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As radio frequency identification (RFID) applications become ubiquitous, security and privacy issues have been addressed with universal acceptances. This paper proposes a lightweight Zero-Knowledge Authentication Protocol (ZKAP) based on alternative mode to address such severe problems. In ZKAP, dual zero-knowledge proofs are randomly chosen to provide anonymity and mutual authentication without revealing any sensitive identifiers. Pseudo-random flags and access lists employed for quick search and check ensure high efficiency and scalability. Meanwhile, formal proof model based on reasonable mathematical assumptions is established to prove the adaptive completeness, soundness and zero-knowledgeness, and the attack models are adopted to analyze the resilience and resistance for malicious attacks. It indicates that ZKAP owns no obvious design defects theoretically and is robust enough to resist major attacks (e.g., forgery, replay, Man-in-the-Middle, and tracking). The protocol is attractive and appropriate for low-cost and resource-restricted RFID systems.