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current-item"><span property="name">CARAMBA</span></a><meta property="position" content="4"></span> </div>  <div class="col-sm-3"> <div class="nav-widget widget menu"><h1>Navigation</h1>   <ul> <a class='navigation-back' href='https://www.loria.fr/fr/la-recherche/les-equipes/'>< Les 茅quipes</a><br /> </ul> </div> </div> <div class="col-sm-6 news-content page-content the-content">  <h1 class="the-title">CARAMBA</h1> <div id="parent-fieldname-description" class="documentDescription">Cryptologie, arithm茅tique : mat茅riel et logiciel</div> <p><span id="more-1402"></span></p> <div id="parent-fieldname-text" class=""> <p style="text-align: center;"><i>D茅partement 1 : Algorithmique, calcul, image et g茅om茅trie</i></p> <p><strong><br /> Responsable de l’茅quipe : Emmanuel Thom茅</strong><br /> T茅l. : +33 3 54 95 86 59<br /> Mail : emmanuel.thome@loria.fr</p> <p><a href="https://caramba.inria.fr/fr/">Site de l’茅quipe</a></p> <h2>Pr茅sentation</h2> <p>Nos travaux visent le domaine g茅n茅ral d’application de la cryptographie et de la cryptanalyse, d’un point de vue algorithmique. Nous 茅tudions tous les aspects algorithmiques, du bagage math茅matique fondamental jusqu’脿 l’impl茅mentation optimis茅e dans un contexte de calcul 脿聽haute performances. Plusieurs types d’objets sont fr茅quemment utilis茅s dans nos travaux. Certains sont r茅ellement omnipr茅sents: entiers, corps finis, polyn么mes, nombres r茅els et complexes. Nous manipulons 茅galement des objets plus structur茅s comme des corps de nombres, des courbes alg茅briques, ou des syst猫mes polynomiaux. Dans tous les cas, notre travail est orient茅 vers le fait de rendre les calcul avec ces objets efficace.</p> <p>Les objets math茅matiques que nous manipulons sont de premi猫re importance pour les applications 脿 la cryptologie, puisqu’ils constituent l’assise des primitives cryptographiques les plus r茅pandues, tels le cryptosyst猫me RSA ou l’茅change de cl茅s de Diffie–Hellman. Les deux aspects de la cryptologie (cryptographie et cryptanalyse) jouent un r么le central dans notre recherche. Les d茅fis principaux sont d’une part l’茅valuation de la s茅curit茅 offerte par les primitives cryptographiques, via l’茅tude des probl猫mes essentiels que sont la factorisation d’entiers et le probl猫me du logarithme discret, et d’autre part le travail d’optimisation (math茅matique, algorithmique, et aussi de plus bas niveau) rendant possible des impl茅mentations 脿 la fois efficaces et s没res.</p> <h2 class="Default">Axes th茅matiques</h2> <p>Parmi les axes de recherche mis en avant par Caramba, deux sont guid茅s par les objets math茅matiques les plus importants utilis茅s dans la cryptographie actuelle, deux autres sont plut么t guid茅s par le bagage technologique qui nous est n茅cessaire pour aborder les probl猫mes qui y ont trait.</p> <ul> <li>La famille 茅tendue du crible alg茅brique. Un cadre algorithmique commun, le crible alg茅brique (NFS), s’applique 脿 la fois 脿 la factorisation d’entiers ainsi qu’au probl猫me du logarithme discret sur les corps finis. Nous avons de nombreuses contributions algorithmiques dans ce contexte, et nous d茅veloppons des logiciels qui les illustrent.Nous souhaitons am茅liorer l’茅tat de l’art dans ce domaine par le d茅veloppement de nouveaux algorithmes, par l’am茅lioration de la performance des logiciels. Notre travail est naturellement assorti de d茅monstrations pratique de la port茅e des avanc茅es obtenues, sous la forme de calculs record.</li> <li>Les courbes alg茅briques et leurs jacobiennes. Nous d茅veloppons des algorithmes et des logiciels pour calculer les propri茅t茅s essentielles des courbes alg茅briques pour la cryptologie, rendant possible <em>in fine</em> leur d茅ploiement le plus large possible.Un des d茅fis que nous abordons est celui du comptage de points. Dans une perspective plus vaste, nous 茅tudions 茅galement le lien entre les vari茅t茅s ab茅liennes sur des corps finis, et les vari茅t茅s ab茅liennes principalement polaris茅es sur des corps de caract茅ristique nulle, et leur anneau d’endomorphismes. En particulier, nous travaillons 脿 rendre ce lien explicite d’un point de vue pratique. Nous 茅tudions 茅galement plusieurs approches pour attaquer au probl猫me du logarithme discret sur les jacobiennes de courbes alg茅briques.</li> <li>L’arithm茅tique. Notre travail repose de fa莽on essentielle sur l’efficacit茅 des op茅rations arithm茅tiques sous-jacentes, qu’il s’agisse d’op茅rations sur des objets de petite ou de grande tailles. Nous travaillons 脿 l’am茅lioration des algorithmes et des impl茅mentations pour les calculs qui sont en lien avec nos domaines d’application.</li> <li>Les syst猫mes polynomiaux. De fa莽on naturelle dans le contexte des courbes alg茅briques, mais 茅galement dans les probl猫mes li茅s au crible alg茅briques, de nombreux probl猫mes importants peuvent 锚tre exprim茅s dans le cadre des syst猫mes polynomiaux. Les syst猫mes ainsi obtenus ont des sp茅cificit茅s structurelles fortes, en lien avec le probl猫me d’origine. Nous d茅veloppons des algorithme et des outils qui, lorsque c’est possible, tirent parti de ces sp茅cificit茅s structurelles.</li> </ul> <h2>Logiciels</h2> <ul> <li>GNU MPFR/MPC</li> <li>GMP-ECM</li> <li>CADO-NFS</li> <li>AVIsogenies</li> </ul> <h2>Collaborations</h2> <p>EPFL,<br /> University of Kaiserslautern,<br /> Western University,<br /> University of Calgary<br /> Nos travaux sont repris par les organismes gouvernementaux 茅mettant des recommandations cryptographiques (ANSSI, BSI, NIST).</p> <h2>Mots-cl茅s</h2> <p>Cryptographie – Factorisation d’entiers – Logarithme discret – Cryptographie elliptique et hyperelliptique – Th茅orie des nombres – Syst猫mes polynomiaux – Arithm茅tique</p> </div>  </div>  <div class="col-sm-3"> <div class="sidebar widget"> <div> <div><h1>Colloquium 2024</h1> <div class="textwidget"><p><img loading="lazy" decoding="async" class="alignnone wp-image-27428 size-medium" src="https://www.loria.fr/wp-content/uploads/Colloquium_couleur-300x282.png" alt="" 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