Selection of unitary operations in quantum secret sharing without entanglement

Transcription

1 . RESEARCH PAPERS. SCIENCE CHINA Information Sciences September 2011 Vol. 54 No. 9: doi: /s Selection of unitary operations in quantum secret sharing without entanglement XU Juan 1,CHENHanWu 1,LIUWenJie 1,2 &LIUZhiHao 1 1 School of Computer Science and Engineering, Southeast University, Nanjing , China; 2 School of Computer and Software, Nanjing University of Information Science and Technology, Nanjing , China Received June 24, 2010; accepted September 15, 2010; published online April 30, 2011 Abstract We propose a substitute-bell-state attack strategy for quantum secret sharing schemes without entanglement, as well as a definition of the minimum failure probability of such attack strategy. A quantitative analysis of security degrees corresponding to different unitary operations is also provided, when the secret sharing schemes without entanglement are stricken by substitute-bell-state attack. As a result, the relation between the selection of unitary operations and the effect of substitute-bell-state attack is shown, which can serve as an important guidance for the selection of unitary operations in designing and implementing quantum secret sharing schemes. quantum secret sharing, substitute-bell-state attack, unitary operation, minimum failure proba- Keywords bility Citation Xu J, Chen H W, Liu W J, et al. Selection of unitary operations in quantum secret sharing without entanglement. Sci China Inf Sci, 2011, 54: , doi: /s Introduction Due to the unconditional security and ability of detecting eavesdropping, quantum cryptography has stimulated great interest in the scientific and industrial communities [1 35], and becomes a significant research direction of mainstream computer science and physics over the last some twenty years. Combining quantum mechanics with classical cryptography, quantum cryptography exploits the principles of quantum physics to achieve provably secure communication. Quantum secret sharing (QSS) is an important branch of quantum cryptography. It mainly deals with the splitting and distributing of an arbitrary secret among n sharers using quantum resources such that m authorized sharers (m n) can acquire the secret only when they cooperate. In the past decade, QSS has received widespread attention [1 30]. The existing QSS schemes can be classified by purpose into two categories: One is to share classical information [1 12, 25], and the other is to share quantum information [13 18, 26 28]. Secret sharing of quantum information, also known as quantum information splitting, inevitably needs entangled states due to the accustomed adoption of controlled teleportation [15, 16, 18] or quantum error correcting code [19], while QSS of classical information (QSSCI) can be with Corresponding author ( hw chen@seu.edu.cn) c Science China Press and Springer-Verlag Berlin Heidelberg 2011 info.scichina.com

