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：我们基于刘、杨和李提出的量子多方私有集合并集（PSU）协议，提出了一种用于计算最大公约数（GCD）的安全多方量子计算（MPQC）方案。作为第一步，我们通过构建一个高效精确的量子周期查找算法（EQPA）作为子程序，替代标准（概率性）Shor量子周期查找算法（QPA），从而改进了刘和李提出的最小公倍数（LCM）MPQC协议的安全性。使用EQPA而非标准QPA可确保协议无需重复即可保证正确性。LCM协议的改进也进一步优化了基于LCM计算的私有集合并集协议。最终，借鉴PSU协议的思想，我们通过将私有集合交集（PSI）问题转化为GCD计算问题，构建了量子多方私有集合交集（PSI）协议。性能分析表明，该协议的正确性及在半诚实模型下的无条件安全性可直接由子程序协议（LCM和PSU协议）的正确性与安全性保证。此外，所提出协议的复杂度在秘密输入规模与参与方数量上均呈多项式级增长。