We present a password‐authenticated (2, 3)‐threshold group key share (PATS) mechanism. Although PATS resembles threshold secret sharing schemes, it has a different structure. The innovative perspective of the PATS mechanism that makes a difference from the standard secret‐sharing schemes is that it involves parties in the generation of the shares. PATS allows parties to communicate securely to establish their shares over insecure channels. Parties (shareholders) construct a secret (key) using shares obtained at the end of the protocol. PATS takes advantage of zero‐knowledge proofs compared to well‐known threshold key exchange schemes and will tolerate the existence of semi‐trusted parties. We present two variants of PATS, centralized and distributed, and then generalize PATS to ( t , n )‐threshold scheme. PATS supports the distributed operation and optionally facilitates group key verification by a trusted third party, which may also partake in group key sharing. In this paper, we present PATS, which employs finite fields and elliptic curves, along with its security and complexity analyses.

Zero-knowledge protocols (ZKPs) allow a party to prove the validation of secret information to some other party without revealing any information about the secret itself. Appropriate, effective, and efficient use of cryptographic ZKPs contributes to many novel advances in real-world privacy-preserving frameworks. One of the most important type of cryptographic ZKPs is the zero-knowledge range proofs (ZKRPs). Such proofs have wide range of applications such as anonymous credentials, cryptocurrencies, e-cash schemes etc. In many ZKRPs the secret is represented in binary then committed via a suitable commitment scheme. Though there exist different base approaches on bilinear paring-based and RSA-like based constructions, to our knowledge there is no study on investigating the discrete logarithm-based constructions. In this study, we focus on a range proof construction produced by Mao in 1998. This protocol contains a bit commitment scheme with an OR-construction. We investigate the effect of different base approach on Mao's range proof and compare the efficiency of these basis approaches. To this end, we have extended Mao's range proof to base-3 with a modified OR-proof. We derive the number of computations in modulo exponentiations and the cost of the number of integers exchanged between parties. Then, we have generalized these costs for the base-u construction. Here, we mainly show that comparing with other base approaches, the base-3 approach consistently provides approximately 12% efficiency in computation cost and 10% efficiency in communication cost. We implemented the base-3 protocol and demonstrated that the results are consistent with our theoretical computations.