## 1 Introduction  
Synchronization is the process of controlling the output of a chaotic dynamical slave system in order to force its variables to match those of a corresponding master system in time [1]. The subject of synchronization has been around for about 30 years. Over the course of this period, the subject has attracted the interest of researchers from a variety of fields including but not limited to engineering, natural sciences, social sciences, physics, chemistry, and many more. Although the amount of literature related to chaotic dynamical systems is vast, an exact definition of such systems is not easy to find. The general consensus, however, is that a chaotic system is one with an extremely high sensitivity to small variations in the initial conditions. The trajectory of the solutions is seemingly random and difficult to pretend. However, if the initial conditions are known, the trajectory can be exactly reproduced. Chaotic systems and their synchronization are of particular interest in the field of secure communications due to the many similarities between chaos and the encryption of data being transmitted. Synchronization in general requires some form of control strategy. Various kinds of control schemes can be found in the literature aimed at synchronizing integer-order chaotic systems [2]. However, most of the existing studies consider only the simple case where both systems have exact same order and the same dimensions. Attempts have been dedicated to synchronize systems with different dimensions such as those shown in Table 1. The importance of different dimensional dynamic system synchronization stems from its wide range of applications as well as its enrichment of control theory. In addition to integer-order systems, the research community has also been looking at fractional-order ones due to the added flexibility chaotic nature which they add to the mix [12]. The same progress made with integer-order systems has been attempted here. Some studies have examined the synchronization of fractional-order systems with identical dimensions [13] and others considered the more general case of arbitrary dimensions [14; 15]. This study presents a novel contribution to the topic. We investigate **F-M** synchronization, which combines generalized synchronization based on a functional relationship **F** with inverse matrix projective synchronization based on a matrix **M** with synchronization index \(d\), which basically represents the dimension of the synchronization error. By exploiting the fractional Laplace transform along with the stability theory related to linear systems with integer orders, the **F-M** synchronization of fractional-order systems is proven for the case \(d=m\) showing that the zero solution of the error system is globally asymptotically stable. The case \(d<m\) is also considered and the synchronization demonstrated. The proposed scheme is rather general with the only restriction on the scaling functions being that they must be differentiable. Chaos has attracted considerable attention in the field of wireless and optical communications over the last two decades. The main reason for this attraction is the many similarities between chaos and encryption, which is a necessary part of any modern communications system [16; 17]. Traditionally, encryption was performed at higher layers within the Open Systems Interconnection (OSI) communications model. With chaos, security concerns have shifted to the physical layer where the carriers of information, which were always sine and cosine waves, are replaced with chaotic or hyperchaotic signals. The amount of literature concerning the use of chaos in communications is vast [18]. However, they may be generally classified into five main categories: masking schemes, modulations schemes, multiple access schemes, multicarrier schemes, and secret/public key encryption schemes. In this paper, we employ the proposed **F-M** synchronization strategy to form a multiple input multiple output (MIMO) secure communications system based on message masking.