2 1838 Xu J, et al. Sci China Inf Sci September 2011 Vol. 54 No. 9 or without entangled states. The original schemes of QSS are based on the characteristic of entangled states [20, 21]. Since the efficiency of preparing multi-particle entangled state is very low and the cost is high at present [22], QSS schemes based on single particles or product states are proposed [4, 10, 11, 23, 25]. In this paper, we focus on QSSCI schemes based on single particles. Unitary operations are usually employed in these schemes, which are used to encode the information on quanta and (or just) to smash the quantum states so that the eavesdropper cannot reliably distinguish the quantum states by measurement. The key steps with unitary operations can be described as follows: Bob i sends the sequence of N particles to Bob i +1; Bob i + 1 implements unitary operations on each particle, and then sends these particles Ψ i+1 to Bob i + 2. Suppose Bob i is dishonest during the above procedure, he can make an attack like this: i) Bob i holds the sequence Ψ i 1 from Bob i 1, and does not do any unitary Ψ i = N k=1 ψi k operation on it; at the same time, he prepares N Bell states Φ 12 = N k=1 φ k 12,where φ k 12 is one of the four Bell states: φ ± = 1 2 ( 00 ± 11 ), ψ ± = 1 2 ( 01 ± 10 ). (1) Without loss of generality, suppose φ k is (1/ 2)( ). ii) Bob i keeps the first particles Φ 1 of the Bell states, and then sends the second ones Φ 2 to Bob i + 1. iii) While Bob i + 1 sends the single-particle sequence to Bob i+2, Bob i intercepts and captures the sequence Ψ i+1, and sends Ψ i 1 to Bob i +2. iv)bobi combines Φ 1 with Ψ i+1 to make the new N Bell states Φ 12 = N k=1 φ k 12, and tries to acquire all the unitary operations Bob i + 1 has made via measuring Φ 12, that is, to gain the encoding information. v) During sampling detection, Bob i broadcasts the same unitary operations as Bob i + 1 to prevent the eavesdropping from being detected. This kind of attack is called substitute- Bell-state attack in this paper, whose process is depicted in Figure 1. Substitute-Bell-state attack is inspired by [11] and [24]; however, it is different from the attacks presented in the two papers, and is more general. There are two points that need to be supplemented: i) Bob i can intercept and capture Ψ j 1 when Bob j sends the single-particle sequence to Bob j +1(j>i+ 1), and then sends Ψ i 1 to Bob j +1;thus,Bobi gains the integrated encoding information from Bob i +1toBobj. ii) When the eavesdropper is another party Eve, she can also make such an attack. In addition, if the identity or message authentication is introduced between each pair of correspondents, this kind of attacks could be detected. However, if the authentication is not completely effective or perfect, there is a probability that the attack is not discovered. This problem will not be studied here. In this paper we mainly discuss the relation between the selection of unitary operations and the effect of substitute-bell-state attack. 2 Relation between unitary operations and substitute-bell-state attack The state of a single particle can be expressed by a qubit, qutrit or qudit, depending on which system the particle belongs to: two-level, three-level or multi-level. Two-level system, namely qubit, is concerned here. All the effective unitary operations performed on a qubit can be represented by a 2 2matrixU. The matrix U is evidently unitary, i.e., U U = I. The most common unitary operations are as follows: I = = , X = = , Y = = , (2) Z = = , 0 1

3 Xu J, et al. Sci China Inf Sci September 2011 Vol. 54 No Figure 1 H = 1 2 [ Schematic diagram of substitute-bell-state attack. ] = 1 ( ) ( 0 1 ) The existing QSSCI schemes, especially the schemes with qubits, always use the above unitary operations. On the other hand, after intercepting Ψ i+1,bobicombines it with the first particles Φ 1 of the Bell states and obtains N new Bell states Φ 12.IfBobicould definitely distinguish each state in Φ 12,he could successfully acquire the encoding information of Bob i + 1. Obviously, if all the states in Φ 12 are mutually orthogonal, Bob i is able to know all encoding information of Bob i + 1 successfully. At this rate, Bob i can get the shared secret without the cooperation of Bob i + 1. In [36], the upper bound for the success probability of unambiguous discrimination among quantum states is given by D 1 1 pi p j ψ i ψ j. (3) n 1 i j In (3), n denotes the number of quantum states to be distinguished, ψ represents each state to be discriminated, and p stands for the appearance probability of each state. To analyze the effect of substitute- Bell-state attack, referring to the success probability D, the minimum failure probability of substitute- Bell-state attack F min is given as follows. Definition 1. Suppose an eavesdropper makes a substitute-bell-state attack on a QSSCI scheme. The eavesdropper intercepts the particle sequence and obtains N two-particle states φ k (1 k N). The prior probability of φ k is denoted by p k. Define the minimum failure probability of substitute-bell-state attack F min as F min = 1 p i n 1 p j φ i φ j. (4) i j Apparently, F min stands for the minimum probability of non-explicit discrimination among quantum states, and its value represents the security degree of QSS schemes against substitute-bell-state attack. Therefore, the larger F min is, the worse effect substitute-bell-state attack has, and the higher security of this QSS scheme is verified. According to our research on existing QSSCI schemes, it is found that unitary operations can be executed by one step [23] or two steps [9, 12], as will be discussed separately below. Above all, suppose the selection of unitary operations has an independent and identical distribution (i.i.d.), and the state of each particle is one of the followings: 0, 1, (1/ 2)( ) = +, (1/ 2)( 0 1 ) =. (5) Definition 2 (one-step unitary operation). Before sending the particle sequence to Bob i + 2, Bob i + 1 implements only one random unitary operation on each particle, which is entitled one-step unitary operation. The unitary operations, randomly selected by Bob i + 1, are called a unitary-operation set, denoted by {}. For instance, Charlie s operations in [23] are one-step unitary operations.

