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A construction of bent functions with optimal algebraic degree and large symmetric group

This work was supported by National Science Foundation of China (Grant No. 61672330 and 61602287) and the State Scholarship Fund no.201808370069 from China Scholarship Council.

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  • As maximal, nonlinear Boolean functions, bent functions have many theoretical and practical applications in combinatorics, coding theory, and cryptography. In this paper, we present a construction of bent function $ f_{a,S} $ with $ n = 2m $ variables for any nonzero vector $ a\in \mathbb{F}_{2}^{m} $ and subset $ S $ of $ \mathbb{F}_{2}^{m} $ satisfying $ a+S = S $. We give a simple expression of the dual bent function of $ f_{a,S} $ and prove that $ f_{a,S} $ has optimal algebraic degree $ m $ if and only if $ |S|\equiv 2 (\bmod 4)  $. This construction provides a series of bent functions with optimal algebraic degree and large symmetric group if $ a $ and $ S $ are chosen properly. We also give some examples of those bent functions $ f_{a,S} $ and their dual bent functions.

    Mathematics Subject Classification: 03G05, 06E25.

    Citation:

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