The locally crooked permutations and the complete permutations over F2n

The locally crooked permutations and the complete permutations over F2n | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article The locally crooked permutations and the complete permutations over F 2 n Li Shuai, Miao Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4347830/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The crooked function has a fixed point $0$ and all its difference set is the complement of the hyperplane. In this paper, we proposed the locally crooked permutation, at least one of its differential sets is the complement of a hyperplane. We found a close relationship between the locally crooked permutation and the complete permutation. Specifically, the complete permutations over $\mathbb{F}_{2^{n}}$ can be obtained from the locally crooked permutations over $\mathbb{F}_{2^{n+1}}$, and vice versa. In particular, we construct the complete permutations with best-known differential uniformity and nonlinearity and the locally crooked permutations with differential uniformity of $4$ over $\mathbb{F}_{2^{2n}}$. Besides, we also found that the existence of the APN permutation that is locally crooked over $\mathbb{F}_{2^{2n+2}}$ is closely related to the nonlinear complete permutation over $\mathbb{F}_{2^{2n+1}}$. MSC Classification: 05A05 , 11T06 , 11T55 Crooked functions Complete permutations APN permutations Differentially 4-uniform permutations c-differential uniformity Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4347830","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":300475574,"identity":"07599f96-3eca-4c27-9efe-b5bb681df004","order_by":0,"name":"Li Shuai","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5klEQVRIiWNgGAWjYBACPmYGhgMfKv7z2B9vAHINLAhrYWNmYDw44wyzHMOZAyAtEkRoYWBgPszbxmzMcCMBxCdGCzv7g8M8Z9gSG2c+v7rhR4EEA397dwIBh/EYHJxTwZPYLJ1TdrMH6DCJM2c3ENLCcODNGYnENumctBs8QC0GErmEtLA/OMDbZpDYI3km7eYf4rQwGBzkbUswlpBgP3abSFuAfplx5oCcAU8O220ZAwkegn7h5z/++MOHigM8BuzHn91888dGjr+9F78WJMBjACaJVQ4C7A9IUT0KRsEoGAUjCAAAXSZHRf5rc0MAAAAASUVORK5CYII=","orcid":"","institution":"Ningxia University","correspondingAuthor":true,"prefix":"","firstName":"Li","middleName":"","lastName":"Shuai","suffix":""},{"id":300475576,"identity":"e6087472-9054-4653-8cfd-f226b54b8f72","order_by":1,"name":"Miao Li","email":"","orcid":"","institution":"Ningxia University","correspondingAuthor":false,"prefix":"","firstName":"Miao","middleName":"","lastName":"Li","suffix":""}],"badges":[],"createdAt":"2024-04-30 09:23:22","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4347830/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4347830/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":70948158,"identity":"3716b0fc-9398-4148-b662-424881f43d93","added_by":"auto","created_at":"2024-12-09 13:17:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":389286,"visible":true,"origin":"","legend":"","description":"","filename":"snarticletemplate.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4347830/v1_covered_fcffbfbf-7a19-4ab0-9f7b-bd9cc48e7fa0.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eThe locally crooked permutations and the complete permutations over F\u003csub\u003e2\u003c/sub\u003en\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Crooked functions, Complete permutations, APN permutations, Differentially 4-uniform permutations, c-differential uniformity","lastPublishedDoi":"10.21203/rs.3.rs-4347830/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4347830/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe crooked function has a fixed point $0$ and all its difference set is the complement of the hyperplane. In this paper, we proposed the locally crooked permutation, at least one of its differential sets is the complement of a hyperplane. We found a close relationship between the locally crooked permutation and the complete permutation. Specifically, the complete permutations over $\\mathbb{F}_{2^{n}}$ can be obtained from the locally crooked permutations over $\\mathbb{F}_{2^{n+1}}$, and vice versa. In particular, we construct the complete permutations with best-known differential uniformity and nonlinearity and the locally crooked permutations with differential uniformity of $4$ over $\\mathbb{F}_{2^{2n}}$. 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