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首页> 外文期刊>IEEE Transactions on Information Theory >Constructing Differentially 4-Uniform Permutations Over ${BBF}_{2^{2k}}$ via the Switching Method
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Constructing Differentially 4-Uniform Permutations Over ${BBF}_{2^{2k}}$ via the Switching Method

机译：通过切换方法在$ {BBF} _ {2 ^ {2k}} $上构造差分4均匀置换

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摘要

Many block ciphers use permutations defined on ${BBF}_{2^{2k}}$ with low differential uniformity, high nonlinearity, and high algebraic degree as their S-boxes to provide confusion. It is well known that, for a function on ${BBF}_{2^{n}}$ , the lowest differential uniformity is 2 and the functions achieving this lower bound are called almost perfect nonlinear (APN) functions. However, due to the lack of knowledge on APN permutations on ${BBF}_{2^{2k}}$, differentially 4-uniform permutations are usually chosen as S-boxes. For example, the currently endorsed Advanced Encryption Standard chooses one such function, the multiplicative inverse function, as its S-box. By a recent survey on differentially 4-uniform permutations over ${BBF}_{2^{2k}}$, there are only five known infinite families of such functions, and most of them have small algebraic degrees. In this paper, we apply the powerful switching method to discover many CCZ-inequivalent infinite families of such functions on ${BBF}_{2^{2k}}$ with optimal algebraic degree, where $k$ is an arbitrary positive integer. This greatly expands the list of differentially 4-uniform permutations and hence provide more choices for the S-boxes. Furthermore, lower bounds for the nonlinearity of the functions obtained in this paper are presented and they imply that some infinite families have high nonlinearity.
机译：许多分组密码使用在 $ {BBF} _ {2 ^ {2k}} $ 上定义的置换，且差分均匀性低，非线性和高代数度为其S盒提供混淆。众所周知，对于 $ {BBF} _ {2 ^ {n}} $ 上的函数，最低微分均匀性为2，达到此下界的函数称为近乎完美非线性（APN）函数。但是，由于缺乏有关 $ {BBF} _ {2 ^ {2k}} $ 的APN排列的知识，通常选择差分4均匀排列作为S盒。例如，当前认可的高级加密标准选择一种这样的函数，即乘法逆函数，作为其S盒。根据最近对 $ {BBF} _ {2 ^ {2k}} $ 的4均匀排列的调查，这类函数只有五个已知的无限族，并且大多数具有小代数度。在本文中，我们应用了功能强大的切换方法，在 $ {BBF} _ {2 ^ {2k}}上发现了许多CCZ等价的此类函数的无限族。具有最佳代数程度的$ ，其中 $ k $ 是任意正整数。这极大地扩展了差分4均匀排列的列表，因此为S盒提供了更多选择。此外，给出了本文获得的函数非线性的下界，这表明某些无限族具有很高的非线性。

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