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International Conference on the Theory and Application of Cryptology and Information Security

ASIACRYPT 2012: Advances in Cryptology – ASIACRYPT 2012 pp 61–79Cite as

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On the (Im)possibility of Projecting Property in Prime-Order Setting

On the (Im)possibility of Projecting Property in Prime-Order Setting

  • Jae Hong Seo18 
  • Conference paper
  • 3655 Accesses

  • 10 Citations

Part of the Lecture Notes in Computer Science book series (LNSC,volume 7658)

Abstract

Projecting bilinear pairings have frequently been used for designing cryptosystems since they were first derived from composite order bilinear groups. There have been only a few studies on the (im)possibility of projecting bilinear pairings. Groth and Sahai showed that projecting bilinear pairings can be achieved in the prime-order group setting. They constructed both projecting asymmetric bilinear pairings and projecting symmetric bilinear pairings, where a bilinear pairing e is symmetric if it satisfies e(g,h) = e(h,g) for any group elements g and h; otherwise, it is asymmetric.

In this paper, we provide impossibility results on projecting bilinear pairings in a prime-order group setting. More precisely, we specify the lower bounds of

  • the image size of a projecting asymmetric bilinear pairing

  • the image size of a projecting symmetric bilinear pairing

  • the computational cost for a projecting asymmetric bilinear pairing

  • the computational cost for a projecting symmetric bilinear pairing

in a prime-order group setting naturally induced from the k-linear assumption, where the computational cost means the number of generic operations.

Our lower bounds regarding a projecting asymmetric bilinear pairing are tight, i.e., it is impossible to construct a more efficient projecting asymmetric bilinear pairing than the constructions of Groth-Sahai and Freeman. However, our lower bounds regarding a projecting symmetric bilinear pairing differ from Groth and Sahai’s results regarding a symmetric bilinear pairing results; We fill these gaps by constructing projecting symmetric bilinear pairings.

In addition, on the basis of the proposed symmetric bilinear pairings, we construct more efficient instantiations of cryptosystems that essentially use the projecting symmetric bilinear pairings in a modular fashion. Example applications include new instantiations of the Boneh-Goh-Nissim cryptosystem, the Groth-Sahai non-interactive proof system, and Seo-Cheon round optimal blind signatures proven secure under the DLIN assumption. These new instantiations are more efficient than the previous ones, which are also provably secure under the DLIN assumption. These applications are of independent interest.

Keywords

  • Random Oracle
  • Impossibility Result
  • Bilinear Pairing
  • Homomorphic Encryption
  • Overwhelming Probability

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Author information

Authors and Affiliations

  1. National Institute of Information and Communications Technology, 4-2-1, Nukui-kitamachi, Koganei, Tokyo, 184-8795, Japan

    Jae Hong Seo

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  1. Jae Hong Seo
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Editor information

Editors and Affiliations

  1. Tsinghua University, 30 Shuangqing Road, 100084, Beijing, China

    Xiaoyun Wang

  2. Central Research Laboratories, NEC, 1754 Shimonumabe Nakahara, 211-8666, Kawasaki, Japan

    Kazue Sako

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© 2012 International Association for Cryptologic Research

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Seo, J.H. (2012). On the (Im)possibility of Projecting Property in Prime-Order Setting. In: Wang, X., Sako, K. (eds) Advances in Cryptology – ASIACRYPT 2012. ASIACRYPT 2012. Lecture Notes in Computer Science, vol 7658. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34961-4_6

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