Abstract:
At EUROCRYPT '10, van Dijk, Gentry, Halevi and Vaikuntanathan presented simple fully-homomorphic encryption (FHE) schemes
based on the hardness of approximate integer common divisors problems,
which were introduced in 2001 by Howgrave-Graham.
There are two versions for these problems: the partial version (PACD) and the general version (GACD).
The seemingly easier problem PACD was recently used by Coron, Mandal, Naccache and Tibouchi at CRYPTO '11 to build a more efficient variant of the FHE scheme
by van Dijk {\em et al.}. We present a new PACD algorithm whose running time is essentially the ``square root''
of that of exhaustive search, which was the best attack in practice. This allows us to experimentally break the FHE challenges proposed by Coron {\em et al.}
Our PACD algorithm directly gives rise to a new GACD algorithm, which is exponentially faster than exhaustive search:
namely, the running time is essentially the $3/4$-th root of that of exhaustive search.
Interestingly, our main technique can also be applied to other settings, such as noisy factoring, fault attacks on CRT-RSA signatures and attacking low-exponent RSA encryption.

Bibtex:
@inproceedings{ChNg12,
author = {Y. Chen and P. Q. Nguyen},
title = {Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers},
booktitle= "Advances in Cryptology -- Proceedings of EUROCRYPT '12",
publisher= {Springer},
volume = "7237",
series = {LNCS},
year = 2012
}