Sequential Quantum Secret Sharing Using a Single Qudit

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1 Commun. Theor. Phys. 69 ( Vo. 69, No. 5, May 1, 2018 Sequenta Quantum Secret Sharng Usng a Snge Qut Chen-Mng Ba ( 白晨明, 1 Zh-Hu L ( 李志慧, 1, an Yong-Mng L ( 李永明 2 1 Coege of Mathematcs an Informaton Scence, Shaanx Norma Unversty, X an , Chna 2 Coege of Computer Scence, Shaanx Norma Unversty, X an , Chna (Receve December 11, 2017; revse manuscrpt receve January 30, 2018 Abstract In ths paper we propose a nove an effcent quantum secret sharng protoco usng -eve snge partce, whch t can reaze a genera access structure va the thought of concatenaton. In aton, Our scheme ncues a avantages of Tavako s scheme [Phys. Rev. A 92 ( (R]. In contrast to Tavako s scheme, the effcency of our scheme s 1 for the same stuaton, an the access structure s more genera an has avantages n practca sgnfcance. Furthermore, we aso anayze the securty of our scheme n the prmary quantum attacks. PACS numbers: a, U DOI: / /69/5/513 Key wors: quantum secret sharng, access structure, thresho scheme 1 Introucton Secret sharng, frsty ntrouce by Shamr [1] an Bakey, [2] s a cryptographc prmtve whch pays a sgnfcant roe n the varous secure mutparty computaton tasks an management of keys. Secret sharng has extene to the quantum fe. [3 9] Quantum secret sharng (QSS s aso a cryptographc protoco to strbute ether a cassca secret (strng of bts or a quantum secret (unknown quantum state to a group of payers P such that ony authorze subsets of P can coaboratvey recover the secret. The securty of most cassca cryptographc systems s base on the assumptons of computatona compexty, whch mght be broken by the strong power of avance agorthms, such as Shor agorthm. In aton, another man rawback of cassca secret sharng schemes s that they are not perfecty secure from an eavesropper attack. Compare to the cassca one, QSS can enforce the securty for cryptographc tasks ue to the quantum no-conng theorem an uncertanty propertes of quantum mechancs. [10 11] In 1999, Hery et a. [3] frsty ntrouce a protoco of QSS by usng GHZ states. At the same year, Karsson et a. [12] showe how a QSS protoco can be mpemente usng two-partce quantum entangement an scusse how to etect eavesroppng or a shonest partcpant. In 2004, Xao et a. [13] generaze the QSS of Hery et a. nto arbtrary mutpartte entange state. From then on, varous QSS schemes have been propose. [14 23] Many schoars gave QSS schemes base on fferent prncpes, such as quantum error-correctng coe [24] an oca stngushabty of quantum states. [25 29] In these works, t has been shown that QSS s not ony a mere theoretca concept, but aso an expermenta possbty. [30] Wth the eveopment an appcaton of quantum communcaton, t s a great ea to esgn a quantum protoco wth a snge state. Recenty, Tavako et a. [31] propose a moe of (n, n-thresho secret sharng wth a snge -eve quantum system (for any prme. Ths moe has huge avantages n scaabty an can be reaze wth state-of-the-art technoogy. However, n ths scheme, there are two efcences. Frsty, the successfu probabty of each roun s epenent on measurement bases, that s, the effcency s 1/. Wth the ncrease of, the effcency of ths scheme w be very ow. In orer to mprove ths probem, Karmpour et a. [32] propose a QSS wth a ranom wak n a attce of states for arbtrary. Wth ths ranom wak pcture at han, they can mprove ther scheme effcency to reach 1/2. Athough the effcency has been mprove, Ln et a. [33] foun that ther scheme was not secure for one shonest who can recover