Weight of Polynomial Products Mod (Xn+1)-Application to the HQC Cryptosystem

Weight of Polynomial Products Mod (Xn+1)-Application to the HQC Cryptosystem | 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 Weight of Polynomial Products Mod (X n +1)-Application to the HQC Cryptosystem Laila El Aimani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8254043/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 We consider the following problem: given two random polynomials $x$ and $y$ in the ring $\F_2[X]/(X^n+1)$, our goal is to compute the expectation and variance of the weight of their product $x\cdot y$, where the weight of a binary polynomial is defined as the number of its nonzero coefficients. We consider two models for random polynomials $x$ and $y$: the \emph{uniform slice case} with fixed weights $w_x,w_y$, and the \emph{binomial case} where their coefficients are independent Bernoulli variables with success probabilities $p_x$ and $p_y$ respectively. Our work finds a direct application in the accurate analysis of the decryption failure rate for the code-based encryption scheme HQC \cite{HQC2025}. The original construction \cite{HQC2025} relied on heuristic arguments supported by experimental data. Later, \cite{Kawachi2020} provided a formally proven security bound, albeit a much weaker one than the heuristic estimate in \cite{HQC2025}. A fundamental limitation of both analyses is their restriction to the binomial case, a simplification that compromises the resulting security guarantees. Our analysis provides the first precise computation of the expectation and variance of $\weight(x\cdot y)$ across both the uniform slice and binomial models. The results confirm the soundness of the HQC security guarantees and allow for a more informed choice of the scheme parameters that optimizes the trade-off security and efficiency. Weight of a polynomial Expectation/variance of random variables Tail probability Decryption error of HQC 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-8254043","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":554945354,"identity":"41538a9b-69f8-4ff1-98cb-12085a0e6aac","order_by":0,"name":"Laila El Aimani","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABBElEQVRIiWNgGAWjYBACPgbGxgNAWoaBGcQ1sAESEBGcgI2BsQGkgAeqJQ2kpYGAFgYGiBYIOAwm8WthP9xw4AeDDQ9/O3fi54KC83Zr24EiDDU20Ti18CQ2HOxhSOOROMy7WXqGwe3kbWcSgVqOpeU24HQYUAEPw2Eg4t0gzQPUYnYAKMLYcBi3Fv6HDQf/MPznkQfa8pvH4Fyy2fmHBLRIJDYArTjAY3CYdxvQlgN2ZjcI2SLxsOGwjEEyjyFQizWPQXKC2Q2gLQl4/MLPn/7w4ZsKOzm582c33+b5Y2dvdj794YMPNTY4tUCAAYKZCFaZgFc5GrAnRfEoGAWjYBSMDAAA+PheJenWRngAAAAASUVORK5CYII=","orcid":"","institution":"Cadi Ayyad University","correspondingAuthor":true,"prefix":"","firstName":"Laila","middleName":"El","lastName":"Aimani","suffix":""}],"badges":[],"createdAt":"2025-12-01 21:38:11","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8254043/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8254043/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":97568676,"identity":"06c3c635-bbd8-4910-87b6-4c9c63e8fa50","added_by":"auto","created_at":"2025-12-06 02:54:15","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":392142,"visible":true,"origin":"","legend":"","description":"","filename":"main.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8254043/v1/7d39afd4532e9e22bd792286.pdf"},{"id":97568675,"identity":"5d3de335-c348-46be-8316-209b20a4d513","added_by":"auto","created_at":"2025-12-06 02:54:15","extension":"json","order_by":1,"title":"","display":"","copyAsset":false,"role":"acdc-reference","size":3355,"visible":true,"origin":"","legend":"","description":"","filename":"4861f5c39d454acd860a56e6aaddb7c6.json","url":"https://assets-eu.researchsquare.com/files/rs-8254043/v1/1d0f2b7ff2ae96f9358397bd.json"},{"id":105051123,"identity":"ce92cfbe-76e7-4120-b472-9deae7565e2a","added_by":"auto","created_at":"2026-03-20 10:24:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":485317,"visible":true,"origin":"","legend":"","description":"","filename":"main.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8254043/v1_covered_e20700ab-dddf-43bc-9ade-b72d980cc5c3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eWeight of Polynomial Products Mod (X\u003csup\u003en\u003c/sup\u003e+1)-Application to the HQC Cryptosystem\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":"Weight of a polynomial, Expectation/variance of random variables, Tail probability, Decryption error of HQC","lastPublishedDoi":"10.21203/rs.3.rs-8254043/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8254043/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u0026nbsp;We consider the following problem: given two random polynomials $x$ and $y$ in the ring $\\F_2[X]/(X^n+1)$, our goal is to compute the expectation and variance of the weight of their product $x\\cdot y$, where the weight of a binary polynomial is defined as the number of its nonzero coefficients. \u0026nbsp; We consider two models for random polynomials $x$ and $y$: the \\emph{uniform slice case} with fixed weights $w_x,w_y$, and the \\emph{binomial case} where their coefficients are independent Bernoulli variables with success probabilities $p_x$ and $p_y$ respectively. \u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Our work finds a direct application in the accurate analysis of the decryption failure rate for the code-based encryption scheme HQC \\cite{HQC2025}. The original construction \\cite{HQC2025} relied on heuristic arguments supported by experimental data. Later, \\cite{Kawachi2020} provided a formally proven security bound, albeit a much weaker one than the heuristic estimate in \\cite{HQC2025}. A fundamental limitation of both analyses is their restriction to the binomial case, a simplification that compromises the resulting security guarantees.\u0026nbsp;\u0026nbsp;\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;Our analysis provides the first precise computation of the expectation and variance of $\\weight(x\\cdot y)$ across both the uniform slice and binomial models. The results confirm the soundness of the HQC security guarantees and allow for a more informed choice of the scheme parameters that optimizes the trade-off security and efficiency.\u003c/p\u003e","manuscriptTitle":"Weight of Polynomial Products Mod (Xn+1)-Application to the HQC Cryptosystem","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-12-06 02:54:10","doi":"10.21203/rs.3.rs-8254043/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":"5b2ee806-40b4-4a8b-9293-b905c110a4f9","owner":[],"postedDate":"December 6th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-03-20T10:23:08+00:00","versionOfRecord":[],"versionCreatedAt":"2025-12-06 02:54:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8254043","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8254043","identity":"rs-8254043","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}