# Continuous Variable Quantum Secret Sharing with Fairness
## 1 Introduction  
The secret sharing (SS) plays a significant role in cryptography. Since 1999, [PERSON] et al. [1] firstly invited SS to the quantum domain by applying three-particle and four-particle GHZ states, more and more QSS scheme have been proposed [2; 3; 4]. Based on the quantum mechanics, the secret distribution can be ensured unconditional safety.  
Conventionally, a \((t,n)\)-threshold secret sharing scheme is built to prohibit \((t-1)\) or fewer dishonest participants conspiring for secret. At the same time, the participant has more advantages to steal the secret than outside eavesdroppers. Hence, compared with other protocols, such as quantum key distribution [5; 6; 7], quantum signature [8], Quantum anonymous voting [9] and so on, the QSS protocols need to analysis the attack from both inside and outside. But in the previous literatures, it is rarely discussed how the secret is revealed securely against inner attack [10; 11; 12; 13]. Until now, many SS protocols have been improved to verify participants and check the validity of shares in the recovery phase [14; 15; 16], but the participant who is the last one to release share, would desire to obtain the secret alone by sending fake share or keeping silence. So, in order to solve this problem without the simultaneously releasingconstraint, [PERSON] et al. [17] proposed a fair reconstruction, in which [PERSON], in addition to the secret shadow, distributes a check vector, which is used to verify the validity of other participants' shadows in the reconstructing process and a certificate vector, which is used to prove the validity of his own shadow to each participant in classical scenario. Then, in quantum domain, [PERSON] et al. [18] designed a QSS protocol based on partially and maximally entangled states, in which a secure and fair reconstruction mechanism is firstly organized to realize each participant can learn or cannot learn the secret simultaneously. Later, [PERSON] et al. [19] proposed a rational secret sharing scheme for the first time, in which the rational participant tries to maximize his or her utility by obtaining the secret alone, but it is impossible to occur, because the protocol is usually fair (everyone gets the secret).  
The above-mentioned schemes are primarily based on discrete variable quantum entanglement states, which emerge some choke points as the extreme fragility, the low channel capacity and the difficulty of the preparation in experiment. So, in this paper, the continuous variable quantum information theory is invited to distribute shares [20; 21]. The two-mode squeezed vacuum state is well done at preparation, operation and detection [22; 23]. What's more, the modulation performed on the two-mode squeezed vacuum state is not only binary modulation, but also multiple modulation, which can improve the the channel capacity. Furthermore, the quantum channel based on two-mode squeezing states provides secure communication, which is proved that it can send shares successfully, as long as setting proper the squeezing and modulation parameters according to the quantum channel transmission efficiency and the Shannon information of shares. In order to ensure every participant learn or do not learn the secret simultaneously without the simultaneous channel, a fair construction is designed, in which a check sequence is used to hide real secret sequence, a determine pointer is used to find the hidden secret and a verify sequence is used to verify the recovered message. Furthermore, this fair protocol can be incorporated with other sharing schemes.  
The organization of this paper is as follows. In Section 2, we design the \((2,2)\)-threshold CVQSS scheme with fairness. Section 3 explicates the security analysis of the scheme. At last, in Section 4, the conclusion is given.