In this paper, we propose a family of light-weight provably-secure authentication protocols called the RCHB family, which is a variant of the HB protocol. The HB protocol uses the complexity of decoding linear codes for security against passive attacks. The security for the RCHB protocol family to passive attacks is enhanced by the introduction of both the rotation and complement on the secret key. We demonstrate that the existing passive attacks on the HB protocol family, which have contributed to considerable reduction in its effective key-size, do not work against RCHB. Thus, smaller-key sizes are sufficient for RCHB protocols to achieve the same level of passive attack security as the HB Protocol. Furthermore, we provide implementation instances of RCHB family protocols for which the Prover/Verifier complexity is much lower than the HB protocol, enabling authentication on very low-cost devices like RFID tags.