frm QKD, QSDC allws the sender t transmit directly the secret message (nt a randm key) t the receiver in a deterministic and secure manner. If it is designed carefully, a QSDC prtcl can als attain uncnditinal security (Deng, 003). The main bjective f ur research is t intrduce a new prtcl that guarantees mre security f the transmissin than the QKD and als saves mre time, cst and gives mre efficiency fr the transmissin, as it is using the super dense cding technique that transmit tw classical bits by sending ne quantum bit. In ur prtcl f the quantum secure direct cmmunicatin we use the maximally entangled Bell states t encde the message bits n the basis f the super dense cding therem, and then transmitting them n tw quantum channels t the ther side with less prbability f the eavesdrpping, and with n need fr a pre-shared key that in turn needs many runds t distribute, and als a public discussins t verify the crrectness f the key. BACKGROND The mst imprtant and interesting characteristics f the quantum mechanics is that the quantum state culd nt be measured withut disturbing and changing the state f the particles (phtns). S the use f quantum phenmenn will help in vercming ne f the mst imprtant eavesdrpping prblems; that is measuring the infrmatin withut being discvered, s any attempt f Eve t measure the data during transmissin will be knwn t Alice and Bb. 75

2 SECRYPT Internatinal Cnference n Security and Cryptgraphy Als, anther interesting feature f the quantum phenmenn is that any arbitrary quantum state cannt be clned r cpied and that is knwn as N- Clning therem (Nielsen, 000). Of curse, this will help in vercming anther eavesdrpping prblem which is cpying the transmitted signal, s Eve cannt take a cpy f the message during transmissin. These tw characteristics f the quantum phenmenn make it a strnger mechanism in securing the transmissin path mre than the classical transmissin. Quantum mechanics vilates everyday intuitin nt nly because the measured data can nly be predicted prbabilistically but als because f a quantum specific crrelatin called entanglement. Entanglement can be used t cause nn lcal phenmenn. States pssessing such crrelatins are called entangled states. Amng these states, the states with the highest degree f entanglement (crrelatin) are called maximally entangled states r EPR states, as histrically, the idea f a nn lcal effect due t entanglement was pinted ut by Einstein, Pdlsky, and Rsen (Hayashi, 006). The pure quantum nature f entanglement is the prperty f nn-lcal crrelatins between widely separated particles which have interacted in the past. T make particles entangled, it is necessary fr them t interact at a pint. In ther wrds, the nn-lcal prperty f entanglement is arisen frm the lcal prperty f interactin (Lee, 005). 3 THE SPER DENSE CODING PROCEDRE The super dense cding is a simple example f the applicatin f quantum entanglement cmmunicatin. The gal f this prcedure is t transmit tw classical bits by sending ne quantum bit (qubit), s increasing the efficiency f the transmissin. Befre starting the transmissin, it is assumed that a third party has generated an entangled state, ne f the Bell entangled state, fr example ( 00 ), and then sends ne f the tw pairs f the entangled qubits t the sender Alice and the ther t the receiver Bb. When starting the transmissin, Alice culd send the single qubit in her pssessin t Bb after perating n it in such a way t encde tw bits f the classical infrmatin t Bb. As there are fur pssible values f the tw classical bits Alice wishes t send t Bb: 00, 0, 0 and, then if Alice wants t send the tw bits 00, she des nthing t her qubit just simply send it as it is. If she wants t send 0, she applies the phase flip Z t her qubit. If she wants t send 0, she applies the quantum NOT gate X, t her qubit. If she wants t send, she applies the iy gate t her qubit. The fur quantum gates that are used here are: the Pauli matrices I, Z, X, iy, and cmbinatins f them are applied as the unitary peratin that Alice perfrms n her half f the EPR pairs accrding t the diagram in fig.() (Benenti, 004). Message classical bits H H Figure : A quantum circuit implementing the super dense cding. The use f these fur different transfrmatins results in the fur states f Bell states as in the fllwing equatins: 00 ( 00 ) () 0 ( 00 ) () 0 ( 0 0 ) (3) ( 0 0 ) (4) Of curse, the kind f the transfrmatin f the peratr will change accrding t the state that is generated in the first half f the circuit befre the peratr bx, as we tk an example f the state, but it culd be any ther Bell state. In all cases the generated state after the peratr (the encding circuit) will be als any ther ne f the Bell states but in different rder accrding t the classical bits that will be sent. 76