the secret wthout the hep of other partcpants. Secony, Tavako s scheme s just an (n, n-thresho an not sutabe for a genera access structure. For the probem, Lu et a. [34] propose a (t, n-thresho QSS scheme base on a snge -eve quantum system. In ths scheme, they empoye the cassca Shamr s (t, n-thresho secret sharng. However, t cannot be appe to the genera access structure. The genera access structure of a secret sharng scheme s a famy of a authorze sets. In genera, access structures are consere to be monotone,.e., any superset of an authorze set must be authorze. Because each person has fferent weght n rea fe, the abty to recover the secret s fferent. Therefore, ths genera access structure s of practca sgnfcance an pays a major roe n secret sharng schemes. In ths paper, we propose a new QSS scheme, whch Sponsore by the Natona Natura Scence Founaton of Chna uner Grant Nos an , an Inustra Research an Deveopment Project of Scence an Technoogy of Shaanx Provnce uner Grant No. 2013k0611 E-ma: zhhu@snnu.eu.cn c 2018 Chnese Physca Socety an IOP Pubshng Lt

2 514 Communcatons n Theoretca Physcs Vo. 69 can sove the above two probems. For the frst probem, the strbutor Ace n our scheme vountary chooses the measurement bases an sens them to a partcpants through quantum secure rect communcaton. [35] Ths operaton can not ony mprove the effcency of the protoco but aso reuce the possbty of nformaton eakage. Through ths metho, the effcency of our scheme can reach 1. For the secon probem, we treat each authorze set as a sma thresho scheme. By cascang metho, we appy our scheme to the genera access structure, whch makes t more practca. In partcuar, f the access structure contans ony one authorze set, our scheme w be the same as Tavako s. Therefore, our scheme s more extensve an meanngfu. The structure of the paper s organze as foows. In Sec. 2, we gve some premnares. In Sec. 3, we propose the quantum secret sharng scheme for a genera access structure an show an exampe. Secton 4 anayzes the securty an compares our scheme wth the exstng scheme. Fnay, the concuson s gven n Sec Premnares In ths secton we brefy reca some basc concepts such as the access structure an mutuay unbase (orthonorma bases. 2.1 Access Structure Let P be a set of payers, then the authorze set s a subset of P, whch can reconstruct the orgna secret. The access structure Γ of a secret sharng scheme s a famy of a authorze sets. In genera, access structures are consere to be monotone ncreasng,.e., any superset of an authorze set must be authorze. For A, A P, f A Γ an A A, then A Γ. The bass of Γ s the coecton of a mnma authorze sets of P. We assume that every partcpant beongs to at east one mnma authorze set. 2.2 Mutuay Unbase Bases an Untary Operaton Mutuay unbase bases (MUBs are the mportant too n many quantum nformaton processng. As note n Refs. [36 37], t s possbe to fn + 1 MUBs n -mensona quantum system ony f s (any power of prme numbers an o prme. At frst, the computatona bass s enote by { k k D}, where D = {0,..., 1}. For consstency, we aso restrct to o prme number n the whoe artce. Beses the computatona bass, the expct forms of the remanng sets of MUBs are e (j = 1 1 ω k(+jk k, (1 where ω = e 2π/, j D abes the bass an D enumerates the vectors of the gven bass. They are mutuay unbase because the overap s e (j e (j = 1 for j j. (2 In Ref. [31], the encong operaton conssts of two untary operators, X an Y, whch are epcte as foows: 1 1 X = ω n n n, Y = ω n2 n n. (3 Usng Eqs. (1 an (3, we can get ( 1 ( X x Y y 1 = X x ω yn2 n