[Paper Review] Attack on Fully Homomorphic Encryption over the Integers

Chunsheng Gu|arXiv (Cornell University)|Feb 15, 2012

Cryptography and Data Security25 references3 citations

TL;DR

This paper presents a heuristic lattice-based attack on the fully homomorphic encryption (FHE) scheme over the integers proposed by DGHV (2010), demonstrating that the scheme is insecure under certain parameter settings. By applying lattice reduction techniques, the attack recovers the secret key, exposing a critical vulnerability in the original construction, and further proposes a modified scheme to resist such attacks.

ABSTRACT

This paper presents a heuristic attack on the fully homomorphic encryption over the integers by using lattice reduction algorithm. Our result shows that the FHE in [DGHV10] is not secure for some parameter settings. We also present an improvement scheme to avoid the lattice attack in this paper.

Motivation & Objective

• To analyze the security of the fully homomorphic encryption scheme over the integers proposed by DGHV (2010).
• To identify parameter settings that make the scheme vulnerable to cryptanalysis.
• To develop a practical attack using lattice reduction algorithms on the secret key structure.
• To propose a revised scheme that resists the identified lattice-based attack.

Proposed method

• The attack exploits the structure of the secret key in the DGHV FHE scheme, which is a small odd integer.
• It models the secret key recovery as a closest vector problem (CVP) in a lattice constructed from the public keys.
• Lattice reduction algorithms, particularly LLL and its variants, are applied to approximate the secret key from the public parameters.
• The attack uses the fact that the public keys are linear combinations of the secret key and small noise terms.
• The method evaluates the feasibility of the attack under various parameter settings, identifying weak configurations.
• An improved scheme is proposed by modifying the key generation process to increase the lattice hardness and prevent key recovery.

Experimental results

Research questions

• RQ1 Can the secret key in the DGHV FHE scheme over the integers be recovered using lattice reduction techniques under specific parameter settings?
• RQ2 What structural properties of the DGHV scheme make it vulnerable to lattice-based cryptanalysis?
• RQ3 Which parameter ranges in the DGHV scheme lead to insecure lattice instances?
• RQ4 How can the FHE scheme be modified to resist lattice attacks while preserving homomorphic functionality?

Key findings

• The attack successfully recovers the secret key for certain parameter settings using lattice reduction, demonstrating that the DGHV scheme is not secure under those conditions.
• The vulnerability arises because the secret key is small and can be approximated via lattice basis reduction of the public key matrix.
• The attack is heuristic but effective in practice for the parameter ranges tested, particularly when the secret key is small relative to the public key size.
• The paper identifies that the original DGHV scheme's key generation process creates a lattice with exploitable structure, enabling key recovery.
• An improved scheme is proposed that increases the secret key size and modifies the key generation to resist lattice attacks.
• The results show that the security of FHE over the integers critically depends on parameter selection, and naive choices can lead to total breakage.