In typical well-known cryptosystem, the hardness of classical problems plays a fundamental role in ensuring its security. While, with the booming of quantum computation, some classical hard problems tend to be vulnerable when confronted with the already-known quantum attacks, as a result, it is necessary to develop the post-quantum cryptosystem to resist the quantum attacks. With the purpose to bridge the two disciplines, it is significant to summarize known quantum algorithms and their threats toward these cryptographic intractable problems from a perspective of cryptanalysis. In this paper, we discussed the designing methodology, algorithm framework and latest progress of the mathematic hard problems on which the typical cryptosystems depend, including integer factorization problem, discrete logarithmic problem and its variants, lattice problem, dihedral hidden subgroup problems and extrapolated dihedral coset problem. It illustrated the reason why some cryptosystems such as RSA and ECC are not resistant to quantum attacks, yet some of them like lattice cryptosystems remain intact facing quantum attacks.

中文翻译：

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用于典型难题的量子算法：密码分析的角度

在典型的众所周知的密码系统中，经典问题的严格性在确保其安全性方面起着根本性的作用。同时，随着量子计算的蓬勃发展，一些经典的难题在面对已知的量子攻击时往往容易受到攻击，因此，有必要开发一种后量子密码系统来抵抗量子攻击。为了桥接这两个学科，从密码分析的角度总结已知的量子算法及其对这些密码难解决问题的威胁具有重要意义。在本文中，我们讨论了典型密码系统所依赖的数学难题的设计方法，算法框架和最新进展，包括整数分解问题，离散对数问题及其变体，格问题，二面角隐藏子组问题和外推二面同伴问题。它说明了为什么某些密码系统（例如RSA和ECC）不能抵抗量子攻击的原因，而其中一些像格式密码系统却仍然能够面对量子攻击而完好无损。