n e(j 1 ω k(+jk k = ω xn n n ω k[+(j+yk] k = 1 1 ω k[(+x+(j+yk] k = e (j+y (+x. (4 For convenence, the operator X xy y s wrtten as U x,y,.e., U x,y e (j = e (j+y (+x. 3 The QSS Scheme on Access Structure In ths secton, we propose our scheme usng a snge qut to reaze a genera access struture. Assume that the strbutor Ace wants to share secret messages to a group P wth n partcpants, an the genera access structure s enote by Γ = {A 1, A 2,..., A r }, (5 where A ( = 1, 2,..., r s the mnma authorze set. By cascang metho, we utze our new scheme to mpement the genera access structure Γ, whch every mnma authorze set A can be reaze by a sma thresho scheme (see Fg. 1. In orer to factate the escrpton of our scheme, suppose the mnma authorze set A s expresse as A = {Bob (1, Bob (2,..., Bob (t }, ( = 1, 2,..., r. Step 1 Preparaton: Accorng to the authorze set A, the strbutor Ace ranomy chooses some numbers, y (1, y (2,..., y (t D, an sens y (ν to the partcpant Bob (ν (ν = 1, 2,..., t through quantum secure rect communcaton. Note that f Bob (ν appears n a fferent authorze set, Ace sens ony one y (ν. For the authorze set A, Ace generates two ranom numbers x (0, y (0 D an prepares a state e (j, (, j D an ony Ace knows, j. Then she performs the operaton U (0 x on e (j,y (0 an obtans a new sgna state e (j+y(0, enote by ψ (0. +x (0 Step 2 Dstrbuton: Ace sens ths state ψ (0 to the frst partcpant Bob (1 n the authorze set A. After recevng the state ψ (0 sent by Ace, f Bob (1 generate a prvate key x (1 has not, then he ranomy chooses a

3 No. 5 Communcatons n Theoretca Physcs 515 number x (1. Otherwse, the operaton s skppe. Combne wth y (1 sent by Ace, he performs the corresponng operaton U (1 x,y (1 on ths partce. Bob (1 sens the new state ψ (1 to the next partcpant Bob (2 n A. Then Bob (2 oes the same operaton as Bob (1. In turn, the ast partcpant Bob (t n A sens the new state to Ace unt he foows the above metho. Fg. 1 (Coor onne A fow chart for the access structure Γ = {A 1, A 2,..., A r }, where A ( = 1, 2,..., r s the mnma authorze set. Bue an re ots represent a partcpants, where re ots ncate that partcpants appear n fferent mnma authorze sets. Arrows enote the recton of partce transport. Every mnma authorze set A can be reaze by a thresho scheme. For authorze set A ( = 1, 2,..., r, the protoco works because after a the transformatons the fna state reas ( t ψ fna = ν=1 U (ν x,y (ν ψ (0. Step 3 Measurement: After Ace has receve the sgna state, she etermnes whch bass to measure the partce accorng to j, y (0, y (1,..., y (t. Through her cacuaton, f they satsfy the foowng conton j + y (0 + y (1 + + y (t = J(mo, (6 then she chooses the bass M J = { e (J D} to make a measurement on the ast partce an recors the measurement resut a. It mpes that the prvate ata of a partcpants n the authorze set A, {x (1,..., x (t }, satsfy gobay consstency conton + x (0 + x (1 + + x (t = a (mo. (7 Step 4 Detecton: In orer to check the securty, for a ranomy chosen (by Ace subset of the rouns, a partcpants Bob (1, Bob (2,..., Bob (t n A sen ther vaues of ther prvate ata x (ν (ν = 1,..., t to her, an Ace checks conton (7. If Eq. (7 oes not ho, she aborts the scheme an starts agan wth a new set of resources. Step 5 Reconstructon: If no eavesropper s etecte, accorng to x (0, an the measurement resut a, Ace can euce that x (1 + + x (t = (a x (0 (mo = b. (8 For fferent authorze sets, there may be fferent b s. Therefore, Ace epens on b to ajust the parameter α. In orer to restore the orgna secret, each partcpant n A sens the prvate key x (ν to the truste esgne combner (TDC. Afterwars, Ace aso sens the parameter α assocate wth the authorze set A to the TDC. After the TDC has receve a ata, t performs an operator O, whch s enote by O(α, x (1,..., x (t = { s, f (α + t ν=1 x(ν mo = s ; No, otherwse. (9

