Advanced persistent threat (APT) for data theft poses a severe threat to cloud storage systems (CSSs). An APT actor may steal valuable data from the target CSS even in a strategic fashion. To protect a CSS from APT, the cloud defender has to dynamically allocate the limited security resources to recover the compromised storage servers, aiming at mitigating his total loss. This paper addresses this dynamic cloud storage recovery (DCSR) problem by employing differential game theory. First, by introducing an expected state evolution model capturing the CSS's expected state evolution process under a combination of attack strategy and recovery strategy, we measure the APT attacker's net benefit and the cloud defender's total loss. On this basis and in the worst-case situation where the cloud defender assumes that the APT attacker has full knowledge of his expected loss, we reduce the DCSR problem to a differential game-theoretic problem (the DCSR * problem) to characterize the strategic interactions between the two parties. Second, we derive a necessary condition for Nash equilibrium of the DCSR * problem and thereby introduce the concept of competitive strategy profile. Next, we study the structural properties of the competitive strategy profile, followed by some numerical examples. Then, we conduct extensive comparative experiments to exhibit that the competitive strategy profile is superior to a large number of randomly generated strategy profiles in the sense of Nash equilibrium solution concept. Finally, we briefly analyze the practicability (scalability and feasibility) of this paper. Our findings will be helpful to enhance the APT defense capabilities of the cloud defender. INDEX TERMS Advanced persistent threat, cloud storage recovery, state evolution model, differential game, Nash equilibrium, necessity system, competitive strategy profile. using social engineering attacks, an APT attacker can always infiltrate the CSS, leading to severe data leakage. Due to the fast rate at which the APTs are evolving, it is almost impossible to perfectly protect a CSS from APTs only by employing traditional defense mechanisms such as Intrusion Detection System (IDS) and firewall. A. MOTIVATION Consider a CSS consisting of multiple storage servers. The cloud defender as the owner of the CSS is responsible for the decision-making of the whole system. Every day a substantial amount of data will be uploaded to or downloaded from the CSS by individual devices or organizations [4]. See Fig. 1 for the diagram of such a CSS. In this setting, an APT attacker can apply social engineering attacks to the cloud defender to compromise the storage servers and establish footholds. Once having footholds in the target CSS, the APT attacker will be able to encrypt and exfiltrate the valuable data to his remote command-and-control (C&C) server through an established communication channel which applies mainstream protocols such as HTTP, HTTPS, FTP, P2P, and others. See Fig. 2 for the diagram of the APT for data theft on the CSS shown in Fig. 1....