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Fuzzing is one of the most effective approaches for identifying security vulnerabilities. As a state-of-the-art coverage-based greybox fuzzer, AFL is a highly effective and widely used technique. However, AFL allocates excessive energy (i.e., the number of test cases generated by the seed) to seeds that exercise the high-frequency paths and can not adaptively adjust the energy allocation, thus wasting a significant amount of energy. Moreover, the current Markov model for modeling coverage-based greybox fuzzing is not profound enough. This paper presents a variant of the Adversarial Multi-Armed Bandit model for modeling AFL’s power schedule process. We first explain the challenges in AFL's scheduling algorithm by using the reward probability that generates a test case for discovering a new path. Moreover, we illustrated the three states of the seeds set and developed a unique adaptive scheduling algorithm as well as a probability-based search strategy. These approaches are implemented on top of AFL in an adaptive energy-saving greybox fuzzer called EcoFuzz. EcoFuzz is examined against other six AFL-type tools on 14 real-world subjects over 490 CPU days. According to the results, EcoFuzz could attain 214% of the path coverage of AFL with reducing 32% test cases generation of that of AFL. Besides, EcoFuzz identified 12 vulnerabilities in GNU Binutils and other software. We also extended EcoFuzz to test some IoT devices and found a new vulnerability in the SNMP component.