## Cryptology ePrint Archive: Report 2007/424

**When e-th Roots Become Easier Than Factoring**

*Antoine Joux and David Naccache and Emmanuel Thomé*

**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.

**Category / Keywords: **public-key cryptography /

**Publication Info: **RSA, NFS, factoring, digital signatures

**Date: **received 12 Nov 2007

**Contact author: **Emmanuel Thome at normalesup org

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | PDF | BibTeX Citation

**Version: **20071118:222051 (All versions of this report)

**Discussion forum: **Show discussion | Start new discussion

[ Cryptology ePrint archive ]