The fairness of secret sharing guarantees that, if either participant obtains the secret, other participants obtain too. The fairness can be threatened by cheaters who was hidden in the participants. To efficiently and accurately identify cheaters with guaranteeing fairness, this paper proposes a fair (t, n)-threshold secret sharing scheme with an efficient cheater identifying ability. The scheme consists of three protocols which correspond to the secret distribution phase, secret reconstruction phase, and cheater identification phase respectively. The scheme's secret distribution strategy enables the secret reconstruction protocol to detect the occurrence of cheating and trigger the execution of the cheater identification protocol to accurately locate cheaters. Moreover, we prove that the scheme is fair and secure, and show that the cheater identification algorithm has higher efficiency by comparing with other schemes.