## 1 Introduction
Because the operation of bilinear mapping involves too much computational overhead [16; 17], many bilinear  
Figure 1: Data collection in marine wireless sensor networks (WSNs). mapping techniques are not suitable for the battery-limited sensors in marine WSNs (denoted as **Problem 2**). As a consequence, there is great interest for cryptographic researchers to design pairing-free identity-based cryptographic schemes [18]. The first non-pairing IBMS scheme was proposed in [19] with three-round interactive communications and under [PERSON], [PERSON], [PERSON] (RSA) assumptions. Later, a communication efficiency-improved IBMS scheme under RSA assumptions was presented in [20] with two-round interactive communications. [PERSON] et al. [21] proposed an efficient improved IBMS scheme that aims to save the computational resources and communication bandwidth. Even if the RSA assumption approaches the integer factorization assumptions, unfortunately, the RSA assumption has not yet been proved equal to the factorization assumption (denoted as **Problem 3**). To satisfy the application requirements and to avoid security concerns in cryptography, it is common practice to construct alternative cryptographic schemes under a weaker assumption--integer factorization. Recently, cryptographic researchers have been focused on finding a new construction that is proved to be secure directly on the basis of factorization. [PERSON] [22] gave an instance of an identity-based digital signature relying on the quadratic residue assumption. Following this, [PERSON] et al. [6] proposed IBMS schemes using quadratic residue assumptions, under weaker assumptions and a strengthened security model, achieving advantages in the computational consumption and transmission overhead. [PERSON] [23] and [PERSON] [24] presented identity-based signature schemes under the cubic residue assumptions. [PERSON] proposed several signature variants relying on cubic residues, including identity-based ring signature [25], _identity-based proxy multi-signature_ (**IBPMS**) [26] and threshold ring signature [27]. [PERSON] [28] considered an _identity-based multi-proxy signature_ (**IBMPS**) scheme for use in a cloud-based data authentication protocol. [PERSON] [29] proposed a secure multi-entity delegated authentication protocol based on an _identity-based multi-proxy multi-signature_ (**IBMPMS**) for mobile cloud computing. Unfortunately, none considered constructing IBMS schemes directly based on cubic residues (denoted as **Problem 4**). Facing the above problems, this work constructs IBMS schemes relying on the cubic residue assumption equal to integer factoring. Our schemes have merits not only in the efficiency aspect, where we do not rely on the bilinear pairing maps or over exponentiations, but also in the security aspect, where we prove them to be secure under a weaker assumption of factoring to achieve stronger security. The contributions for this paper can be summarized as follows. 1. We have proposed two efficient IBMS schemes, denoted as \(\mathbf{IBMSC^{CR}}\mathbf{-1}\) and \(\mathbf{IBMSC^{CR}}\mathbf{-2}\), which are suitable for data aggregation among the sensors and collectors in marine WSNs. 2. We formally define the security of IBMS and prove \(\mathbf{IBMSC^{CR}}\mathbf{-1}\) to be secure, relying on the cubic residues in a random oracle model. The computational cost of \(\mathbf{IBMSC^{CR}}\mathbf{-1}\) is lower, as the exponentiations are cubic exponentials. 3. To enhance efficiency, the total computational cost of \(\mathbf{IBMSC^{CR}}\mathbf{-2}\) is almost four-fifths that of \(\mathbf{IBMSC^{CR}}\mathbf{-1}\) in implementation. We also prove the security of \(\mathbf{IBMSC^{CR}}\mathbf{-2}\) on the basis of the cubic residues equalling integer factoring in the random oracle model. The organization of this paper is as follows. Section 2 gives necessary preliminaries, and Section 3 gives the formal definition of the security model.