3 QANTM SECRE DIRECT COMMNICATION SING ENTANGLEMENT AND SPER DENSE CODING 4 THE NEW PROPOSED QSDC PROTOCOL SING SPER-DENSE CODING Our new prtcl f the quantum secure direct cmmunicatin uses the maximally entangled Bell states t encde the message bits using super dense cding that was mentined abve, and then transmit them n tw quantum channels t the ther side with less prbability f the eavesdrpping and mre efficiency by sending tw classical bits using ne quantum bit (qubit). 4. Basic Idea f the Prtcl ne f the Bell states which is randmly generated. ) Then after applying the unitary peratr n the qubit f Alice, the result that is the utput f the Alice encding circuit will be sent n ne f the public quantum channel and n a spatially separated quantum channel the Bb s qubit (the half f Bell state). 3) S accrding t the different generated Bell states we will have the functin matrix f peratr accrding t the fllwing table and analysis: a) In case f the carrier Bell state : sing the idea f the super dense cding therem, the message bits are input t the peratr selectr, then accrding t their value (ne f the fur pssible values 00, 0, 0 r ), and als accrding t the state f the entangled pair that will be generated randmly (ne f the Bell states) due t the inputs f the quantum selectr ( i, i0 ), the inputs t the Pauli peratr will drive it t perfrm ne f the fur unitary peratins mentined abve, the Pauli peratrs culd be I, X, iy r Z and their matrix representatins are as fllws: I,, 0, X 0 Y i Z 0 i 0 0 These steps are applied using the blck diagram f figure. 4. Assumptins f the Prtcl Alice will be the ne wh will prduce the EPR pairs (Bell states carriers) in her side nt a third party. Then she keeps ne half fr herself (and apply the encding f peratr n it) and sends the ther half t Bb. The Bell states that will be generated by Alice will be chsen n randm basis using a randm generatr, s we can get ne f the different fur maximally entangled Bell states,,,, and. 4.3 Steps f the Prtcl ) Alice inputs the message bits (M), -bits by - bits, as ne input t select the peratr as mentined abve with the ther input that is -bit element 00 I ( I I ) 0 X ( X I ) 0 Z ( Z I ) iy ( iy I ) Accrding t the fllwing analysis: ( I I ) ( X I ) ( Z I ) ( iy I ) (5) (6) (7) (8) 77

4 SECRYPT Internatinal Cnference n Security and Cryptgraphy b) In case f the carrier Bell state -bit element : 00 Z ( Z I ) 0 XZ ( XZ I ) 0 I ( I I ) iyz ( iyz I ) Accrding t the fllwing analysis: ( Z I ) ( XZ I) ( I I ) ( iyz I ) (9) (0) () () And similar t the same analyses the fllwing carriers will take the fllwing peratrs t get the same results. c) In case f the carrier Bell state : -bit element 00 X ( X I ) 0 I ( I I ) 0 iy ( iy I ) Z ( Z I ) d) In case f the carrier Bell state : -bit element 00 XZ ( XZ I ) 0 Z ( Z I ) 0 iyz ( iyz I ) I ( I I ) 4) After reached Bb, he starts t apply the apprpriate unitary peratins n the Bell states, measuring the tw qubits and btaining the -bit message element. 5) Bb perfrms the reverse peratin f the encding circuit, (decding circuit) as: ( CNOT ( H I ) ) ( H I ) CNOT (3) That is having the matrix representatin: B Therefre: 4.4 Cmments (4) (5) In the implementatin f the abve prtcl, it is essential that the tw quantum channels used shuld be spatially separated all the way frm Alice t Bb. This prevents an eavesdrpper frm accessing the tw channels in ne lcatin and using the same prcedure that shuld be used by Bb t get the riginal message. The fllwing analysis illustrates the effectiveness f the prtcl in cunteracting the effrts f the eavesdrpper. All the Bell states used are pure maximally entangled states since if we cnsider ne f them; 00, then its density matrix ρ is: 78

