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Codes Cryptogr."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In the independent works by Kalgin and Idrisova and by Beierle, Leander and Perrin, it was observed that the Gold APN functions over <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathbb {F}_{2^5}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mn>5<\/mml:mn>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> give rise to a quadratic APN function in dimension 6 having maximum possible linearity of <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^5$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (that is, minimum possible nonlinearity <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^4$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mn>4<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). In this article, we show that the case of <jats:inline-formula><jats:alternatives><jats:tex-math>$$n \\le 5$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is quite special in the sense that Gold APN functions in dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$n&gt;5$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>&gt;<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> cannot be extended to quadratic APN functions in dimension <jats:inline-formula><jats:alternatives><jats:tex-math>$$n+1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> having maximum possible linearity. In the second part of this work, we show that this is also the case for APN functions of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$x \\mapsto x^3 + \\mu (x)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\u21a6<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>\u03bc<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mu $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03bc<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> being a quadratic Boolean function.<\/jats:p>","DOI":"10.1007\/s10623-022-01111-6","type":"journal-article","created":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T12:02:47Z","timestamp":1664366567000},"page":"433-449","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Gold functions and switched cube functions are not 0-extendable in dimension n &gt; 5"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5558-0722","authenticated-orcid":false,"given":"Christof","family":"Beierle","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6118-7927","authenticated-orcid":false,"given":"Claude","family":"Carlet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,28]]},"reference":[{"issue":"1","key":"1111_CR1","doi-asserted-by":"publisher","first-page":"670","DOI":"10.1109\/TIT.2021.3120698","volume":"68","author":"C Beierle","year":"2022","unstructured":"Beierle C., Leander G.: New instances of quadratic APN functions. 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