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**2020/835**

**Title:**

On the Maximum Nonlinearity of De Bruijn Sequence Feedback Function

**Authors:**
Congwei Zhou, Bin Hu, Jie Guan

**Abstract:**

The nonlinearity of Boolean function is an important cryptographic criteria in the Best Affine Attack approach. In this paper, based on the definition of nonlinearity, we propose a new design index of nonlinear feedback shift registers. Using the index and the correlative necessary conditions of de Bruijn sequence feedback function, we prove that when n \ge 9, the maximum nonlinearity Nl{(f)_{\max }} of arbitrary $n - $order de Bruijn sequence feedback function f satisfies 3 \cdot {2^{n - 3}} - ({Z_n} + 1) < Nl{(f)_{\max }} \le {2^{n - 1}} - {2^{\frac{{n - 1}}{2}}} and the nonlinearity of de Bruijn sequence feedback function, based on the spanning tree of adjacency graph of affine shift registers, has a fixed value. At the same time, this paper gives the correlation analysis and practical application of the index.

**ePrint:**
https://eprint.iacr.org/2020/835

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