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Codes Cryptogr."],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>In Pasalic et al. (IEEE Trans Inf Theory 69:2702\u20132712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235\u2013249, 2018), two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {F}}_2^n$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msubsup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> to the cyclic group <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {Z}}_{2^k}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. It is shown how to obtain nega-<jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {Z}}_{2^k}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-bent functions from <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {Z}}_{2^k}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {Z}}_8$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:mn>8<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-bent functions employing permutations with the <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$({\\mathcal {A}}_m)$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>m<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> property, and more generally we show that the inverse permutation gives rise to <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {Z}}_{2^k}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>Z<\/mml:mi>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>-bent functions.<\/jats:p>","DOI":"10.1007\/s10623-024-01454-2","type":"journal-article","created":{"date-parts":[[2024,7,5]],"date-time":"2024-07-05T16:02:02Z","timestamp":1720195322000},"page":"899-921","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Vectorial negabent concepts: similarities, differences, and generalizations"],"prefix":"10.1007","volume":"93","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4600-5088","authenticated-orcid":false,"given":"Nurdag\u00fcl","family":"Anbar","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sadmir","family":"Kudin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wilfried","family":"Meidl","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6343-8796","authenticated-orcid":false,"given":"Enes","family":"Pasalic","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0461-4793","authenticated-orcid":false,"given":"Alexandr","family":"Polujan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,7,5]]},"reference":[{"key":"1454_CR1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2024.3359260","author":"S Alkan","year":"2024","unstructured":"Alkan S., Anbar N., Kalayc\u0131 T., Meidl W.: Bent partitions and LP-packings. 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