4 1840 Xu J, et al. Sci China Inf Sci September 2011 Vol. 54 No. 9 Table 1 Relation between one-step unitary operations and substitute-bell-state attack (omitting normalization and Dirac symbol) Select unitary operations Obtained quantum states F min I,X,H φ +,ψ +,φ + ψ + I,Y,H φ +,ψ,φ + ψ + 0 I,Z,H φ +,φ,φ + ψ + I,X,Y,H φ +,ψ +,ψ,φ + ψ + 2/12 I,X,Z,H φ +,φ,ψ +,φ + ψ + I,Y,Z,H φ +,φ,ψ,φ + ψ + 2/12 I,X,Y,Z,H φ +,φ,ψ +,ψ,φ + ψ + 2/10 Table 2 Relation between two-step unitary operations and substitute-bell-state attack (omitting normalization and Dirac symbol) Select unitary operations Obtained quantum states F min I,X; I,H φ +,ψ +,φ + ψ +,φ + ψ I,Y ; I,H φ +,ψ,φ + ψ +,φ ψ + 0 I,Z; I,H φ +,φ,φ + ψ +,φ + + ψ I,X,Y ; I,H φ +,ψ +,ψ,φ + ψ +,φ + ψ 2 2/15 I,X,Z; I,H φ +,φ,ψ +,φ + ψ +,φ + = ψ 2 2/15 I,Y,Z; I,H φ +,φ,ψ,φ + ψ +,φ + = ψ 2 2/15 I,X,Y,Z; I,H φ +,φ,ψ +,ψ,φ + ψ +,φ + ψ,φ ψ +,φ + + ψ 2/7 The success probability of substitute-bell-state attack by Bob i + 1 varies with the different unitary operations Bob i + 1 has chosen. According to several common sets of unitary operations shown in (2), the minimum failure probability F min of each set is calculated one by one, and the results are given in Table 1. From Table 1, we can conclude that in the listed unitary-operation sets, {I,Y,H} cannot be chosen; {I,X,H}, {I,Z,H} and {I,X,Z,H} are the best sets, followed by {I,X,Y,Z,H}. Definition 3 (two-step unitary operation). Before sending the particle sequence to Bob i + 2, Bob i + 1 implements one random unitary operation on each particle, and then implements another, which may be different from the first unitary operation (the possibility is never zero). This is called two-step unitary operation. A set of two-step unitary operation is denoted by { ; } ; the first unitary operations available are shown before the semicolon, followed by the second ones. For example, Alice s operations in [9] and [12] are two-step unitary operations. The minimum failure probabilities of substitute-bell-state attack based on different two-step unitary operations that Bob i + 1 may choose are presented in Table 2 (only several common sets are listed). By the unitary-operation sets in Table 2, {I,Y ; I,H} cannot be chosen; {I,X; I,H} and {I,Z; I,H} are optimal, {I,X,Y,Z; I,H} in the second place. From the results in Tables 1 and 2, some conclusions can be drawn as follows: i) In the several common unitary-operation sets listed in Tables 1 and 2, {I,Y,H} and {I,Y ; I,H} can never be employed in QSS schemes. ii) Among the unitary-operation sets in Tables 1 and 2, {I,X,H}, {I,Z,H}, {I,X,Z,H}, {I,X; I,H} and {I,Z; I,H} are optimal, and {I,X,Y,Z; I,H} are second optimal. iii) The security against substitute-bell-state attack is not inevitably higher when more kinds of unitary operations are included in a QSS scheme. iv) If a two-step unitary operation contains the same species of unitary operations as a one-step unitary operation, the security of the former against substitute-bell-state attack is not always greater than or equal to that of the latter. For example, if {I,X,Z; I,H} is chosen, the minimum failure probability of substitute-bell-state attack is less than the one that selects {I,X,Z,H}. v) Generally, encoding and decoding become more complex with the increasing varieties of unitary operations. Therefore, in the case of the same security degree against substitute-bell-state attack, the QSS scheme with fewer varieties of unitary operations should be considered first.

