This paper studies the limitations of the generic approaches to solving cryptographic problems in classical and quantum settings in various models.
- In the classical generic group model (GGM), we find simple alternative proofs for the lower bounds of variants of the discrete logarithm (DL) problem: the multiple-instance DL and one-more DL problems (and their mixture). We also re-prove the unknown-order GGM lower bounds, such as the order finding, root extraction, and repeated squaring.
- In the quantum generic group model (QGGM), we study the complexity of variants of the discrete logarithm. We prove the logarithm DL lower bound in the QGGM even for the composite order setting. We also prove an asymptotically tight lower bound for the multiple-instance DL problem. Both results resolve the open problems suggested in a recent work by Hhan, Yamakawa, and Yun.
- In the quantum generic ring model we newly suggested, we give the logarithmic lower bound for the order-finding algorithms, an important step for Shor's algorithm. We also give a logarithmic lower bound for a certain generic factoring algorithm outputting relatively small integers, which includes a modified version of Regev's algorithm.
- Finally, we prove a lower bound for the basic index calculus method for solving the DL problem in a new idealized group model regarding smooth numbers.
The quantum lower bounds in both models allow certain (different) types of classical preprocessing. All of the proofs are significantly simpler than the previous proofs and are through a single tool, the so-called compression lemma, along with linear algebra tools. Our use of this lemma may be of independent interest.
Protecting Quantum Computers against Untrusted Users
Joint Center for Quantum Information and Computer Science, University of Maryland and NIST（马里兰大学与美国国家标准与技术研究院联合量子信息与计算中心）；Electrical & Computer Engineering Department, The University of California, Los Angeles（加州大学洛杉矶分校电气与计算机工程系）；DIENS, École Normale Supérieure, PSL University, CNRS, INRIA（巴黎高等师范学院、PSL大学、法国国家科学研究中心、法国国家信息与自动化研究所联合信息系）；QuEra Computing Inc.（QuEra计算公司）；QuEra Computing UK Ltd.（QuEra计算英国有限公司）
在此之前，Bennet和Feynman问海森堡的不确定原理是否对量子计算机有限制（Quantum Mechanical Computers，Richard P. Feynman，Foundations of Physics，Vol. 16，No. 6，p597-531，1986）。费曼的回答是否定的。在本文中，我们将重新讨论离散时间傅里叶变换的不确定原理的相同问题。我们将表明，离散时间傅立叶变换不确定原理起着根本性的作用，表明肖尔的量子算法的类型具有有效的运行时间，并得出结论，离散时间不确定原理是一个援助，在我们目前的制定和理解肖尔的量子算法的类型。事实证明，对于这些算法，测量某个集合$T$中的元素的概率（在算法结束时）可以用有限傅立叶分析的时间限制和频带限制算子来表示。与这些算子相联系的是有限傅里叶变换不确定原理。不确定原理为上述概率提供了一个下限。我们将推导出这些类型的概率的一般下界。我们将这些下界称为量子算法不确定原理或QAUP。QAUP之所以重要，是因为它们让我们对衡量某些可取之处的概率有了一些了解。我们将使用这些下界来推导Shor的因子分解和离散对数算法。
Minimizing CNOT-count in quantum circuit of the extended Shor's algorithm for ECDLP
Institute of Information Engineering, CAS（中国科学院信息工程研究所）；School of Cyber Security, University of Chinese Academy of Sciences（中国科学院大学网络安全学院）；Institute of Information Engineering, Chinese Academy of Sciences（中国科学院信息工程研究所）