4 516 Communcatons n Theoretca Physcs Vo. 69 If Ace s secret s more than one bt, Ace an a partcpants n the authorze set A execute Steps 1 5 repeatey. In orer to save costs, the measurement bass nformaton, y (1, y (2,..., y (t D, can be reuse. At the same tme, Ace ony nees to change y (0 to ensure the securty of every roun. Exampe 1 In orer to expan our scheme more ceary, we w gve an exampe n the foowng. Suppose the access structure s enote by Γ = {P 1 P 4 P 5, P 2 P 3 P 5, P 1 P 2 } wth fve honest partcpants an Ace wants to share the secret message s = 2 F 5. We gve a strbutve fow chart for the access structure (see Fg. 2. Fg. 2 (Coor onne A strbutve fow chart for the access structure Γ = {P 1P 2, P 2P 3P 5, P 1P 4P 5}, where U x,y represents everyone operaton, the re ne represents the authorze set P 1P 2, the bue ne represents the authorze set P 2 P 3 P 5 an the back ne represents the authorze set P 1 P 4 P 5. In accorance wth the above steps a partcpants have ther prvate keys. We assume that x 1 = 0, x 2 = 2, x 3 = 3, x 4 = 1, x 5 = 3 respectvey. For the authorze set A 1 = {P 1 P 4 P 5 }, Ace can cacuate that x 1 + x 4 + x 5 = 4. Then she can choose parameter α 1 = 3 an sen t to the TDC. In orer to restore the secret, P 1, P 4 an P 5 sen ther prvate keys to the TDC. At ast, the TDC can output the secret s = 2 (see Fg. 3. Fg. 3 Reconstructon for the authorze set P 1 P 4 P 5. For the authorze set A 2 = {P 2 P 3 P 5 }, Ace can choose parameter α 2 = 4 an sen t to the TDC. P 2, P 3 an P 5 sen ther prvate keys to the TDC. Then, the TDC can cacuate α 2 + x 2 + x 3 + x 5 = 12 mo 5 = 2 an output the secret s = 2. For the authorze set A 3 = {P 1 P 2 }, Ace can choose parameter α 3 = 0. The TDC can cacuate α 3 + x 1 + x 2 an output the secret. 4 Anayss of Our Scheme 4.1 Securty Dscusson In Ref. [31], Tavako et a. have gven the securty anayss of ther scheme. However, we have mprove ther scheme an mpemente a genera access structure usng cascang metho. In orer to check the securty of our scheme, we anayse the attacks of our scheme,.e. externa attack an nterna attack. Externa attack The frst strategy for the eavesropper Eve s the ntercept-an-resen attack. For ths attack, we conser two cases. (a Eve may ntercept the qut, n the state e (j, on the way from Bob (k to Bob (k+1 n the authorze set A. (b Eve may ntercept the qut sent by the strbutor Ace an sen a qut of her own to the frst partcpant Bob (1 n A n ts stea. Eve coects her qut once t s sent by the ast partcpant Bob (t n A. However, for two cases, she oes not have any nformaton about the measurement bass because y (1,..., y (t n our protoco are sent to partcpants through quantum secure rect communcaton wthout pubcaton. In orer to get the orgna secret, she can ony choose one of reevant bases to measure. Obvousy, Eve can obtan the correct measurement resut ony when she happens to choose the true bass j = j. Therefore, the successfu probabty s 1/. The eavesroppng, to some extent epenng on, causes nconsstences between the prvate ata an conton (7. In aton, f Ace sens n-bt secret, the successfu probabty to obtan nformaton s (1/ n. Thus when the numbers of n get arger, the probabty s (1/ n 0. Therefore, ths ntercept-an-resen attack oes not work n our scheme.