5 Xu J, et al. Sci China Inf Sci September 2011 Vol. 54 No A supplement to the above is that when any participant utilizes unitary operations to encode the information, there must be at least one unitary operation which can realize the transformation between two different measuring bases, so as to avoid the complete success of substitute-bell-state attack. In QSS schemes based on two-level single particles, if participants only employ unitary operations to flip bit or (and) phase (say, I, X, Y and Z), but not use the operations that can transit the quantum state between the two conjugated bases { 0, 1 } and { +, } (e.g., H), the two-particle states that an eavesdropper can obtain after substitute-bell-state attack must be φ ± or ψ ±, i.e., the four Bell states. It is easy to figure out that the four Bell states are mutually orthogonal and they are a complete set in a fourdimensional Hilbert space. Thus an eavesdropper can distinguish them using Bell-basis measurement. In other words, the eavesdropper can know the exact unitary operations that the wiretapped participant has made. Evidently this is not allowed. If participants employ H operation simultaneously, the quantum states that an eavesdropper can attain at last still have the possibility to be mutually orthogonal, such as {I,Y,H} and {I,Y ; I,H}. In a word, if the QSS scheme based on single particles is secure against substitute-bell-state attack, the existence of unitary operations to realize basis transformation is a necessary condition, but not a sufficient condition. In [23], Alice encodes a special message on quanta via I or Y operations. Obviously, an eavesdropper obtains φ + or ψ after substitute-bell-state attack. Thus the minimum failure probability of the attack F min = 0, so there is a hidden safety flaw in the scheme proposed in [23]. 3 Conclusions In this paper, we first introduce a novel attack strategy named substitute-bell-state attack for QSS schemes without entanglement. Based on the definition of the minimum failure probability of this attack, we also provide a quantitative analysis of the security degrees corresponding to several common sets of unitary operations that QSS schemes may use. As a result, the relation between the effect of substitute- Bell-state attack and the selection of unitary operations is revealed. In the process of designing and implementing QSS schemes, the work of this paper can serve as an important guidance for the selection of unitary operations. Moreover, we analyze several common unitary-operation sets concerning only two-level quantum states and two conjugated bases. Further research on more complex situations, such as more varieties of unitary operations and higher level quantum states, would be conducted in the near future. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No ) and Jiangsu Natural Science Foundation (Grant Nos. BK , BK ). References 1 Gottesman D. Theory of quantum secret sharing. Phys Rev A, 2000, 61: Yang C, Gea-Banacloche J. Teleportation of rotations and receiver-encoded secret sharing. J Opt B: Quantum Semiclass Opt, 2001, 3: Karimipour V, Bahraminasab A, Bagherinezhad S. Entanglement swapping of generalized cat states and secret sharing. Phys Rev A, 2002, 65: Guo G P, Guo G C. Quantum secret sharing without entanglement. Phys Lett A, 2003, 310: Xiao L, Long G L, Deng F G, et al. Efficient multiparty quantum-secret-sharing schemes. Phys Rev A, 2004, 69: Deng F G, Long G L, Wang Y, et al. Increasing the efficiencies of random-choice-based quantum communication protocols with delayed measurement. Chin Phys Lett, 2004, 21: Deng F G, Zhou H Y, Long G L. Bidirectional quantum secret sharing and secret splitting with polarized single photons. Phys Lett A, 2005, 337: Yang Y G, Wen Q Y, Zhu F C. An efficient quantum secret sharing protocol with orthogonal product states. Sci China Ser G-Phys Mech Astron, 2007, 50:

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