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Implicit Quadratic Property of Differentially 4-Uniform Permutations

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Progress in Cryptology – INDOCRYPT 2016 (INDOCRYPT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10095))

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Abstract

Substitution box (S-box) is an important component of block ciphers for providing nonlinearity. It is often constructed from differentially 4-uniform permutation. In this paper, we examine all (to the best of our knowledge) the differentially 4-uniform permutations that are known in the literature and determine whether they are implicitly quadratic. We found that all of them are implicitly quadratic, making them vulnerable to algebraic attack [10, 12–14]. This leads to an open question of whether there exists a differentially 4-uniform permutation over \(\mathbb {F}_{2^n}\) that is not implicitly quadratic. We provide a partial answer to this question by solving it for the special cases of \(n=11\) and \(n=13\).

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Notes

  1. 1.

    The differential spectrum [a, b, c] represents the multi-set in which 0 appears a times, 2 appears b times and 4 appears c times.

  2. 2.

    We have tested the Li-Wang functions [17] in Magma for \(n \le 12\). The computation shows that all of them are implicitly quadratic. This suggests that it is likely that all the Li-Wang functions are implicitly quadratic.

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Authors and Affiliations

  1. Temasek Laboratories, National University of Singapore, 5A Engineering Drive 1, #09-02, Singapore, 117411, Singapore

    Theo Fanuela Prabowo & Chik How Tan

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  1. Theo Fanuela Prabowo
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  2. Chik How Tan
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Correspondence to Theo Fanuela Prabowo .

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Editors and Affiliations

  1. University of Haifa , Haifa, Israel

    Orr Dunkelman

  2. Indraprashtha Institute of Information Technology (IIIT-D), New Delhi, India

    Somitra Kumar Sanadhya

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Prabowo, T.F., Tan, C.H. (2016). Implicit Quadratic Property of Differentially 4-Uniform Permutations. In: Dunkelman, O., Sanadhya, S. (eds) Progress in Cryptology – INDOCRYPT 2016. INDOCRYPT 2016. Lecture Notes in Computer Science(), vol 10095. Springer, Cham. https://doi.org/10.1007/978-3-319-49890-4_20

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  • DOI: https://doi.org/10.1007/978-3-319-49890-4_20

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