Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security

Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security | 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 Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security Taylan Demir, Hasan Hüseyin Tosunoğlu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8242571/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 Classical and dynamic Caesar ciphers are mainly vulnerable to frequency attacks, since in both schemes the shift applied to each character position is memoryless. This is not the case for the proposed fractional differential shift cipher, where the running shift sequence is generated by a Caputo fractional system D t α x(t) = F(x(t), k) with α ∈(0,1) and k denotes the secret key vector. The discrete fractional integrator associated with this Caputo fractional system yields a long memory, non-uniform sequence of shifts, which induces a generalized Caesar mapping on any chosen character set or ASCII set. Qualitative analysis shows that relatively small variations in either α or k lead to clearly distinct shift patterns and substantially enlarge the effective key space. Numerical tests on textual data demonstrate an approximately uniform symbol distribution, a very low degree of local correlation and, conversely, a high level of entropy, while linear-time complexity and a minimal implementation footprint are preserved. Consequently, taken together, these characteristics indicate that Caputo based fractional differential shift encryption can serve as a practical alternative to traditional substitution ciphers in resource-constrained environments. Applied Mathematics Fractional differential shift encryption Caputo derivative dynamic Caesar cipher substitution cipher fractional dynamical systems lightweight cryptography. Full Text Additional Declarations The authors declare no competing interests. Supplementary Files 11.png Figure 1 21.png Figure 2 FractionalDifferentialShiftEncryptionACaputoBasedGeneralizationofDynamicCaesarCiphering16.pdf Article pdf main4.tex Latex file 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. 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4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eClassical and dynamic Caesar ciphers are mainly vulnerable to frequency attacks, since in both schemes the shift applied to each character position is memoryless. This is not the case for the proposed \u0026nbsp;fractional differential shift cipher, \u0026nbsp;where the running shift sequence \u0026nbsp;is generated by a Caputo fractional system D\u003csub\u003et\u003c/sub\u003e\u003csup\u003eα \u0026nbsp;\u003c/sup\u003e\u003cem\u003ex(t) = F(x(t), k) \u003c/em\u003ewith \u0026nbsp;\u003cem\u003eα ∈(0,1) \u003c/em\u003eand \u003cem\u003e\u0026nbsp;k \u003c/em\u003edenotes the secret key vector. The discrete fractional \u0026nbsp;integrator associated with this Caputo fractional system yields a long memory, non-uniform sequence of shifts, which induces a generalized \u0026nbsp;Caesar mapping on any chosen character set or ASCII set. Qualitative \u0026nbsp;analysis shows that relatively small variations in either \u003cem\u003eα\u003c/em\u003e or \u003cem\u003ek\u003c/em\u003e lead to \u0026nbsp;clearly distinct shift patterns and substantially enlarge the effective \u0026nbsp;key space. Numerical tests on textual data demonstrate an approximately uniform symbol distribution, a very low degree of local correlation and, conversely, a high level of entropy, while linear-time complexity and a minimal implementation footprint are preserved. Consequently, taken together, these characteristics indicate that Caputo based fractional differential shift encryption can serve as a practical \u0026nbsp;alternative to traditional substitution ciphers in resource-constrained \u0026nbsp;environments.\u003c/p\u003e","manuscriptTitle":"Caputo Fractional Differential Shift Encryption: A Generalized Dynamic Caesar Cipher with Enhanced Statistical Security","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-02 07:08:55","doi":"10.21203/rs.3.rs-8242571/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","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}}],"origin":"","ownerIdentity":"14da3057-d7e8-4411-b6ad-a993671390f8","owner":[],"postedDate":"December 2nd, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":58839105,"name":"Applied Mathematics"}],"tags":[],"updatedAt":"2025-12-02T07:08:55+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-02 07:08:55","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8242571","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8242571","identity":"rs-8242571","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}