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More accurate results on the provable security of AES against impossible differential cryptanalysis

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Abstract

Whether there exist longer impossible differentials than existing ones for a block cipher, is an important problem in the provable security evaluation of a block cipher against impossible differential cryptanalysis. In this paper, we give more accurate results for this problem for the AES. After investigating the differential properties of both the S-box and the linear layer of AES, we theoretically prove that there do not exist impossible concrete differentials longer than 4 rounds for AES by proving that any concrete differential is possible for the 5-round AES, under the only assumption that the round keys are independent and uniformly random. We use a tool, called “(w, d)-Dependent Tree (DT)”, to show how any concrete differential \(\varDelta X \rightarrow \varDelta Z\) can be connected in the middle of the 5-round AES by some DTs. Our method might shed some light on bounding the length of impossible differentials with the differential properties of the S-boxes considered for some SPN block ciphers.

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Notes

  1. Note that we only need to consider (truncated) differentials that are non-trivial and we will take this as a default setting.

  2. Note that for any fixed-key cipher, it always has impossible differentials for arbitrary rounds.

  3. Here, (*) represents the nonzero byte.

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Acknowledgements

We would like to thank anonymous reviewers for their comments on this paper, which greatly simplify the proof of the Lemma 1 and improve this paper, as well as give us some guidance on the impact of the key schedule.

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  1. Information Science and Technology Institute, Zhengzhou, China

    Qian Wang & Chenhui Jin

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  1. Qian Wang
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Correspondence to Qian Wang.

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Communicated by C. Carlet.

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This work was supported by National Natural Science Foundation of China (Grant Nos. 61272488, 61402523, 61772547, 61802438 and 61602514).

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Wang, Q., Jin, C. More accurate results on the provable security of AES against impossible differential cryptanalysis. Des. Codes Cryptogr. 87, 3001–3018 (2019). https://doi.org/10.1007/s10623-019-00660-7

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