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In this paper we give the distributions of Walsh spectrum and the distributions of autocorrelation functions for (\n                    <jats:italic>n<\/jats:italic>\n                    + 1)-bit S-boxes in [12]. We obtain the nonlinearity of (\n                    <jats:italic>n<\/jats:italic>\n                    + 1)-bit S-boxes, and one necessary and sufficient conditions of (\n                    <jats:italic>n<\/jats:italic>\n                    + 1)-bit S-boxes satisfying\n                    <jats:italic>m<\/jats:italic>\n                    -order resilient. Meanwhile, we also give one characterization of (\n                    <jats:italic>n<\/jats:italic>\n                    + 1)-bit S-boxes satisfying\n                    <jats:italic>t<\/jats:italic>\n                    -order propagation criterion. Finally, we give one relationship of the sum-of-squares indicators between an\n                    <jats:italic>n<\/jats:italic>\n                    -bit S-box\n                    <jats:italic>S<\/jats:italic>\n                    <jats:sub>0<\/jats:sub>\n                    and the (\n                    <jats:italic>n<\/jats:italic>\n                    + 1)-bit S-box\n                    <jats:italic>S<\/jats:italic>\n                    (which is constructed by\n                    <jats:italic>S<\/jats:italic>\n                    <jats:sub>0<\/jats:sub>\n                    ).\n                  <\/jats:p>","DOI":"10.1515\/jmc-2020-0004","type":"journal-article","created":{"date-parts":[[2020,12,14]],"date-time":"2020-12-14T04:56:17Z","timestamp":1607921777000},"page":"258-265","source":"Crossref","is-referenced-by-count":1,"title":["On cryptographic properties of (\n                    <i>n<\/i>\n                    + 1)-bit S-boxes constructed by known\n                    <i>n<\/i>\n                    -bit S-boxes"],"prefix":"10.1515","volume":"15","author":[{"given":"Yu","family":"Zhou","sequence":"first","affiliation":[{"name":"Science and Technology on Communication Security Laboratory , Chengdu , 610041 , China"}]},{"given":"Daoguang","family":"Mu","sequence":"additional","affiliation":[{"name":"Science and Technology on Communication Security Laboratory , Chengdu , 610041 , China"}]},{"given":"Xinfeng","family":"Dong","sequence":"additional","affiliation":[{"name":"Science and Technology on Communication Security Laboratory , Chengdu , 610041 , China"},{"name":"Center for Cyber Security, School of Computer Science and Engineering , University of Electronic Science and Technology of China , Chengdu , 611731 , China"}]}],"member":"374","published-online":{"date-parts":[[2020,12,8]]},"reference":[{"key":"2025120600333848549_j_jmc-2020-0004_ref_001","doi-asserted-by":"crossref","unstructured":"Y. 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