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Title:
Lattice Attacks on the DGHV Homomorphic Encryption Scheme
Authors: Abderrahmane Nitaj, Tajjeeddine Rachidi
Abstract:In 2010, van Dijk, Gentry, Halevi, and Vaikuntanathan described the first fully homomorphic encryption over the integers, called DGHV. The scheme is based on a set of m public integers c_i=pq_i+r_i, i=1,\cdots,m, where the integers p, q_i and r_i are secret. In this paper, we describe two lattice-based attacks on DGHV. The first attack is applicable when r_1=0 and the public integers c_i satisfy a linear equation a_2c_2+\ldots+a_mc_m=a_1q_1 for suitably small integers a_i, i=2,\ldots,m. The second attack works when the positive integers q_i satisfy a linear equation a_1q_1+\ldots+a_mq_m=0 for suitably small integers a_i, i=1,\ldots,m. We further apply our methods for the DGHV recommended parameters as specified in the original work of van Dijk, Gentry, Halevi, and Vaikuntanathan.
ePrint: https://eprint.iacr.org/2015/1145
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