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Quantum security of Grain-128/Grain-128a stream cipher against HHL algorithm

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Abstract

HHL algorithm is a quantum algorithm for solving linear equation system. It can achieve an exponential improvement over the best classical algorithm. In this paper, we analyze the quantum security of Grain-128/Grain-128a stream cipher by using the HHL algorithm. Our algorithm is based on Chen and Gao’s research on solving nonlinear equation system in Chen et al. (Quantum algorithm for optimization and polynomial system solving over finite field and application to cryptanalysis, 2018. arXiv:1802.03856) and Chen et al. (Quantum algorithms for Boolean equation solving and quantum algebraic attack on cryptosystems, 2017. arXiv:1712.06239). Firstly, we build a nonlinear Boolean equation system by choosing any keystream. Then, the nonlinear equation system is transformed into a special linear equation system that can be solved with the HHL algorithm. Finally, we solve the system by the HHL quantum algorithm. Our attack requires \( N > {2^8} \)-bit keystream, and the complexity is \(O(2^{21} N^{3.5} \kappa ^2 e^\epsilon /\epsilon ^{0.5})\) for Grain-128, and \(O(2^{21.5} N^{3.5} \kappa ^2 e^\epsilon /\epsilon ^{0.5})\) for Grain-128a where \(\kappa \) is the condition number of the matrix of the corresponding linear systems and \(\epsilon \) is a given error bound. Then we give a toy example of Grain family to estimate \(\kappa \) and briefly analyze the security of Grain-128/Grain-128a against HHL algorithm.

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Acknowledgements

This work is supported in part by the Key Research and Development Program of Shaanxi (No. 2021ZDLGY06-04), the Natural Science Foundation of China (No. 61303217, 61502372), Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS201802).

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  1. School of Telecommunications Engineering, Xidian University, P.O. 106, No. 2 South Taibai Road, Xi’an, Shaanxi 710071, China

    Weijie Liu & Juntao Gao

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  1. Weijie Liu
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  2. Juntao Gao
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Correspondence to Juntao Gao.

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Liu, W., Gao, J. Quantum security of Grain-128/Grain-128a stream cipher against HHL algorithm. Quantum Inf Process 20, 343 (2021). https://doi.org/10.1007/s11128-021-03275-x

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