**A New Algorithm for Solving the General Approximate Common Divisors Problem and Cryptanalysis of the FHE Based on the GACD problem**

*Jintai Ding, Chengdong Tao *

**Abstract: **In this paper, we propose a new algorithm for solving the general approximate common divisors (GACD) problems, which is based on lattice reduction algorithms on certain special lattices and linear equation solving algorithms over integers. Through both theoretical arguments and experimental data, we show that
our new algorithm works in polynomial time but under roughly the following
condition:

\begin{itemize} \item There is a positive integer $t$ such that

$$\frac{\gamma+\eta}{t} + \frac{t}{H}+\rho < \eta;$$

\item We have more than $t$ GACD samples.

\end{itemize} or equivalently \begin{itemize} \item $$H(\eta-\rho)^{2}-4(\gamma+\eta)>0$$ \item We have more than $t=\lceil\frac{H(\eta-\rho)-\sqrt{H^{2}(\eta-\rho)^{2}-4H(\gamma+\eta)}}{2}\rceil$ GACD samples. \end{itemize}

Here $\gamma$, $\eta$ and $\rho$ are parameters describing a GACD problem, $H =1/ \log_{2}F$ and $F$ is the Hermite Factor. In our experiments, $H$ is shown to be roughly $40$ when using the LLL reduction algorithm and it should be around $80$ when using Deep or BKZ algorithms. % We use our algorithm to solve concrete problems that no other algorithm could solve before. We show how our algorithm can be applied to attack the fully homomorphic encryption schemes which are based on the general approximate common divisors problem and its limitations.

**Category / Keywords: **General approximate common divisors problems; Fully homomorphic encryption; Lattice; LLL; BKZ; Hermite Factor

**Date: **received 16 Jan 2014, last revised 2 Mar 2014

**Contact author: **jintai ding at gmail com

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

**Version: **20140303:022136 (All versions of this report)

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