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Commun."],"published-print":{"date-parts":[[2020,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider <jats:italic>n<\/jats:italic>-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence for all <jats:italic>n<\/jats:italic> =\u20093 and <jats:italic>n<\/jats:italic> \u2265\u20095 based on a construction in Alsalami (Cryptogr. Commun. <jats:bold>10<\/jats:bold>(4): 611\u2013628, 2018). In this context, we also show that 4-uniform 2-1 functions obtained from <jats:italic>admissible sequences<\/jats:italic>, as defined by Idrisova in (Cryptogr. Commun. <jats:bold>11<\/jats:bold>(1): 21\u201339, 2019), exist in every dimension <jats:italic>n<\/jats:italic> =\u20093 and <jats:italic>n<\/jats:italic> \u2265\u20095. Such functions fulfill some necessary properties for being subfunctions of APN permutations. Finally, we use the 4-uniform permutations with null nonlinearity to construct some 4-uniform 2-1 functions from <jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathbb {F}_{2}^{n}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:msubsup>\n<mml:mrow>\n<mml:mi>F<\/mml:mi>\n<\/mml:mrow>\n<mml:mrow>\n<mml:mn>2<\/mml:mn>\n<\/mml:mrow>\n<mml:mrow>\n<mml:mi>n<\/mml:mi>\n<\/mml:mrow>\n<\/mml:msubsup>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> to <jats:inline-formula><jats:alternatives><jats:tex-math>$\\mathbb {F}_{2}^{n-1}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<mml:msubsup>\n<mml:mrow>\n<mml:mi>F<\/mml:mi>\n<\/mml:mrow>\n<mml:mrow>\n<mml:mn>2<\/mml:mn>\n<\/mml:mrow>\n<mml:mrow>\n<mml:mi>n<\/mml:mi>\n<mml:mo>\u2212<\/mml:mo>\n<mml:mn>1<\/mml:mn>\n<\/mml:mrow>\n<\/mml:msubsup>\n<\/mml:math><\/jats:alternatives><\/jats:inline-formula> which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.<\/jats:p>","DOI":"10.1007\/s12095-020-00434-2","type":"journal-article","created":{"date-parts":[[2020,4,18]],"date-time":"2020-04-18T07:02:20Z","timestamp":1587193340000},"page":"1133-1141","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["4-uniform permutations with null nonlinearity"],"prefix":"10.1007","volume":"12","author":[{"given":"Christof","family":"Beierle","sequence":"first","affiliation":[]},{"given":"Gregor","family":"Leander","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2020,4,18]]},"reference":[{"issue":"4","key":"434_CR1","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s12095-017-0246-5","volume":"10","author":"Y Alsalami","year":"2018","unstructured":"Alsalami, Y.: Constructions with high algebraic degree of differentially 4-uniform (n, n - 1)-functions and differentially 8-uniform (n, n - 2)-functions. 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