[论文解读] Stronger impossibility results for quantum string commitment

Rahul Jain|arXiv (Cornell University)|May 31, 2005

Quantum Computing Algorithms and Architecture参考文献 11被引用 6

一句话总结

本文通过利用子态定理，为量子字符串承诺（QSC）协议建立了更强的绑定-隐藏权衡，改进了Buhrman等人先前的结果。该分析适用于单次执行和并行执行两种情形，表明权衡边界更紧，并在各类情形下具有普遍适用性。

ABSTRACT

String commitment schemes are similar to the well studied bit commitment schemes in cryptography with the difference that the committing party, say Alice is supposed to commit a long string instead of a single bit, to another party say Bob. Similar to bit commitment schemes, such schemes are supposed to be binding, i.e Alice cannot change her choice after committing and concealing i.e. Bob cannot find Alice’s committed string before Alice reveals it. Strong impossibility results are known for bit commitment schemes both in the classical and quantum settings, for example due to Mayer [13] and Lo and Chau [11, 12]. In fact for approximate quantum bit commitment schemes, trade-offs between the degrees of cheating of Alice and Bob, referred to as binding-concealing trade-offs are known as well for example due to Spekkens and Terry [15]. Recently, Buhrman, Christandl, Hayden, Lo and Wehner [1] have shown similar bindingconcealing trade-offs for quantum string commitment schemes (QSC), both in the scenario of single execution of the protocol and in the asymptotic regime of sufficiently large number of parallel executions of the protocol. We show stronger trade-off in the scenario of single execution of a QSC protocol which also immediately imply the trade-off shown by Buhrman et al. in the asymptotic regime of multiple parallel executions of a QSC protocol. We show our results by making a central use of an important information theoretic tool called the substate theorem due to Jain, Radhakrishnan and Sen [6]. Our techniques are quite different from that of [1] and may be of independent interest.

研究动机与目标

• 在先前界限的基础上，进一步强化量子字符串承诺（QSC）协议的不可能性结果。
• 弥合单次执行与渐近并行执行情形下QSC安全权衡中的差距。
• 提出一种新颖的信息理论方法，利用子态定理推导出更紧的边界。
• 证明新权衡关系可推出先前已知的渐近结果，统一分析框架。

提出的方法

• 以Jain、Radhakrishnan和Sen提出的子态定理作为核心信息理论工具。
• 将子态定理应用于分析QSC协议中绑定性与隐藏性之间的关系。
• 推导出单次执行中Alice（绑定）与Bob（隐藏）欺骗概率的更紧边界。
• 通过直接推论，将单次执行的权衡关系扩展至并行执行的渐近情形。
• 采用量子信息理论技术，避免依赖先前工作（如Buhrman等人）所使用的方法。
• 建立一个可能对分析其他量子承诺方案亦具独立价值的框架。

实验结果

研究问题

• RQ1 在单次执行情形下，能否为量子字符串承诺导出更强的绑定-隐藏权衡？
• RQ2 新权衡边界是否蕴含先前为并行执行所建立的渐近权衡？
• RQ3 子态定理能否有效应用于强化量子承诺方案中的不可能性结果？
• RQ4 所提出的方法是否独立于且优于先前用于QSC的分析技术？
• RQ5 与Buhrman等人结果相比，权衡边界的定量改进程度如何？

主要发现

• 本文推导出的单次执行量子字符串承诺协议的绑定-隐藏权衡，强于此前已知的结果。
• 所推导的权衡关系蕴含了Buhrman等人针对多次并行执行所建立的渐近权衡边界。
• 子态定理的应用使得对Alice与Bob欺骗概率的边界更加紧密。
• 结果通过一种与Buhrman等人方法根本不同的新颖技术获得。
• 该方法提供了一个统一框架，适用于单次执行与并行执行两种情形。
• 研究结果强化了量子字符串承诺的不可能性结果，进一步凸显了此类量子协议在此场景下的局限性。