Private set union (PSU) allows several parties to obtain the union of their private sets without disclosing each party’s private information. Existing PSU protocols often have polynomial complexity for the complete set size or complicated process. In this paper, a quantum multi-party PSU protocol based on least common multiple (LCM) and Shor’s algorithm is proposed, which enables the union of multiple sets to be computed all at once. In order to increase the one-time success probability of the protocol, we first improved Shor’s period-finding algorithm, which is used in LCM computation and integer factoring. Each party’s private set is encoded into an integer obtained by multiplying several prime numbers, thus the PSU problem is transformed into an LCM problem. The LCM of these integers is computed by using the improved Shor’s period-finding algorithm, and then factored to derived the union set. We prove the correctness of the proposed protocol, and its unconditional security against semi-honest attacks. Complexity analysis shows that our protocol has logarithmic complexity for the complete set size.
Private set union (PSU) allows several parties to obtain the union of their private sets without disclosing each party’s private information. Existing PSU protocols often have polynomial complexity for the complete set size or complicated process. In this paper, a quantum multi-party PSU protocol based on least common multiple (LCM) and Shor’s algorithm is proposed, which enables the union of multiple sets to be computed all at once. In order to increase the one-time success probability of the protocol, we first improved Shor’s period-finding algorithm, which is used in LCM computation and integer factoring. Each party’s private set is encoded into an integer obtained by multiplying several prime numbers, thus the PSU problem is transformed into an LCM problem. The LCM of these integers is computed by using the improved Shor’s period-finding algorithm, and then factored to derived the union set. We prove the correctness of the proposed protocol, and its unconditional security against semi-honest attacks. Complexity analysis shows that our protocol has logarithmic complexity for the complete set size.
We present a secure multiparty quantum computation (MPQC) for computing greatest common divisor (GCD) based on quantum multiparty private set union (PSU) by Liu, Yang, and Li. As the first step, we improve the security of the MPQC protocol for computing least common multiple (LCM) by Liu and Li by constructing an efficient exact quantum period-finding algorithm (EQPA) as a subroutine instead of the standard (probabilistic) Shor's quantum period-finding algorithm (QPA). The use of EQPA instead of the standard QPA guarantees the correctness of the protocol without repetitions. The improvement of LCM protocol also improves the private set union protocol which is based on computing LCM. Finally, using the same idea of the PSU protocol, we construct a quantum multiparty private set intersection (PSI) by transforming the PSI problem into the problem of computing GCD. Performance analysis shows that the correctness and the unconditional security in the semihonest model are guaranteed directly from the correctness and the security of the subroutine protocols (LCM and PSU protocols). Moreover, we show that the complexity of the proposed protocols is polynomial in the size of the secret inputs and the number of parties.
Private set intersection (PSI) is a cryptographic primitive that allows two or more parties to learn the intersection of their input sets and nothing else. In this paper, we introduce a new secure multi-party quantum computation for the greatest common divisor (GCD) and a PSI protocol based on it. The protocol extends the recent least common multiple (LCM)-based quantum private set union protocol by Liu et al. Performance analysis verifies the correctness and demonstrates the entire security of the suggested protocols in the semi-honest model. Additionally, it has been demonstrated that the complexity is efficient in terms of the input set size.
In this paper, we present a secure multiparty computation (SMC) protocol for least common multiple (LCM) based on Shor's quantum period-finding algorithm (QPA). Our protocol is based on the following principle: the connection of multiple periodic functions is also a periodic function whose period is exactly the least common multiple of all small periods. Since QPA is a probabilistic algorithm, we also propose a one-vote-down vote protocol based on the existing secure multi-party quantum summation protocol, which is used to verify the results of the proposed LCM protocol. Security analysis shows that under the semi-honest model, the proposed protocol is secure with high probability, while the computational consumption remains at polynomial complexity. The protocol proposed in this paper solves the problem of efficient and secure multiparty computation of LCM, demonstrating quantum computation potential.
Private set intersection (PSI) has important application value, however, current quantum PSI protocols are either unsuitable for multi-party scenarios or inefficient. Recently, Imran (arXiv: 2303.17196v3 , 2023) proposed two quantum secure multi-party greatest common divisor (GCD) protocols that can be used for PSI, but with the downside of information leakage and resource consumption. In this paper, we propose a novel quantum secure multi-party GCD protocol that has higher security and lower complexity. To hide privacy, each party randomly selects a coefficient within a range determined by his input integer, and with the assistance of a semi-honest third party TP, all parties secretly calculate the linear combination of their inputs under these coefficients. Once enough linear combinations are collected, TP calculates the GCD of these combinations, which is equal to the GCD of all input integers. To verify the honesty of participants, a quantum zero-knowledge proof sub-protocol is designed. Analysis shows that our GCD protocol is correct and has security against malicious attacks. Moreover, its complexity is polynomial level and lower than Imran’s. Furthermore, we demonstrate the scalability of our GCD protocol in private set operations, such as private set intersection, private set intersection cardinality, private multi-set intersection, etc.
Private Set Intersection Cardinality (PSI-CA) and Private Set Union Cardinality (PSU-CA) are two cryptographic primitives whereby two or more parties are able to obtain the cardinalities of the intersection and the union of their respective private sets, and the privacy of their sets is preserved. In this paper, we propose a three-party protocol to finish these tasks by using quantum resources, where every two, as well as three, parties can obtain the cardinalities of the intersection and the union of their private sets with the help of a semi-honest third party (TP). In our protocol, GHZ states play a role in encoding private information that will be used by TP to compute the cardinalities. We show that the presented protocol is secure against well-known quantum attacks. In addition, we analyze the influence of six typical kinds of Markovian noise on our protocol.
Abstract Card-based cryptography aims to realize secure multiparty computation with physical cards. This paper is the first to address Private Set Intersection (PSI) and Private Set Union (PSU) in card-based cryptography. PSI and PSU are well-studied secure computation protocols to compute the set intersection and the set union, respectively. We show two-party PSI and PSU protocols in each of the two operation models: one is the shuffle-based model in which parties perform all operations publicly, and the other is the private-permutation-based model that allows parties to perform some operations privately. In the shuffle-based model, we show PSI and PSU protocols can be realized with existing secure AND and OR protocols, respectively. However, these protocols have an issue of increasing the number of shuffles depending on the size of the universal set. To resolve the issue, we further propose PSI and PSU protocols with only one shuffle at the cost of increasing the number of cards. In the private-permutation-based model, we show PSI and PSU protocols can be achieved with existing secure AND and OR protocols, respectively, as in the shuffle-based protocols. These protocols have an advantage of requiring only one private permutation and one communication. We further show that the number of cards of these protocols can be reduced at the cost of increasing the number of private permutations and communications.