## Cryptology ePrint Archive: Report 2011/591

**A Unified Framework for Small Secret Exponent Attack on RSA**

*Noboru Kunihiro and Naoyuki Shinohara and Tetsuya Izu*

**Abstract: **We address a lattice based method on small secret exponent
attack on RSA scheme. Boneh and Durfee reduced the attack into
finding small roots of a bivariate modular equation: $x(N+1+y)+1
¥equiv 0 mod e)$, where $N$ is an RSA moduli and $e$ is the RSA
public key. Boneh and Durfee proposed a lattice based algorithm
for solving the problem. When the secret exponent $d$ is less than
$N^{0.292}$, their method breaks RSA scheme. Since the lattice used
in the analysis is not full-rank, the analysis is not easy.
Bl¥"omer and May gave an alternative algorithm. Although their
bound $d ¥leq N^{0.290}$ is worse than Boneh--Durfee result,
their method used a full rank lattice. However, the proof for
their bound is still complicated. Herrmann and May gave an
elementary proof for the Boneh--Durfee's bound: $d ¥leq N^{0.292}$.
In this paper, we first give an elementary proof for achieving the
bound of Bl¥"omer--May: $d ¥leq N^{0.290}$. Our proof employs
unravelled linearization technique introduced by Herrmann and May
and is rather simpler than Bl¥"omer--May's proof. Then, we
provide a unified framework to construct a lattice that are used
for solving the problem, which includes two previous method:
Herrmann--May and Bl¥"omer--May methods as a special case. Furthermore, we prove that the bound of Boneh--Durfee: $d ¥leq
N^{0.292}$ is still optimal in our unified framework.

**Category / Keywords: **public-key cryptography / lattice techniques, RSA, cryptanalysis

**Publication Info: **This is a full version of SAC2011 paper.

**Date: **received 30 Oct 2011

**Contact author: **kunihiro at k u-tokyo ac jp

**Available format(s): **PDF | BibTeX Citation

**Version: **20111103:101812 (All versions of this report)

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