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难问题，例如RSA中的整数分解和Diffie-Hellman中的离散对数。这些算法在面对容错量子计算机时将会变得脆弱。同时，量子比特的不确定性原理（例如基于量子不确定性原理的量子密钥分发或QKD）能够提供信息理论安全性。从经典计算到量子计算的范式转变中，一个有趣的启示在于：用于经典密码学的NP难性可能会转变为用于量子密码学的广义不确定性原理，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的启示。我们识别出三种能提供量子安全的广义不确定性原理：置换门之间的非对易性、相干态位移与相位算符之间的非对易性，以及后量子密码学中一般线性代数下的模丢番图方程问题。