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Tight Security for Key-Alternating Ciphers with Correlated Sub-keys

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Advances in Cryptology – ASIACRYPT 2021 (ASIACRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13092))

Abstract

A substantial effort has been devoted to proving optimal bounds for the security of key-alternating ciphers with independent sub-keys in the random permutation model (e.g., Chen and Steinberger, EUROCRYPT ’14; Hoang and Tessaro, CRYPTO ’16). While common in the study of multi-round constructions, the assumption that sub-keys are truly independent is not realistic, as these are generally highly correlated and generated from shorter keys.

In this paper, we show the existence of non-trivial distributions of limited independence for which a t-round key-alternating cipher achieves optimal security. Our work is a natural continuation of the work of Chen et al. (CRYPTO ’14) which considered the case of \(t = 2\) when all-subkeys are identical. Here, we show that key-alternating ciphers remain secure for a large class of \((t-1)\)-wise and \((t-2)\)-wise independent distribution of sub-keys.

Our proofs proceed by generalizations of the so-called Sum-Capture Theorem, which we prove using Fourier-analytic techniques.

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Notes

  1. 1.

    In fact, Chen and Steinberger [6] already noted that their result holds in the case where the underlying subkeys are t-wise independent. The tight concrete bound proved by Hoang and Tessaro [16] also extends to t-wise independent setting.

  2. 2.

    Actually, Chen et al. [5] also addressed reducing the number of keys and permutations in parallel. They showed that a 2-round KAC is secure against \(q = \varOmega (N^{2/3})\) queries when instantiated by a single permutation and a single key with a key schedule built over a linear orthomorphism.

  3. 3.

    In fact, the Frobenius norm and 2-norm can have up to \(\sqrt{N}\) multiplicative gap for \(N\times N\) matrix (e.g. the identity matrix), and we believe that a large gap exists in our Frobenius norm bound. However, to get a better 2-norm bound, it requires a much better understanding to our defined matrix for analyzing (2) than we do.

  4. 4.

    Since we are taking the natural field interpretation over \(\{0,1\}^n\), in which the field addition is the bit-wise xor operation, we have the i-th column of \(A_c\) defined as the n-dimension vector representation of field element \(c\cdot \nu _i\), in which \(\nu _i\) is the field element that has the corresponding representation to be a basis vector with the i-th position being one and the rest positions being zero.

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Acknowledgements

We thank the anonymous reviewers for sharing many helpful suggestions that improved the paper. Stefano Tessaro and Xihu Zhang were partially supported by NSF grants CNS-1930117 (CAREER), CNS1926324, CNS-2026774, a Sloan Research Fellowship, and a JP Morgan Faculty Award.

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  1. University of Washington, Seattle, USA

    Stefano Tessaro & Xihu Zhang

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  1. Stefano Tessaro
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  2. Xihu Zhang
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Correspondence to Xihu Zhang .

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  1. NTT Corporation, Tokyo, Japan

    Mehdi Tibouchi

  2. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Tessaro, S., Zhang, X. (2021). Tight Security for Key-Alternating Ciphers with Correlated Sub-keys. In: Tibouchi, M., Wang, H. (eds) Advances in Cryptology – ASIACRYPT 2021. ASIACRYPT 2021. Lecture Notes in Computer Science(), vol 13092. Springer, Cham. https://doi.org/10.1007/978-3-030-92078-4_15

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