Implementing Attacks on the Approximate Greatest Common Divisor Problem

The security of many fully homomorphic encryption (FHE) schemes is guaranteed by the difficulty of the approximate greatest common divisor (AGCD) problem. Therefore, the study of AGCD problem is of great significance to the security of the fully homomorphic encryption. This paper surveys three kinds of attacks on the AGCD problem, i.e. exhaustive search attack, simultaneous Diophantine approximation (SDA) attack and the orthogonal lattice (OL) attack. We utilize the Number Theory Library (NTL) to implement the SDA attack and the optimized OL attack on the AGCD problem. Comparisons are performed based on the experimental results to illustrate that the exhaustive search attack can be easily defended just by increasing the size of ρ. And increasing the length of the public key is the most effective way to defend SDA attack and OL attack. Meanwhile, we concluded that the success rate of SDA attack and OL attack can be improved by increasing the dimension of lattice at the expense of a certain time efficiency. In addition, the analysis and experiments show that the fully homomorphic computing efficiency of FHE scheme can’t be improved by simply increasing the private key without appropriately increasing the size of public key. Otherwise, the FHE scheme is vulnerable to OL and SDA attack. Besides, experimental results show that optimized OL attack performs better than both classical OL attack and SDA attack in terms of attack success rate and the time efficiency.