5 QANTM SECRE DIRECT COMMNICATION SING ENTANGLEMENT AND SPER DENSE CODING ρ (6) Since Tr(ρ ), then this is a pure state. The partial trace ver the first qubit is: ρ I/ Since Tr((ρ ) ) ½ which is less than, then the first qubit is in a mixed state. Similarly, fr the secnd qubit the same cnclusin will be held. And as lng as there is n unique mixed state fr each separate quantum channel, then let assume each f them will be represented by ne f these, called privileged mixed state. This culd be btained frm the eigenvalues and eigenvectrs f ρ r ρ. The eigenvalues are equal t ½ and the eigenvectrs are 0 and, s if we chse this specific case as the mixed state, then ρ r ρ 0 0 (7) If the eavesdrpper Eve has access t ne quantum channel nly, and makes a measurement she gets 0 r with prbability ½ fr each case. Classical cmmunicatin link Alice Tw spatially separated quantum channels Figure : Synchrnized attack with a classical link. Let us, therefre assume that Eve can use the fllwing attack which is rather difficult t implement. We call this attack Synchrnized attack tgether with a classical cmmunicatin link, fig (3). If Eve measure the qubit n quantum channel she gets 0 with a prbability ½, and assuming that a synchrnized measurement is perfrmed n quantum channel she gets 0 with a prbability ½. Therefre, fr the Bell state, she gets 00 n the tw Eve Eve Bb quantum channels with prbability ¼. Assuming that the Bell states used are unifrmly distributed then each will have a prbability f ¼. Then fr message bits 00 Eve will get this result with prbability /6. Similarly, fr the ther three cmbinatins Eve will get the same results with the same prbability. S if the message has length N, then the prbability that Eve get the crrect result N N is EXAMPLES OF QANTM NOISE AND ITS EFFECT ON THE QANTM CHANNEL In this sectin we examine sme examples f quantum nise that culd affect a quantum channel. These mdels are imprtant in understanding the practical effects f the nise n quantum systems, and hw nise can be cntrlled by techniques such as errr-crrectin. Thse mdels are bit flip; phase flip and bth tgether (bit flip and phase flip). Ofcurse thse mdels d nt include all kinds f nise that culd affect the quantum channel, there are thers, but we chse these t analyze as they are mre likely t ccur. In ur prtcl we have tw quantum channels, s the mdels f nise we mentined abve will be applied n bth channels at randm, i-e, we cannt knw which mdel will affect which channel at a time, therefre we will study all different cmbinatins f different mdels n the tw channels, and then will analyze the last ne in detail as it cntains the greater cmbinatins f the tw ther kinds. In the first mdel (bit flip); the first qubit f Alice n the first channel after encding the classical bits, culd be flipped with prbability (p), with the secnd qubit transmitted crrectly. The secnd case is when the secnd qubit n the secnd channel, the half qubit f Bb, culd be flipped with prbability (p) where the first qubit transmitted crrectly. The third case, if bth qubits n bth channels are flipped with prbability (p ). In the secnd mdel (phase flip); als we have three cases as abve, i-e, (anyne f the qubits will flip with prbability (p), where the secnd will nt), r the tw qubits will flipped with prbability (p ). In the third and last mdel (bth bit and phase flip); all different cmbinatins culd happen; fr example the first qubit culd have a bit flip when the 79

6 SECRYPT Internatinal Cnference n Security and Cryptgraphy secnd qubit has a phase flip; r vice versa, and each f which will ccur with prbability (p ), etc. s we will intrduce the analysis f this ne as the mst general ne. Anyway, as we are using maximally entangled Bell states, all the mdels f quantum nise will just change the transmitted state t anther ne f the Bell states als, which makes it mre cnfusing and harder t discver. T prtect the quantum state frm the effect f the nise we wuld like t develp quantum errrcrrecting cdes based upn similar idea f the classical errr crrecting cdes. This idea is the repetitin cde, as that used by shr cde (Nielsen, 000). In the fllwing analysis we will cnsider the Bell 00 state as