Hash functions have always attracted a lot of attention in modern cryptography because of their hard to invert nature. However, all previous constructions of cryptographic primitives face the threat of being broken by the recent advancements in quantum technology. The focus has thus shifted to developing cryptographic primitives on mathematical structures such as lattices that are intractable by quantum algorithms. We review the computational problems defined on lattices and their respective hardness and discuss constructions of hash function families based on both integer and ideal lattices whose security depends on these computational problems on lattices. We provide a comparative analysis of the theoretical security and concrete instantiations claimed by the different hash function families. Finally, we review techniques used in the reductions for the security proofs of constructions of different hash function families.

散列函数由于难以逆转而一直在现代密码学中引起广泛关注自然。但是，所有以前的密码原语构造都面临着被量子技术的最新发展所破坏的威胁。因此，重点已转移到开发量子结构难以处理的数学结构（例如晶格）上的密码原语。我们回顾了定义在晶格上的计算问题及其各自的硬度，并讨论了基于整数和理想晶格的哈希函数族的构造，其安全性取决于晶格上的这些计算问题。我们对不同哈希函数族要求的理论安全性和具体实例进行了比较分析。最后，我们回顾了在归约中使用的技术，以用于不同哈希函数族的构造的安全性证明。