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Codes Cryptogr."],"published-print":{"date-parts":[[2022,4]]},"abstract":"Abstract<\/jats:title>In this work, we study functions that can be obtained by restricting a vectorial Boolean function$$F :\\mathbb {F}_{2}^n \\rightarrow \\mathbb {F}_{2}^n$$<\/jats:tex-math>F<\/mml:mi>:<\/mml:mo>F<\/mml:mi>2<\/mml:mn><\/mml:mrow>n<\/mml:mi><\/mml:msubsup>\u2192<\/mml:mo>F<\/mml:mi>2<\/mml:mn><\/mml:mrow>n<\/mml:mi><\/mml:msubsup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>to an affine hyperplane of dimension$$n-1$$<\/jats:tex-math>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and then projecting the output to an$$n-1$$<\/jats:tex-math>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-dimensional space. We show that a multiset of$$2 \\cdot (2^n-1)^2$$<\/jats:tex-math>2<\/mml:mn>\u00b7<\/mml:mo>(<\/mml:mo>2<\/mml:mn>n<\/mml:mi><\/mml:msup>-<\/mml:mo>1<\/mml:mn>)<\/mml:mo><\/mml:mrow>2<\/mml:mn><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on$$\\mathbb {F}_{2}^n$$<\/jats:tex-math>F<\/mml:mi>2<\/mml:mn><\/mml:mrow>n<\/mml:mi><\/mml:msubsup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Further, for all of the known quadratic APN functions in dimension$$n < 10$$<\/jats:tex-math>n<\/mml:mi><<\/mml:mo>10<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we determine the restrictions that are also APN. Moreover, we construct 6368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function$$F :\\mathbb {F}_{2}^n \\rightarrow \\mathbb {F}_{2}^n$$<\/jats:tex-math>F<\/mml:mi>:<\/mml:mo>F<\/mml:mi>2<\/mml:mn><\/mml:mrow>n<\/mml:mi><\/mml:msubsup>\u2192<\/mml:mo>F<\/mml:mi>2<\/mml:mn><\/mml:mrow>n<\/mml:mi><\/mml:msubsup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>with linearity of$$2^{n-1}$$<\/jats:tex-math>2<\/mml:mn>n<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity$$2^7$$<\/jats:tex-math>2<\/mml:mn>7<\/mml:mn><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>up to EA-equivalence.<\/jats:p>","DOI":"10.1007\/s10623-022-01024-4","type":"journal-article","created":{"date-parts":[[2022,3,11]],"date-time":"2022-03-11T12:09:01Z","timestamp":1647000541000},"page":"1009-1036","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Trims and extensions of quadratic APN functions"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5558-0722","authenticated-orcid":false,"given":"Christof","family":"Beierle","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2579-8587","authenticated-orcid":false,"given":"Gregor","family":"Leander","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4722-7005","authenticated-orcid":false,"given":"L\u00e9o","family":"Perrin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,11]]},"reference":[{"issue":"7","key":"1024_CR1","doi-asserted-by":"publisher","first-page":"4863","DOI":"10.1109\/TIT.2021.3071533","volume":"67","author":"C Beierle","year":"2021","unstructured":"Beierle C., Brinkmann M., Leander G.: Linearly self-equivalent APN permutations in small dimension. 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