This paper presents a formal logic for analyzing authentication protocols, which are essential for security in distributed systems. The authors argue that the design of these protocols is error-prone, often containing redundancies or security flaws. They introduce a logic that describes the beliefs of trustworthy parties involved in authentication protocols and how these beliefs evolve through communication. This logic allows for the formal verification of protocols, identifying subtleties and suggesting improvements. The paper includes a detailed formalism, including syntax, semantics, and logical postulates, which are used to transform and analyze protocols. The authors apply this logic to four published protocols: Kerberos, Andrew Secure RPC Handshake, Needham-Schroeder Public-Key Protocol, and CCITT X.509 Protocol. Each protocol is analyzed step-by-step to determine its goals, the type of cryptosystem used, whether secrets are employed, and whether message timeliness is guaranteed. The analysis reveals that Kerberos, Andrew Secure RPC Handshake, and CCITT X.509 have vulnerabilities, such as replay attacks and insufficient message timeliness guarantees. The authors propose solutions to these issues, such as adding timestamps or using more secure message structures. The paper concludes by summarizing the results and highlighting the benefits of using the logic for protocol analysis. It emphasizes the importance of reasoning about knowledge in distributed computation and the need for formal methods to ensure the correctness of authentication protocols.This paper presents a formal logic for analyzing authentication protocols, which are essential for security in distributed systems. The authors argue that the design of these protocols is error-prone, often containing redundancies or security flaws. They introduce a logic that describes the beliefs of trustworthy parties involved in authentication protocols and how these beliefs evolve through communication. This logic allows for the formal verification of protocols, identifying subtleties and suggesting improvements. The paper includes a detailed formalism, including syntax, semantics, and logical postulates, which are used to transform and analyze protocols. The authors apply this logic to four published protocols: Kerberos, Andrew Secure RPC Handshake, Needham-Schroeder Public-Key Protocol, and CCITT X.509 Protocol. Each protocol is analyzed step-by-step to determine its goals, the type of cryptosystem used, whether secrets are employed, and whether message timeliness is guaranteed. The analysis reveals that Kerberos, Andrew Secure RPC Handshake, and CCITT X.509 have vulnerabilities, such as replay attacks and insufficient message timeliness guarantees. The authors propose solutions to these issues, such as adding timestamps or using more secure message structures. The paper concludes by summarizing the results and highlighting the benefits of using the logic for protocol analysis. It emphasizes the importance of reasoning about knowledge in distributed computation and the need for formal methods to ensure the correctness of authentication protocols.