We know the classical public cryptographic algorithms are based on certain NP-hard problems such as the integer factoring in RSA and the discrete logarithm in Diffie-Hellman. They are going to be vulnerable with fault-tolerant quantum computers. We also know that the uncertainty principle for quantum bits or qubits such as quantum key distribution or QKD based on the quantum uncertainty principle offers the information theoretical security. The interesting implication with the paradigm shifts from classical computing to quantum computing is that the NP-hardness used for classical cryptography may shift to the uncertainty principles for quantum cryptography including quantum symmetric encryption, post-quantum cryptography, as well as quantum encryption in phase space for coherent optical communications. This paper would like to explore those so-called generalized uncertainty principles and explain what their implications are for quantum security. We identified three generalized uncertainty principles offering quantum security: non-commutability between permutation gates, non-commutability between the displacement and phase shift operators for coherent states, and the modular Diophantine Equation Problem in general linear algebra for post-quantum cryptography.

翻译：我们知道经典公钥密码算法基于某些NP困难问题，如RSA中的整数分解和Diffie-Hellman中的离散对数。在容错量子计算机面前，它们将变得脆弱。我们还知道，基于量子比特的不确定性原理（如量子密钥分发或QKD）提供了信息论安全性。从经典计算到量子计算的范式转变所带来的有趣含义是：用于经典密码学的NP困难性可能转化为用于量子密码学的不确定性原理，包括量子对称加密、后量子密码学，以及用于相干光通信的相空间量子加密。本文旨在探讨这些所谓的广义不确定性原理，并解释它们对量子安全性的意义。我们确定了三种提供量子安全性的广义不确定性原理：置换门之间的非对易性、相干态位移算符与相位移动算符之间的非对易性，以及后量子密码学中一般线性代数框架下的模狄奥番图方程问题。