Fog computing (fog networking) is known as a decentralized computing infrastructure in which data, applications, compute as well as data storage are scattered in the most logical and efficient place among the data source (i.e., smart devices) and the cloud. It gives better services than cloud computing because it has better performance with reasonably low cost. Since the cloud computing has security and privacy issues, and fog computing is an extension of cloud computing, it is therefore obvious that fog computing will inherit those security and privacy issues from cloud computing. In this paper, we design a new secure key management and user authentication scheme for fog computing environment, called SAKA-FC. SAKA-FC is efficient as it only uses the lightweight operations, such as one-way cryptographic hash function and bitwise exclusive-OR (XOR), for the smart devices as they are resource-constrained in nature. SAKA-FC is shown to be secure with the help of the formal security analysis using the broadly accepted Real-Or-Random (ROR) model, the formal security verification using the widely-used Automated Validation of Internet Security Protocols and Applications (AVISPA) tool and also the informal security analysis. In addition, SAKA-FC is implemented for practical demonstration using the widely-used NS2 simulator.