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Codes Cryptogr."],"published-print":{"date-parts":[[2023,1]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider image sets of differentially <jats:italic>d<\/jats:italic>-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a particularly interesting case of APN maps on binary fields <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:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for <jats:italic>n<\/jats:italic> even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map <jats:italic>f<\/jats:italic>, these results imply <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\frac{2^n+1}{3}+1 \\le |{\\text {Im}}(f)| \\le 2^n-2^{(n-1)\/2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mfrac>\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mn>2<\/mml:mn>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:msup>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mn>3<\/mml:mn>\n                    <\/mml:mfrac>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mtext>Im<\/mml:mtext>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>f<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                        <mml:mo>)<\/mml:mo>\n                        <mml:mo>\/<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-022-01094-4","type":"journal-article","created":{"date-parts":[[2022,8,9]],"date-time":"2022-08-09T12:03:19Z","timestamp":1660046599000},"page":"1-27","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Image sets of perfectly nonlinear maps"],"prefix":"10.1007","volume":"91","author":[{"given":"Lukas","family":"K\u00f6lsch","sequence":"first","affiliation":[]},{"given":"Bj\u00f6rn","family":"Kriepke","sequence":"additional","affiliation":[]},{"given":"Gohar M.","family":"Kyureghyan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,9]]},"reference":[{"key":"1094_CR1","doi-asserted-by":"crossref","unstructured":"Anbar N., Kalayci T., Meidl W.: Determining the Walsh spectra of Taniguchi\u2019s and related APN-functions. 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