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.

翻译：我们提出了一种基于Liu、Yang和Li的量子多方私有集合并集（PSU）的安全多方量子计算（MPQC）协议，用于计算最大公约数（GCD）。作为第一步，我们通过构造一个高效的精确量子周期查找算法（EQPA）作为子程序，替代标准的（概率性）Shor量子周期查找算法（QPA），改进了Liu和Li提出的用于计算最小公倍数（LCM）的MPQC协议的安全性。使用EQPA而非标准QPA确保了协议无需重复即可保证正确性。LCM协议的改进也改进了基于LCM计算的私有集合并集协议。最后，利用PSU协议的相同思路，我们通过将私有集合交集（PSI）问题转化为计算GCD的问题，构建了量子多方私有集合交集协议。性能分析表明，在半诚实模型下，该协议的正确性和无条件安全性直接由子程序协议（LCM和PSU协议）的正确性和安全性保证。此外，我们证明了所提出协议的复杂度在秘密输入大小和参与方数量上呈多项式级。