Universal hash functions (UHFs) have been extensively used in the design of cryptographic schemes. If we consider the related-key attack against these UHF-based schemes, some of them may not be secure, especially those using the key of UHF as a part of the whole key of scheme, due to the weakness of UHF in the related-key setting. In order to solve the issue, we propose a new concept of related-key almost universal hash function, which is a natural extension to almost universal hash function in the related-key setting. We define related-key almost universal (RK-AU) hash function and related-key almost XOR universal (RK-AXU) hash function. However almost all the existing UHFs do not satisfy the new definitions. We construct fixed-input-length universal hash functions such as RH1 and variable-input-length universal hash functions such as RH2, RH3. We show that RH1 and RH2 are both RK-AXU, and RH3 is RK-AU. Furthermore, RH1, RH2 and RH3 are nearly as efficient as previous similar constructions. RK-AU (RK-AXU) hash functions can be used as components in the related-key secure cryptographic schemes. If we replace the universal hash functions in the schemes with our corresponding constructions, the problems about related-key attack can be solved. More specifically, we give four concrete applications of RK-AU and RK-AXU in related-key secure MACs and TBCs.
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