## 1 Introduction  
Continuous-variable quantum key distribution (CV-QKD) is one of the most promising technologies in the field of cryptography [1; 2; 3], which can avoid the risk of being eavesdropped to some extent, thus laying a theoretical and experimental foundation for the establishment of secure quantum information network. Most CV-QKD protocols typically utilize the Gaussian modulation scheme to distribute a shared secret string to the two distant users, but their secure transmission distance is limited, mainly because the reconciliation efficiency is quite low when the transmission distance is long [4; 5]. To improve the performance of the protocol, discrete-modulated CV-QKD protocols were proposed. Thus far, CV-QKD protocols have been demonstrated in both fibers [6; 7; 8] and free-space air [9; 10; 11]. However, underwater CV-QKD has only recently become of interest to researchers. This is mainly because the oceans cover 70% of the Earth, and the completion of the undeveloped quantum communication under the seawater is an indispensable part for the establishment of a global secure communication network. As an alternative to Gaussian modulation, discrete modulation scheme has attracted much attention in recent years. This is mainly because it not only allows us to simplify significantly both the modulation scheme and the key extraction task, but also makes it possible to distill secret keysover much longer distance [12]. In Gaussian modulation, the secret information loaded onto the optical field is discrete data whose amplitude obeys the Gaussian distribution, and the information carrying amount is large, but the subsequent data are not easy to process. In the discrete modulation, the discrete 0 and 1 signals are directly modulated to the two orthogonal components of the optical field, and the information carrying capacity is slightly smaller, but it is easy to realize. When the channel attenuation is very large, the signal-to-noise ratio (SNR) is very low. In this case, if Gaussian modulation is adopted, the bit error rate will be very high, which brings great difficulties to error correction. However, if discrete modulation is adopted, the bit error rate can be reduced to a very low level. In this way, combining with low-density parity-check (LDPC) codes in classical communication, highly efficient error correction can be achieved. All of these indicate that discrete modulation is more suitable for long-distance transmission than its Gaussian counterpart. Similar to quantum communication in free atmosphere [13; 14; 15], underwater secure communication is an indispensable part of the global secure communication network. In view of the complex composition and special optical properties of seawater, it is not easy to analyze the attenuation in the seawater channels and information errors in communication when developing underwater quantum communication. The absorption of light by seawater results in the loss of energy of the signal light reaching the receiver, thus affecting the final key generation. The scattering of light by seawater changes the polarization of photons, which in turn increases the error of information. The feasibility of quantum communication under seawater has been investigated theoretically [16; 17] and demonstrated experimentally [18; 19], but these studies are based on discrete variables. In recent years, with continuous variable quantum key distribution gradually becoming a research hotspot in the field of quantum communication, it has also been applied to the research model based on the seawater channel [20; 21]. In this paper, we propose the discrete-modulated underwater CV-QKD protocol to improve the performance of the previous corresponding protocols that are based on Gaussian modulation. We investigate the optical property of the seawater channel and make an assumption for the security analysis of the protocol. Besides, the security of both the four-state protocol and the eight-state protocol in asymptotic and finite-size scenarios is analyzed. This paper is organized as follows.