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Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity

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Abstract

At PKC 2019, Clear and McGoldrick presented the first identity-based encryption (IBE) scheme that is group homomorphic for addition modulo a poly-sized prime e. Assuming that deciding solvability of a special system of multivariate polynomial equations is hard, they proved that their scheme for \(e>2\) is anonymous. In this paper, we review the classical Galbraith’s test on the anonymity of the first pairing-free IBE scheme due to Cocks. With the eye of the reciprocity law for \(\mathbb {F}_\mathtt {q}[x]\), we can have a profound understanding of the test and naturally extend it to give a practical attack on the anonymity of the Clear–McGoldrick IBE scheme. Furthermore, we believe that our technique plays a crucial role in anonymizing IBE schemes from higher residuosity.

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Notes

  1. It is better to use a small prime e because of its small message-ciphertext expansion factor. In practice, we can use the Chinese Remainder Theorem to support homomorphic addition modulo a “large” square-free modulus, see [13, Sect. 3.5].

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Acknowledgements

We are grateful to the referee for carefully reading our manuscript and for his/her valuable comments. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 61632012 and 61672239), in part by the Peng Cheng Laboratory Project of Guangdong Province (Grant No. PCL2018KP004), and in part by the “Fundamental Research Funds for the Central Universities”.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, China

    Xiaopeng Zhao, Zhenfu Cao & Xiaolei Dong

  2. Cyberspace Security Research Center, Peng Cheng Laboratory, Shenzhen and Shanghai Institute of Intelligent Science and Technology, Tongji University, Shanghai, China

    Zhenfu Cao

  3. School of Computer and Information Engineering, Zhejiang Gongshang University, Hangzhou, China

    Jun Shao

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  1. Xiaopeng Zhao
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Correspondence to Zhenfu Cao.

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Communicated by S. D. Galbraith.

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Zhao, X., Cao, Z., Dong, X. et al. Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity. Des. Codes Cryptogr. 89, 241–253 (2021). https://doi.org/10.1007/s10623-020-00816-w

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