Counterfeit integrated circuits (IC) can be very harmful to the security and reliability of critical applications. Physical unclonable functions (PUFs) have been proposed as a mechanism for uniquely identifying ICs and thus reducing the prevalence of counterfeits. However, maintaining large databases of PUF challenge response pairs (CRPs) and dealing with PUF errors make it difficult to use PUFs reliably. This paper presents an innovative approach to authenticate CRPs on PUF-based ICs. The proposed method can tolerate considerable bit errors from responses of PUFs without the use of error correcting codes. Different types of optimization methods are applied to improve the overall performance. The simulation shows that it is successful in authenticating 99.96% authorized chips and filtering out 99.92% cloned chips by tolerating 12 errors in 128 bits. The results are verified with ring oscillator PUF and arbiter PUF implementations on Kintex-7 FPGA. The approach saves hardware and software resources significantly, compared to those of other authentication solutions.