SCIENCE CHINA Information Sciences, Volume 61, Issue 10: 102501(2018) https://doi.org/10.1007/s11432-017-9468-y

## Quantum key-recovery attack on Feistel structures

• AcceptedMay 8, 2018
• PublishedAug 31, 2018
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### Abstract

Post-quantum cryptography has drawn considerable attention from cryptologists on a global scale. At Asiacrypt 2017, Leander and May combined Grover's and Simon's quantum algorithms to break the FX-based block ciphers, which were introduced by Kilian and Rogaway to strengthen DES. In this study, we investigate the Feistel constructions using Grover's and Simon's algorithms to generate new quantum key-recovery attacks on different rounds of Feistel constructions. Our attacksrequire $2^{0.25nr-0.75n}$ quantum queries to break an $r$-round Feistel construction.The time complexity of our attacks is less than that observed for quantum brute-force search by a factor of $2^{0.75n}$. When compared with the best classical attacks, i.e., Dinur et al.'s attacks at CRYPTO 2015, the time complexity is reduced by a factor of $2^{0.5n}$ without incurring any memory cost.

### Acknowledgment

This work was supported by National Key Research and Development Program of China (Grant No. 2017YFA0303903), National Natural Science Foundation of China (Grant No. 61672019), Fundamental Research Funds of Shandong University (Grant No. 2016JC029), National Cryptography Development Fund (Grant No. MMJJ20170121), Zhejiang Province Key RD Project (Grant No. 2017C01062), and Project Funded by China Postdoctoral Science Foundation (Grant No. 2017M620807).

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• Figure 1

The $i$th round of the Feistel structure.

• Figure 2

FX constructions.

• Figure 3

Quantum key-recovery attack on 5-round Feistel structures.

• Table 1   Summary of the key-recovery attacks on Feistel schemes in classical and qCPA settings
 Classical setting qCPA setting Dinur et al. [14] Trivial bound Ours Rounds Time Memory Time Time 5 $2^{n}$ $2^{0.5n}$ $2^{1.25n}$ $2^{0.5n}$ 7 $2^{1.5n}$ $2^{n}$ $2^{1.75n}$ $2^{n}$ 8 $2^{1.75n}$ $2^{1.25n}$ $2^{2n}$ $2^{1.25n}$ 15 $2^{3.5n}$ $2^{2n}$ $2^{3.75n}$ $2^{3n}$ 31 $2^{7.5n}$ $2^{4n}$ $2^{7.75n}$ $2^{7n}$ 32 $2^{7.75n}$ $2^{7.25n}$ $2^{8n}$ $2^{7.25n}$

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