{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,20]],"date-time":"2025-10-20T18:56:10Z","timestamp":1760986570135,"version":"build-2065373602"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,8,23]],"date-time":"2025-08-23T00:00:00Z","timestamp":1755907200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,23]],"date-time":"2025-08-23T00:00:00Z","timestamp":1755907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"University of Bergen"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Cryptogr. Commun."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>When permutation polynomials over finite fields are used as core components of cryptographic algorithms, one step in reducing the hardware area of their secure implementation is to represent them as a composition of permutation polynomials of lower algebraic degree. In this work, we present a criterion for the existence of a class of decompositions of the inverse power function. We use this criterion to show the existence of such decompositions in the finite fields with <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{2}^{\\varvec{n}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\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:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{2}^{\\varvec{2n}}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> elements when <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{2}^{\\varvec{n}} \\varvec{-1}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\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:msup>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is prime. We further present a search algorithm based on the criterion that can produce novel decompositions for some <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{n}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> between <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{32}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>32<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\varvec{500}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>500<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> and improve the length of some known decompositions. Finally, we discuss how to further reduce the number of power functions in the decomposition.<\/jats:p>","DOI":"10.1007\/s12095-025-00820-8","type":"journal-article","created":{"date-parts":[[2025,8,23]],"date-time":"2025-08-23T11:12:45Z","timestamp":1755947565000},"page":"1721-1737","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Further existence results of decompositions of permutation polynomials"],"prefix":"10.1007","volume":"17","author":[{"given":"Samuele","family":"Andreoli","sequence":"first","affiliation":[]},{"given":"George","family":"Petrides","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,23]]},"reference":[{"key":"820_CR1","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1007\/11935308_38","volume-title":"Information and Communications Security","author":"S Nikova","year":"2006","unstructured":"Nikova, S., Rechberger, C., Rijmen, V.: Threshold implementations against side-channel attacks and glitches. 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