基于多比特量子秘密共享方案的搜索算法

Grover’salgorithmbasedmulti-qubitsecretsharingschemeArtiChamoliandC.M.BhandariIndianInstituteofInformationTechnology,Allahabad,Deoghat,Jhalwa,Allahabad-211011,India.Email:[e-mail],[e-mail]etsharingschemeshavinguniquequantumfeatureslikeparallelismandentanglementaresupposedtoberelativelysecure.Differentschemesproposedbyvariousresearchersovertheyearshavefeatureswhichcouldbespecifictothenatureandneedofasituation.FollowingHsu’sproposedschemeweproposeasecretsharingschemeusingGrover’ssearchalgorithmforafourqubitsystemwithseveralmarkedstates.Further,theschemehasbeengeneralizedtoann-qubitsystem.IntroductionQuantumsecretsharingcanbeaccreditedasoneofthemajorimplicationsofquantuminformationprocessing.Itaddressestheproblemofsecuredtransferofinformationthroughquantumchannelstodistantreceivers.Withtheadventofquantumcomputers,quantumsecretsharingschemesarebeingextensivelystudiedbyseveralresearchers.Thesequantumcounterpartsoftheclassicalsecretsharingschemesexploituniquequantumfeatureslikeparallelismandentanglement.Themessagetobecommunicatedisencryptedinarbitraryquantumstatesandissenttointendedreceiversafterinitialquantummanipulationsbythesender.Thesemanipulationsaresoperformedthatnoneofthereceiverscanretrievethecompletesecretmessagesingle-handedly.TheideaofquantumsecretsharingschemewasoriginallyconceivedandimplementedbyHilleryetal[1]usingthreeandfourparticleGreenberger-Horne-Zeilinger(GHZ)states[2].Similartotheprotocolgivenin[1],Karlsonetal[3]showedtheimplementationofaquantumsecretsharingprotocolusingtwo-particlequantumentanglement.Later,(k,n)-thresholdquantumsecretsharingschemewasproposedbyCleveetal[4]inwhichtheencodedquantumstateissplitamongnpeoplesuchthatanykofthemcanreconstructtheencodedinformation,whilenumberofpeople,iflessthank,canneversucceed.Thishasbeenexperimentallydemonstratedin[5,6].Gottesman[7]demonstratedmixedstatequantumsecretsharingbydiscardingasharefrompurestatescheme,theonlyconstraintsbeingmonotonicity[8]andthenocloningtheorem[9-11].ThisquantumsecretsharingschemeforgeneralaccessstructureswasshownbySmith[12]inasomewhatdifferentmanner.Karimpouretal[13]demonstratedquantumsecretsharingschemesbasedonentanglementswappingofgeneralizedd-levelBellstates.Cabello[14]generalizedquantumsecretsharingschemeston-particles.Recently,Guo-PingGuoetal[15]presentedquantumsecretsharingwithoutentanglement.In2003,anewdimensionwasaddedtoquantumsecretsharingschemeswiththeincorporationofGrover’sunsorteddatabasesearchalgorithm[16].Li-Yi-Hsuproposedatwo-qubitquantumsecretsharingprotocolbasedonGrover’salgorithm[17].Apartfromtheoperationalaspectstheprotocolisdifferentfromtheabovementionedschemesinthesenseofcheatdetection.Withthisprotocolcheatingcouldbedetectedimmediatelywithoutexhaustingaportionofthesequenceofmeasurementsofoutcomes.Inatwo-qubitGrover’salgorithm,themarkedstatecanberetrievedwithfullprobabilityafterasingleiteration.Hsu’sprotocolisbasedonthisproperty.Inthispaperwepresentageneralizationoftwo-qubitquantumsecretsharingprotocolbasedonGrover’salgorithmton-qubitsystem.WewillshowthatmarkedstatecanbefoundwithcertaintyafterasingleiterationforanynumberofqubitsinGrover’ssearchalgorithmifthenumberofmarkedstatesisone-fourthofthetotalnumberofelementsinsearchspace.Thustwo-qubitquantumsecretsharingprotocolbasedonGrover’ssearchalgorithmcanbegeneralizedton-qubitsecretsharingscheme.Theincreaseinthenumberofmarkedstateswiththeincreaseinnumberofqubitsnotonlyreducestheprobabilityoferrorbutalsoenhancesthesecurityaspectofthesecretsharingprotocol.Themessagecanbesplitandencodedintovariousmarkedstates.WestartwithareviewoforiginalschemepresentedbyHsu.TwoqubitssecretsharingprotocolAlice,thesender,randomlypreparesatwo-qubitsuperpositionstate1Softheform()()()()12011201+⊗+Thesuperpositionstatecanbetheproductofanytwoofthefollowingfourstates:()()()()()()()()1201,1201,1201,1201ii+−+−.Themessageisencodedinthemarkedstates01or10,whereasthestates00or11areusedtodetectanypossibleeavesdropping.SheperformsWPoperationon1SwhereWPisoftheformWP=12WW−,(1)Wbeingthemarkedstatecontainsthesecretinformation.Ifthemessageisencryptedinthestate10,thenWP1S=1WS=()()1210101200011011⎡⎤⎡−⎤+++⎣⎦⎣⎦()()1200011011⎡⎤=+−+⎣⎦(2)Thistransformationchangesthephaseofthedesiredstatekeepingtheotherstatesunchanged.AlicethensendsthesetwoqubitstoBobandCharliewhoareatadistantplace.ThequbitssentbyAlicearesuchthatBobreceivesthefirstqubitandCharliethesecondone.Itisassumedthatatmostoneofthemmaybeacheatwhomaytrytocapturebothqubitssothathecanretrievewholeoftheinformationbyhimself.HenceAlicealsoaimstodetectanypossiblecheatingstrategy.Onceshehasconfirmedviaclassicalchannelthateachoneofthemisinreceiptoftheirrespectivequbit,shemakesapublicannouncementofherinitialstate1S.EvenafterAlice’sdeclarationoftheinitialstate1SBobandCharliecanhaveaccesstotheencryptedinformationonlywhentheycombinetheirrespectivequbitsandperform1SP−onthetwoqubitscollectively.1SP−1WS=112SSI⎡−⎤⎣⎦()()1200011011⎡⎤+−+⎣⎦()(){}()(){}()()2120001101112000110111200011011I⎡⎤⎡⎤=++++++−+−+⎣⎦⎣⎦10=(3)Τhusthesecretin