an example since the ther cases culd be analyzed in a similar manner.. if the st bit has bth bit flips and phaseflips and the nd bit remains as it is with 0 0 (8) prbability (p ) s. if the nd bit has bth bit flips and phaseflips and the st bit remains as it is with prbability (p) s 0 0 (9) 3. if bth bits have bit flips with nly ne f them has phase-flips with prbability (p3) s 00 (0) 4. if bth bits have bth bit flips and phaseflips and the with prbability (p4) s 00 () 5. if the st bit has bit flip and the nd bit has phase flip, with prbability (p) s 0 0 () 6. if the st bit has phase flip and the nd bit has bit flip, with prbability (p) s 0 0 (3) Nte that the ( ) sign in all the abve relatins intrduced a glbal phase shift with n bservable effect and culd be drpped. S errrs are intrduced in 4 ut f 6 cases abve with apprpriate prbabilities. An errr-crrecting cde scheme, like Steane cde culd then be used (Nielsen, 000). 6 CONCLSIONS This paper intrduced a new prtcl fr direct quantum cmmunicatin making use f pure maximally entangled Bell states. Als, fr efficiency purpses super dense cding is used, which is als based n entanglement, t duble the transmissin speed by sending tw classical bits ver ne quantum channel. This prtcl uses ne step r ne pass t end the message in a secure manner. It is essential that the tw quantum channels used in the implementatin be spatially separated all the way frm Alice t Bb. T illustrate the security f the prtcl, a hypthesized attack prcedure used by Eve was cnsidered that is called synchrnized attack tgether with a classical cmmunicatin. Analysis was given t indicate that the prbability f Eve getting the message is extremely small. Als, this type f attack is very difficult t implement. The effect f sme quantum nise mdels was als cnsidered, indicating the errrs intrduced. In this case sme frm f errr-crrecting prcedure shuld be used. 7 FTRE WORK There are many aspects that culd be cnsidered t cmplete the abve study. A few f them will be presented here:. Since the prtcl is based n using pure maximally entangled Bell states, then it is essential t study prcedure that culd be used t get such states either frm pure nn entangled states, a prcess called cncentratin, r distillatin and purificatin fr mixed states.. It is essential t study entanglement degradatin which depends n the length f the quantum channel. In particular,study f what is called Entanglement Sudden Death (ESD) phenmenn, which shuld be given apprpriate attentin, since it reduced sharply the distance ver which entanglement is effective. 3. Other quantum nise effect shuld als be given due attentin such as: deplarizing channel, amplitude damping, and phase damping. 80

7 QANTM SECRE DIRECT COMMNICATION SING ENTANGLEMENT AND SPER DENSE CODING Tw spatially separated channels M Operatr selectr i i nifrmly Randm Bell state generatr H Decder circuit M Figure 3: The blck diagram f the cding and decding circuit f the prpsed prtcl. REFERENCES Benenti, G., Casati, G., Strini, G., 004. Princples f Quantum Cmputatin and Infrmatin, Vlume I: Basic Cncepts, Wrld Scientific Publishing C. Bennett, C., Bessette, F., Brassard, G., Salvail, L. and Smlin, J., 99. Experimental Quantum Cryptgraphy, J. Cryptlgy, 5 (99) 3. Bennett, C.H. and Brassard, G., 984. Quantum Cryptgraphy: Public Key Distributin and Cin Tssing. In: Prceedings f IEEE Internatinal Cnference n Cmputers, Systems and Signal Prcessing, p. 75. Bström, K., Felbinger, T., 00. Eavesdrpping n the Tw-Way Quantum Cmmunicatin Prtcls with Invisible Phtns, Phys. Rev. Lett.89 (00) Deng, F.G., Li, X.H., Li, C.Y., et al., 008. Quantum Secure Direct Cmmunicatin Netwrk with Einstein- Pdlsky-Rsen Pairs, quant-ph/ (008). Deng, F.G., Lng, G.L., Liu, X.S Tw-Step Quantum Direct Cmmunicatin Prtcl sing the Einstein-Pdlsky-Rsen Pair Blck, Phys. Rev. A 68 (003) Hayashi, M., 006. Quantum Infrmatin, An Intrductin, Springer. Lee, H. at el, 005. Entanglement Generates Entanglement: Entanglement transfer by interactin, Physics letters A 338 (005), Massey, J.L., 988. An intrductin t cntemprary Cryptlgy. In: Prc. IEEE 76 (988) 533. Nielsen, M.A., Chuang, I.L., 000. Quantum cmputatin and quantum infrmatin, Cambridge niversity press, Cambridge. 8