Improved Key Generation Algorithm for Gentry’s Fully Homomorphic Encryption Scheme

  • Yang Zhang
  • Renzhang Liu
  • Dongdai Lin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10779)


At EUROCRYPT 2011, Gentry and Halevi implemented a variant of Gentry’s fully homomorphic encryption scheme. The core part in their key generation is to generate an odd-determinant ideal lattice having a particular type of Hermite Normal Form. However, they did not give a rigorous proof for the correctness. We present a better key generation algorithm, improving their algorithm from two aspects.

  • We show how to deterministically generate ideal lattices with odd determinant, thus increasing the success probability close to 1.

  • We give a rigorous proof for the correctness. To be more specific, we present a simpler condition for checking whether the ideal lattice has the desired Hermite Normal Form. Furthermore, our condition can be checked more efficiently.

As a result, our key generation is about 1.5 times faster. We also give experimental results supporting our claims. Our optimizations are based on the properties of ideal lattices, which might be of independent interests.


Fully homomorphic encryption Key generation Hermite Normal Form Ideal lattice 



The authors would like to thank all anonymous referees of ISC’2017 and ICISC’2017 for their valuable comments that greatly improve the manuscript. This work is supported by the National Natural Science Foundation of China (No. Y31005A102, No. Y610092302).


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.State Key Laboratory of Information Security, Institute of Information EngineeringChinese Academy of SciencesBeijingChina
  2. 2.School of Cyber SecurityUniversity of Chinese Academy of SciencesBeijingChina

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