5 No. 5 Communcatons n Theoretca Physcs 517 The secon strategy for the eavesropper Eve s the entange-an-measure attack. Assumng that the eavesropper Eve mpements ancary system to obtan the nformaton. Suppose that Eve performs the untary transform U E to entange an auxary partce on the transmtte partce an then measures the auxary partce to stea secret nformaton. Wthout oss of generaty, we conser the bass j = 0,.e., e (0 = (1/ 1 ωk k n the foowng forms, U E k E = 1 m=0 U E e (0 E = U E ( 1 a km m ε km, = m=0 ω k k E = 1 ω k a km ( g=0 ω k( 1 m=0 ω mg e (0 g a km m ε km ε km = m=0 g=0 (10 ω k mg a km e (0 g ε km, (11 where ω = e 2π/ ; E s the nta state of Eve s ancary system; ε km (k, m = 0, 1,..., 1 s the pure auxary state etermne unquey by the untary transform U E, an 1 m=0 a km 2 = 1 (k = 0, 1,..., 1. (12 In orer to avo ntroucng the error rate, Eve has to set: a km = 0, where k m an k, m {0, 1,..., 1}. Therefore, Eq. (10 an Eq. (11 can be smpfe as foows: U E k E = a kk k ε kk, (13 U E e (0 E = g=0 ω k( g a kk e (0 g ε kk. (14 Smary, Eve can obtan that 1 ωk( g a kk ε kk = 0, where g {0, 1,..., 1} an g. Then for any {0, 1,..., 1}, we can get equatons. Accorng to these equatons, we can compute that a 00 ε 00 = a 11 ε 11 = = a 1, 1 ε 1, 1. (15 Therefore, no matter what the usefu state s, Eve can ony obtan the same nformaton from ancary partces. The smar scusson can be appe for the other quantum state e (j = (1/ 1 ωk(+jk k. Therefore, the entange-an-measure attack s unsuccessfu. In Ref. [31], the authors consere that an aternatve attack use by Eve s to sen va the untary gate of payer Bob (k one more qut or even a mutqut puse so that t can be somehow ntercepte by her beyon the gate wthout nterceptng the protoco qut. For the eavesroppng, t s not sutabe for our scheme an she cannot earn the actua untary transformaton,.e., x (1,..., x (t because y (1,..., y (t n our protoco are not announce. Interna attack In the foowng, we w prmary conser the partcpants conspracy attack because t s aways easer an more powerfu than externa attack an the partcpants can get more usefu nformaton than a fourth eavesropper. In the worse case, ony the strbutor Ace an one more partcpant n each authorze set, are honest. For exampe, we assume that Bob (1 s honest n A = {Bob (1, Bob (2,..., Bob (t } an the rest t 1 are consprng payers. If they o not sen ther prvate keys to the TDC, they cannot recover any nformaton because they o not know the TDC s operator. If they sen ther prvate keys to the TDC, they can not restore the orgna nformaton because the TDC oes not start unt t must receve a ata. Even f Bob (1 aso sens hs key, they st can not obtan the secret ue to the ack of Ace s parameter α. Because Ace s parameter settng s epenent on the partcpant keys, f they sen some fake keys, then the TDC w output No, that s, they cannot get the secret. 4.2 Comparson x D} (J D to measure her own partce. Therefore, the probabty of a va roun s 1/. In aton, after y 1,..., y n are announce, the eavesropper may earn x 1,..., x n from the attack wth one more qut or even a mutqut puse through the untary gate. Therefore, the announcement of these ata ncreases the probabty of nformaton eakage. In our protoco, the strbutor vountary chooses y 1,..., y n wthout announcement, that s, she knows the measurement bass nformaton. Therefore, the probabty of a va roun s 1. In ths secton, we compare our protoco wth Tavako s scheme [31] n Tabe 1. In Tavako s scheme, a partcpants ranomy choose the measurement bass nformaton y 1,..., y n, that s, the strbutor Ace oes not know any nformaton about the measurement bass. Before announcng them, she ranomy chooses the bass M J = { e (J x In aton, Tavako et a. propose an (n, n- thresho scheme,.e., the (n, n-thresho access structure. In our protoco, we gve a quantum secret sharng scheme to reaze a genera access structure. If the access structure contans ony one authorze set, then our scheme w become an (n, n-thresho scheme. Therefore, our scheme s more extensve an meanngfu.

