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In this paper, we present the design of the lightweight Ff family of privacy-preserving authentication protocols for RFID-systems. Ff results from a systematic design based on a new algebraic framework focusing on the security and privacy of RFID authentication protocols. Ff offers user-adjustable, strong authentication, and privacy against known algebraic attacks and recently popular SAT-solving attacks. In contrast to related work, Ff achieves these security properties without requiring an expensive cryptographic hash function. Ff is designed for a challenge-response protocol, where the tag sends random nonces and the results of HMAC-like computations of one of the nonces together with its secret key back to the reader. In this paper, the authentication and privacy of Ff is evaluated using analytical and experimental methods.