# When e-th Roots Become Easier Than Factoring

3 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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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Conference papers
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. 〈10.1007/978-3-540-76900-2_2〉
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https://hal.inria.fr/inria-00187782
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Submitted on : Thursday, November 15, 2007 - 11:41:44 AM
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Antoine Joux, David Naccache, Emmanuel Thomé. When e-th Roots Become Easier Than Factoring. 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. 〈10.1007/978-3-540-76900-2_2〉. 〈inria-00187782〉

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