The (t, N) threshold multi-party private set intersection (TMP-PSI) protocol allows a given party’s data element x to appear in the private sets of no fewer than t–1 other parties. The data element x is then output as the intersection result, which is widely applied in scenarios such as proposal voting, financial transaction threat identification, and security assessment. Existing threshold multi-party private set intersection protocols suffer from low efficiency, high communication rounds, and a limitation that only a specific participant can obtain the intersection. To address these issues, this study proposes a threshold testing method based on robust secret sharing (RSS) and a TMP-PSI scheme combined with oblivious key-value store (OKVS), which effectively reduces both computational overhead and the number of communication rounds. To meet the demand for multiple participants to access the intersection information from their private sets, this study also proposes a second extended threshold multi-party private set intersection (ETMP-PSI) protocol, which modifies the share distribution method. Compared to the first scheme, the secret distributor and secret reconstructor do not incur additional communication rounds or computational complexity, allowing multiple participants to obtain the intersection elements from their private sets. The proposed protocol runs in 6.4 seconds (TMP-PSI) and 8.7 seconds (ETMP-PSI) in a three-party scenario with a dataset size of n=2¹⁶. Compared to existing threshold multi-party private set intersection protocols, the communication complexity between the reconstructor and distributor is reduced from O(nNtlognλ) to O(bNλ).