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Quantum algorithm to find invariant linear structure of MD hash functions

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Abstract

In this paper, we consider a special problem. “Given a function \(f\): \(\{0, 1\}^{n}\rightarrow \{0, 1\}^{m}\). Suppose there exists a n-bit string \(\alpha \in \{0, 1\}^{n}\) subject to \(f(x\oplus \alpha )=f(x)\) for \(\forall x\in \{0, 1\}^{n}\). We only know the Hamming weight \(W(\alpha )=1\), and find this \(\alpha \).” We present a quantum algorithm with “Oracle” to solve this problem. The successful probability of the quantum algorithm is \((\frac{2^{l}-1}{2^{l}})^{n-1}\), and the time complexity of the quantum algorithm is \(O(\log (n-1))\) for the given Hamming weight \(W(\alpha )=1\). As an application, we present a quantum algorithm to decide whether there exists such an invariant linear structure of the \(MD\) hash function family as a kind of collision. Then, we provide some consumptions of the quantum algorithms using the time–space trade-off.

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Acknowledgments

WanQing Wu: Supported by the Fundamental Research Funds for the Central Universities (No. 2012211020213). HuanGuo Zhang: Supported by the Major Research Plan of the National Natural Science Foundation of China (No. 91018008), the National Natural Science Foundation of China (No. 61303212, 61202386), and Major State Basic Research Development Program of china (No. 2014CB340600). E-mail: liss@whu.edu.cn

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  1. Computer School of Wuhan University, Wuhan, 430072, People’s Republic of China

    WanQing Wu, HuanGuo Zhang, ShaoWu Mao & HouZhen Wang

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  1. WanQing Wu
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  2. HuanGuo Zhang
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Correspondence to WanQing Wu.

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Wu, W., Zhang, H., Mao, S. et al. Quantum algorithm to find invariant linear structure of MD hash functions. Quantum Inf Process 14, 813–829 (2015). https://doi.org/10.1007/s11128-014-0909-5

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  • DOI: https://doi.org/10.1007/s11128-014-0909-5

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