## 1 Introduction  
In most of the wireless sensor networks (WSNs), the significant issue for data collection or data aggregation always lies in the center of data transmission, both in the academia and in the industry [1; 2; 3]. In most scenarios of marine WSNs, all the nearby wireless sensors send their data, such as the temperature, pressure, salinity, and potential of hydrogen (pH value) in the chemistry of the environmental monitoring ocean, to a central node, which is located at a base station or a buoy for data collection, as shown in Figure 1. The central node further sends the aggregated data through the long-distance data transmission networks, such as vessel-based or satellite-based networks [4]. However, marine sensors are always deployed far away from the security monitoring. Thus, the secure data aggregation for marine sensor networks has emerged and attracted the interest of researchers and engineers. In order to mitigate the malicious attackers injecting false data, it is quite necessary for each central node to authenticate these sensing measurements from the nearby sensors in the ocean observation system [5]. Generally, a digital signature often provides the properties of authenticity and non-repudiation through checking the signed acknowledgments from senders [6]. However, in WSNs, the international standards for broadcasting authentication are very vulnerable to signature verification flooding attacks, as the excessive requests for signature verification must run out of the computational resources of those victims [7]. The scenario seems worse, as the marine wireless sensors are powered by a limited battery and cannot afford these overloaded requests in an oceanic environment. To optimize the communication and computational overhead, a variant of digital signature, named _multi-signature_, permits various signers to sign on a message individually and aggregate partial signatures to a compact signature [8]. A multi-signature can play a significant role in authenticating different sensors' data by checking a single compact signature to cut down the communication bandwidth for marine wireless devices, as the transmission of one-bit data consumes more energy than the arithmetic operations on several bits [9]. This seems a promising way to solve the data authentication in a multi-user scenario. Since the primitive has been proposed, multi-signature schemes have been paid attention to by most of the network designers and industry engineers. However, in the past years, most of the work on multi-signature schemes has been constructed by relying on the assumed existence of _public key infrastructure_ (**PKI**) [10; 11]; the heavy burdens of the digital public key certificate management bring high communication overhead and storage overhead when **PKI** is applied and implemented in the wireless networks. The cases become worse when the sensors are deployed in the marine environments (denoted as **Problem 1**). To overcome the weakness brought by **PKI**, identity-based cryptography emerges as a novel cryptographic primitive and a powerful alternative to traditional certificate-based cryptography, which has been raised early on in [12] and is further specifically designed in [13; 14]. Identity-based cryptography makes some public, known information a public key, such as the device's number, IP address, or a username, to mitigate the management problem for the public key certificates. In the extreme case that the bandwidth is a bottleneck, the identities of the signers often appear in the head of the communication packets, instead of in the transmission of the heavy public keys. Inspired by this concept, the first _identity-based multi-signature_ (**IBMS**) scheme, proposed in [15], uses a mathematical technique named \"bilinear mapping\", such as is used in [13], and is proved to be secure, relying on _discrete logarithm_ (DL) assumptions or _computational Diffie-Hellman_ (CDH) assumptions.