6 518 Communcatons n Theoretca Physcs Vo. 69 Tabe 1 Comparson of Tavako s scheme [31] an our propose one. Tavako s scheme Our propose scheme Quantum state Snge qut Snge qut Va probabty 1/ 1 Access structure (n, n-thresho Genera access structure 5 Concuson In ths paper, we propose a quantum secret sharng scheme wth a snge -eve quantum state to reaze a genera access structure. In the propose protoco, we st appe the cycc property of MUBs because these bases are mportant toos n quantum nformaton processng. In aton, we aso anayse the securty of our scheme aganst prmary quantum attacks an compare our protoco wth Tavako s scheme. For our scheme, the probabty of each va roun s 1. When our access structure contans ony one authorze set, our scheme w be the same as Tavako s. However, snce MUBs are st unknown, we ony conser the o prmes. For even number or power of prme number, we o not gve any scusson. In Refs. [36 27], many agebrac propertes of power of prme numbers have been gven, then we hope that more researchers w stuy these probems an propose more mportant schemes. Acknowegements We want to express our grattue to anonymous referees for ther vauabe an constructve comments. References [1] A. Shamr, Commun. ACM 22 ( [2] G. R. Bakey, n Proceengs of the Natona Computer Conference, (AFIPS, 1979 (1979 pp [3] M. Hery, V. Buzek, an A. Berthaume, Phys. Rev. A 59 ( [4] R. Ceve, D. Gottesman, an H. K. Lo, Phys. Rev. Lett. 83 ( [5] D. Gottesman, Phys. Rev. A 61 ( [6] F. G. Deng, H. Y. Zhou, an G. L. Long, J. Phys. A: Math. Gen. 39 ( [7] A. M. Lance, T. Symu, W. P. Bowen, et a., Phys. Rev. Lett. 92 ( [8] F. G. Deng, X. H. L, C. Y. L, et a., Phys. Rev. A 72 ( [9] G. Goron an G. Rgon, Phys. Rev. A 73 ( [10] W. K. Wootters an W. H. Zurek, Nature (Lonon 299 ( [11] D. Deks, Phys. Lett. A 92 ( [12] A. Karsson, M. Koash, an N. Imoto, Phys. Rev. A 59 ( [13] L. Xao, G. L. Long, F. G. Deng, an J. W. Pan, Phys. Rev. A 69 ( [14] H. Cao an W. P. Ma, IEEE Photoncs J. 9 ( [15] C. M. Ba, Z. H. L, T. T. Xu, an Y. M. L, Int. J. Theor. Phys. 55 ( [16] A. Matra, S. J. De, G. Pau, an A. K. Pa, Phys. Rev. A 92 ( [17] L. H. Gong, H. C. Song, C. S. He, et a., Physca Scrpta 89 ( [18] P. Sarvepa an R. Raussenorf, Phys. Rev. A 81 ( [19] L. Y. Hsu an C. M. L, Phys. Rev. A 71 ( [20] C. M. Ba, Z. H. L, M. M. S, et a., Eur. Phys. J. D 71 ( [21] X. J. Wang, L. X. An, X. T. Yu, et a., Phys. Lett. A 381 ( [22] Y. F. He an W. P. Ma, Int. J. Quantum Inf. 14 ( [23] K. J. Zhang, L. Zhang, T. T. Song, et a., Scence Chna Phys. Mech. & Astron. 6 ( [24] Z. Zhang, W. Lu, an C. L, Chn. Phys. B 20 ( [25] R. Rahaman an M. G. Parker, Phys. Rev. A 91 ( [26] Y. H. Yang, F. Gao, X. Wu, et a., Sc. Rep. 5 ( [27] C. M. Ba, Z. H. L, T. T. Xu, et a., Quantum Inf. Process 16 ( [28] J. Wang, L. L, H. Peng, an Y. Yang, Phys. Rev. A 95 ( [29] J. T. Wang, G. Xu, X. B. Chen, et a., Phys. Lett. A 381 ( [30] H. Lu, Z. Zhang, L. K. Chen, et a., Phys. Rev. Lett. 117 ( [31] A. Tavako, I. Herbauts, M. Zukowsk, an M. Bourennane, Phys. Rev. A 92 ( (R. [32] V. Karmpour an M. Asoueh, Phys. Rev. A 92 ( (R. [33] S. Ln, G. D. Guo, Y. Z. Xu, et a., Phys. Rev. A 93 ( [34] C. Lu, F. Mao, et a., Quantum Inf. Process 17 ( [35] C. Wang, F. Mao, Y. S. L, et a., Phys. Rev. A 71 ( [36] I. D. Ivonovc, J. Phys. A: Math. Gen. 14 ( [37] W. K. Wootters an B. D. Fes, Ann. Phys. 191 (