Implementing Attacks on the Approximate Greatest Common Divisor Problem

  • Leizhang WangEmail author
  • Quanbo Qu
  • Tuoyan Li
  • Yange Chen
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 1105)


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.


Approximate greatest common divisor problem Orthogonal lattice attack Simultaneous diophantine approximation attack Lattice reduction algorithm 



First of all, I would like to thank my mentor Professor Baocang Wang and Professor Hailou Yao. When I was puzzled to solve the AGCD problem, it was Professor Wang’s appropriate advice that guides me. In addition, when I wrote my paper, Professor Wang and Professor Yao also gave me many valuable opinions and suggestions which benefited me a lot. In the end, I would like to express my heartfelt thanks to Professor Wang and Professor Yao for their concern and help.

This work is supported by the National Key R&D Program of China under Grant No. 2017YFB0802000, the National Natural Science Foundation of China under Grant Nos. 61572390, U1736111, the National Cryptography Development Fund under Grant No. MMJJ20180111, the Plan For Scientific Innovation Talent of Henan Province under Grand no. 184100510012, the Program for Science & Technology Innovation Talents in Universities of Henan Province under Grant No. 8HASTIT022, the Innovation Scientists and Technicians Troop Construction Projects of Henan Province.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Leizhang Wang
    • 1
    • 2
    • 3
    Email author
  • Quanbo Qu
    • 1
    • 2
  • Tuoyan Li
    • 3
  • Yange Chen
    • 4
  1. 1.State Key Laboratory of Integrated Service NetworksXidian UniversityXi’anPeople’s Republic of China
  2. 2.Cryptographic Research CenterXidian UniversityXi’anPeople’s Republic of China
  3. 3.College of Applied ScienceBeijing University of TechnologyBeijingPeople’s Republic of China
  4. 4.School of Information EngineeringXuchang UniversityXuchangChina

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