{"title":"利用 $$P_\\tau$ 属性设计弯曲函数,可证明其在已完成的马约拉纳-麦克法兰类之外","authors":"Enes Pasalic, Amar Bapić, Fengrong Zhang, Yongzhuang Wei","doi":"10.1007/s10623-024-01407-9","DOIUrl":null,"url":null,"abstract":"<p>In this article, we identify certain instances of bent functions, constructed using the so-called <span>\\(P_\\tau \\)</span> property, that are provably outside the completed Maiorana–McFarland (<span>\\({\\mathcal{M}\\mathcal{M}}^\\#\\)</span>) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the <span>\\(P_\\tau \\)</span> property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes <span>\\({\\mathcal{M}\\mathcal{M}}^\\#\\)</span>, <span>\\({{\\mathcal {C}}},{{\\mathcal {D}}}\\)</span> and <span>\\({{\\mathcal {D}}}_0\\)</span>, where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form <span>\\(h(y,z)=Tr(y\\pi (z)) + G_1(Tr_1^m(\\alpha _1y),\\ldots ,Tr_1^m(\\alpha _ky))G_2(Tr_1^m(\\beta _{k+1}z),\\ldots ,Tr_1^m(\\beta _{\\tau }z))+ G_3(Tr_1^m(\\alpha _1y),\\ldots ,Tr_1^m(\\alpha _ky))\\)</span> to be bent and outside <span>\\({\\mathcal{M}\\mathcal{M}}^\\#\\)</span> is specified without a strong assumption that the components of the permutation <span>\\(\\pi \\)</span> do not admit linear structures.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Using $$P_\\\\tau $$ property for designing bent functions provably outside the completed Maiorana–McFarland class\",\"authors\":\"Enes Pasalic, Amar Bapić, Fengrong Zhang, Yongzhuang Wei\",\"doi\":\"10.1007/s10623-024-01407-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>In this article, we identify certain instances of bent functions, constructed using the so-called <span>\\\\(P_\\\\tau \\\\)</span> property, that are provably outside the completed Maiorana–McFarland (<span>\\\\({\\\\mathcal{M}\\\\mathcal{M}}^\\\\#\\\\)</span>) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the <span>\\\\(P_\\\\tau \\\\)</span> property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes <span>\\\\({\\\\mathcal{M}\\\\mathcal{M}}^\\\\#\\\\)</span>, <span>\\\\({{\\\\mathcal {C}}},{{\\\\mathcal {D}}}\\\\)</span> and <span>\\\\({{\\\\mathcal {D}}}_0\\\\)</span>, where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form <span>\\\\(h(y,z)=Tr(y\\\\pi (z)) + G_1(Tr_1^m(\\\\alpha _1y),\\\\ldots ,Tr_1^m(\\\\alpha _ky))G_2(Tr_1^m(\\\\beta _{k+1}z),\\\\ldots ,Tr_1^m(\\\\beta _{\\\\tau }z))+ G_3(Tr_1^m(\\\\alpha _1y),\\\\ldots ,Tr_1^m(\\\\alpha _ky))\\\\)</span> to be bent and outside <span>\\\\({\\\\mathcal{M}\\\\mathcal{M}}^\\\\#\\\\)</span> is specified without a strong assumption that the components of the permutation <span>\\\\(\\\\pi \\\\)</span> do not admit linear structures.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-04-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10623-024-01407-9\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10623-024-01407-9","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
Using $$P_\tau $$ property for designing bent functions provably outside the completed Maiorana–McFarland class
In this article, we identify certain instances of bent functions, constructed using the so-called \(P_\tau \) property, that are provably outside the completed Maiorana–McFarland (\({\mathcal{M}\mathcal{M}}^\#\)) class. This also partially answers an open problem in posed by Kan et al. (IEEE Trans Inf Theory, https://doi.org/10.1109/TIT.2022.3140180, 2022). We show that this design framework (using the \(P_\tau \) property), can provide instances of bent functions that are outside the known classes of bent functions, including the classes \({\mathcal{M}\mathcal{M}}^\#\), \({{\mathcal {C}}},{{\mathcal {D}}}\) and \({{\mathcal {D}}}_0\), where the latter three were introduced by Carlet in the early nineties. We provide two generic methods for identifying such instances, where most notably one of these methods uses permutations that may admit linear structures. For the first time, a set of sufficient conditions for the functions of the form \(h(y,z)=Tr(y\pi (z)) + G_1(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))G_2(Tr_1^m(\beta _{k+1}z),\ldots ,Tr_1^m(\beta _{\tau }z))+ G_3(Tr_1^m(\alpha _1y),\ldots ,Tr_1^m(\alpha _ky))\) to be bent and outside \({\mathcal{M}\mathcal{M}}^\#\) is specified without a strong assumption that the components of the permutation \(\pi \) do not admit linear structures.
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.