近年來，電腦資訊技術發展蓬勃，其中量子電腦的發展將扮演一個重要的角色，由於量子電腦具有強大的平行處理及運算能力。因此，一旦量子電腦研發成功，現有的近代密碼系統將會面臨被破解的危機。現今的密碼學系統均是基於數學計算上的難題來達到其安全性，也就是以目前的電腦技術欲破解需花費相當多的時間。然而，由於量子電腦的能力，這些數學上的難題將於短時間內被破解，目前已有研究指出可在多項式時間內破解因數分解問題。這表示現今常用的RSA加密系統在量子電腦下將失去其安全性。因此要如何在量子電腦上設計一個安全的量子密碼技術，也成為近年來密碼學家研究的重要領域。
量子密碼學與近代密碼學不同的是，它是利用量子的物理特性來設計達成其安全性。因此要如何利用這些量子特性來設計一個安全的密碼技術便是量子密碼學的研究重點，而現今也已有許多量子密碼技術被提出，如：量子金鑰分配協定、量子秘密分享協定和量子私密比較協定等。
量子私密比較協定(Quantum Private Comparison)為量子密碼學上一個重要的研究。經由量子的糾纏特性及一semi-honest第三方的協助，讓雙方使用者能在彼此不知道對方私密訊息的情況下，仍然能夠比較彼此的私密訊息是否相同。近年來，許多量子私密比較協定被提出，這些協定植基於量子貝爾態的么正算子或是GHZ糾纏態量子的量測特性等來達到其需求。
現今的量子私密比較協定可能為了避免一些常見的攻擊而導致效率低落，且部分可能還存在一些安全性上的問題。因此，本論文首先針對最近一篇利用GHZ糾纏態量子所設計的量子私密協定比較提出一個攔截重送攻擊，接著提出一個基於利用量子貝爾態來設計一個有效率的量子私密比較協定。

The recent years have witnessed considerable development in computers and the field of information science. The development of the quantum computer is expected to play a key role in the future because it is capable of computation and parallel processing. Further, the current cryptography system would be prone to security breaches by the quantum com-puter. The security of modern cryptography systems is based on the complexity of calcula-tions of mathematical problems involved in different cryptography techniques. These mathematical problems can be solved more efficiently by using the quantum computer (e.g., Shor proposed an efficient quantum algorithm for the factorization of numbers in polynomial time). This implies that the RSA encryption system can be broken by a quantum computer. Therefore, at present, designing a secure quantum cryptography system is an important research topic.
Quantum cryptography differs from modern cryptography in that it achieves its secu-rity by adopting the physical properties of quantum mechanics. Consequently, the basis of quantum cryptography lies in the application of these physical properties in designing se-cure cryptography techniques. Many quantum cryptography protocols have been proposed, such as the quantum key distribution (QKD) protocols, quantum secret sharing (QSS), and quantum private comparison (QPC).
The concept of private comparison has already been a topic of discussion in conven-tional cryptography. Yao proposed a protocol [17] to solve the millionaires’ problem in which two millionaires, Alice and Bob, are interested in knowing which of them is richer without revealing their actual wealth. On the basis of Yao’s millionaires’ problem, Boudot subsequently proposed a protocol [18] to determine whether the two millionaires are equal-ly rich. However, Lo [19] indicated that the equality function used to determine this cannot be securely evaluated in a two-party scenario. Therefore, some additional assumptions (e.g., a semi-honest third party (TP)) should be considered to successfully achieve private com-parison.
Quantum voting [20, 21] and quantum auction [22-24] have necessitated the applica-tion of private comparison in the past. On the basis of quantum mechanics, evaluation functions have been designed to calculate the summation of the votes in quantum voting and determine the winner of a quantum auction. Recently, researchers have been able to re-alize this concept by using the quantum computer and have proposed some QPC protocols.
The concept of QPC is important in the field of quantum cryptography. By introduc-ing the concept of quantum entanglement and involving a semi-honest TP, two users can compare undisclosed information for equality. Several QPC protocols that involve unitary operations to set the Bell state or the entangled quantum GHZ state have been proposed to achieve the goal of comparison for equality between the undisclosed information of both parties.
In order to prevent some common attacks, current QPC protocols suffer from low ef-ficiency. Moreover, many other protocols have security loopholes. This thesis aims to demonstrate the intercept–resend attacks on a QPC protocol, which is designed using the GHZ state, and present a more efficient QPC protocol based on the Bell state.

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