New Bent Functions from Permutations and Linear Translators

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• 关键词：Boolean functions ; Bent functions ; Linear translators ; Permutations
• 刊名：Lecture Notes in Computer Science
• 出版年：2017
• 出版时间：2017
• 年：2017
• 卷：10194
• 期：1
• 页码：282-297
• 丛书名：Codes, Cryptology and Information Security
• ISBN：978-3-319-55589-8
• 卷排序：10194

文摘

Starting from the secondary construction originally introduced by Carlet [“On Bent and Highly Nonlinear Balanced/Resilient Functions and Their Algebraic Immunities”, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 2006], that we shall call “Carlet‘s secondary construction”, Mesnager has showed how one can construct several new primary constructions of bent functions. In particular, she has showed that three tuples of permutations over the finite field \(\mathbb {F}_{2^m}\) such that the inverse of their sum equals the sum of their inverses give rise to a construction of a bent function given with its dual. It is not quite easy to find permutations satisfying such a strong condition \((\mathcal {A}_m)\). Nevertheless, Mesnager has derived several candidates of such permutations in 2015, and showed in 2016 that in the case of involutions, the problem of construction of bent functions amounts to solve arithmetical and algebraic problems over finite fields.