{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T00:30:27Z","timestamp":1764981027245,"version":"3.46.0"},"reference-count":13,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,9]]},"abstract":"Abstract<\/jats:title>\n \n Transparency order (\n \n \n \n \n TO<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ) is one of the indicators used to measure the resistance of\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,m)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -function to differential power analysis. At present, there are three definitions:\n \n \n \n \n TO<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , redefined transparency order (\n \n \n \n \n \u211bTO<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal R} TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ), and modified transparency order (\n \n \n \n \n \u2133TO<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal M} TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ). For the first time, we give one necessary and sufficient condition for\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,m)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -function reaching\n \n \n \n \n TO<\/m:mi>\n =<\/m:mo>\n m<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}=m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and completely characterize\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,m)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -functions reaching\n \n \n \n \n TO<\/m:mi>\n =<\/m:mo>\n m<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}=m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for any\n \n \n \n \n n<\/m:mi>\n <\/m:math>\n n<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n \n \n m<\/m:mi>\n <\/m:math>\n m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . We find that any\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,1)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -function cannot reach\n \n \n \n \n TO<\/m:mi>\n =<\/m:mo>\n m<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}=m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n for odd\n \n \n \n \n n<\/m:mi>\n <\/m:math>\n n<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Based on the matrix product, the necessary conditions for\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,m)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -function reaching\n \n \n \n \n \u2133TO<\/m:mi>\n =<\/m:mo>\n m<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal M} TO}}=m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n or\n \n \n \n \n \u211bTO<\/m:mi>\n =<\/m:mo>\n m<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal R} TO}}=m<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are given, respectively. Finally, it is proved that any balanced\n \n \n \n \n \n (<\/m:mo>\n \n n<\/m:mi>\n ,<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \\left(n,m)<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -function cannot reach the upper bound on\n \n \n \n \n TO<\/m:mi>\n <\/m:math>\n {\\mathcal{TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n (or\n \n \n \n \n \u211bTO<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal R} TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ,\n \n \n \n \n \u2133TO<\/m:mi>\n <\/m:math>\n {\\mathcal{ {\\mathcal M} TO}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n ).\n <\/jats:p>","DOI":"10.1515\/jmc-2023-0040","type":"journal-article","created":{"date-parts":[[2024,7,9]],"date-time":"2024-07-09T11:23:20Z","timestamp":1720524200000},"source":"Crossref","is-referenced-by-count":0,"title":["Characterizing the upper bound on the transparency order of (\n n, m<\/i>\n )-functions"],"prefix":"10.1515","volume":"18","author":[{"given":"Yu","family":"Zhou","sequence":"first","affiliation":[{"name":"National Key Laboratory of Security Communication , Chengdu 610041 , China"}]},{"given":"You","family":"Wu","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Security Communication , Chengdu 610041 , China"}]},{"given":"Bing","family":"Shen","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Security Communication , Chengdu 610041 , China"}]},{"given":"Jinbo","family":"Wang","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Security Communication , Chengdu 610041 , China"}]},{"given":"Rong","family":"Cheng","sequence":"additional","affiliation":[{"name":"National Key Laboratory of Security Communication , Chengdu 610041 , China"}]}],"member":"374","published-online":{"date-parts":[[2024,7,9]]},"reference":[{"key":"2025120600251220786_j_jmc-2023-0040_ref_001","doi-asserted-by":"crossref","unstructured":"Kocher P, Jaffe J, Jun B. 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