When e-th Roots Become Easier Than Factoring
Identifieur interne : 005364 ( Hal/Curation ); précédent : 005363; suivant : 005365When e-th Roots Become Easier Than Factoring
Auteurs : Antoine Joux [France] ; David Naccache [France] ; Emmanuel Thomé [France]Source :
English descriptors
Abstract
We show that computing $e$-th roots modulo $n$ is easier than factoring $n$ with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form $x_i + c$. Here $c$ is fixed and $x_i$ denotes small integers of the attacker's choosing. Several variants of the attack are presented, with varying assumptions on the oracle, and goals ranging from selective to universal forgeries. The computational complexity of the attack is $L_n(\frac{1}{3}, \sqrt[3]{\frac{32}{9}})$ in most significant situations, which matches the {\sl special} number field sieve's ({\sc snfs}) complexity. This sheds additional light on {\sc rsa}'s malleability in general and on {\sc rsa}'s resistance to affine forgeries in particular -- a problem known to be polynomial for $x_i > \sqrt[3]{n}$, but for which no algorithm faster than factoring was known before this work.
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DOI: 10.1007/978-3-540-76900-2_2
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<front><div type="abstract" xml:lang="en">We show that computing $e$-th roots modulo $n$ is easier than factoring $n$ with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form $x_i + c$. Here $c$ is fixed and $x_i$ denotes small integers of the attacker's choosing. Several variants of the attack are presented, with varying assumptions on the oracle, and goals ranging from selective to universal forgeries. The computational complexity of the attack is $L_n(\frac{1}{3}, \sqrt[3]{\frac{32}{9}})$ in most significant situations, which matches the {\sl special} number field sieve's ({\sc snfs}) complexity. This sheds additional light on {\sc rsa}'s malleability in general and on {\sc rsa}'s resistance to affine forgeries in particular -- a problem known to be polynomial for $x_i > \sqrt[3]{n}$, but for which no algorithm faster than factoring was known before this work.</div>
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<idno type="halRefHtml">Kaoru Kurosawa. 13th International Conference on the Theory and Application of Cryptology and Information Security - ASIACRYPT 2007, Dec 2007, Kuching, Malaysia. Springer Berlin / Heidelberg, 4833, pp.13-28, 2007, Lecture Notes in Computer Science; Advances in Cryptology -- ASIACRYPT 2007 13th International Conference on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia, December 2-6, 2007. Proceedings. <10.1007/978-3-540-76900-2_2></idno>
<idno type="halRef">Kaoru Kurosawa. 13th International Conference on the Theory and Application of Cryptology and Information Security - ASIACRYPT 2007, Dec 2007, Kuching, Malaysia. Springer Berlin / Heidelberg, 4833, pp.13-28, 2007, Lecture Notes in Computer Science; Advances in Cryptology -- ASIACRYPT 2007 13th International Conference on the Theory and Application of Cryptology and Information Security, Kuching, Malaysia, December 2-6, 2007. Proceedings. <10.1007/978-3-540-76900-2_2></idno>
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<seriesStmt><idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
<idno type="stamp" n="INRIA">INRIA - Institut National de Recherche en Informatique et en Automatique</idno>
<idno type="stamp" n="INPL">Institut National Polytechnique de Lorraine</idno>
<idno type="stamp" n="ENS-PARIS">Ecole Normale Supérieure de Paris</idno>
<idno type="stamp" n="PRISM">Parallélisme, Réseaux, Systèmes d'information, Modélisation</idno>
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<idno type="stamp" n="LORIA">LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications</idno>
<idno type="stamp" n="UVSQ">Université de Versailles Saint-Quentin-en-Yvelines</idno>
<idno type="stamp" n="UNIV-LORRAINE">Université de Lorraine</idno>
<idno type="stamp" n="INRIA-LORRAINE">INRIA Nancy - Grand Est</idno>
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<notesStmt><note type="commentary">The original publications is available at www.springerlink.com ; ISBN 978-3-540-76899-9 ; ISSN 0302-9743 (Print) 1611-3349 (Online)</note>
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<sourceDesc><biblStruct><analytic><title xml:lang="en">When e-th Roots Become Easier Than Factoring</title>
<author role="aut"><persName><forename type="first">Antoine</forename>
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<author role="aut"><persName><forename type="first">David</forename>
<surname>Naccache</surname>
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<monogr><meeting><title>13th International Conference on the Theory and Application of Cryptology and Information Security - ASIACRYPT 2007</title>
<date type="start">2007-12</date>
<settlement>Kuching</settlement>
<country key="MY">Malaysia</country>
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<name>International Association for Cryptologic Research</name>
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<editor>Kaoru Kurosawa</editor>
<imprint><publisher>Springer Berlin / Heidelberg</publisher>
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<biblScope unit="volume">4833</biblScope>
<biblScope unit="pp">13-28</biblScope>
<date type="datePub">2007</date>
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<idno type="doi">10.1007/978-3-540-76900-2_2</idno>
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<textClass><keywords scheme="author"><term xml:lang="en">rsa</term>
<term xml:lang="en">factoring</term>
<term xml:lang="en">nfs</term>
<term xml:lang="en">roots</term>
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<abstract xml:lang="en">We show that computing $e$-th roots modulo $n$ is easier than factoring $n$ with currently known methods, given subexponential access to an oracle outputting the roots of numbers of the form $x_i + c$. Here $c$ is fixed and $x_i$ denotes small integers of the attacker's choosing. Several variants of the attack are presented, with varying assumptions on the oracle, and goals ranging from selective to universal forgeries. The computational complexity of the attack is $L_n(\frac{1}{3}, \sqrt[3]{\frac{32}{9}})$ in most significant situations, which matches the {\sl special} number field sieve's ({\sc snfs}) complexity. This sheds additional light on {\sc rsa}'s malleability in general and on {\sc rsa}'s resistance to affine forgeries in particular -- a problem known to be polynomial for $x_i > \sqrt[3]{n}$, but for which no algorithm faster than factoring was known before this work.</abstract>
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