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Regularly Lossy Functions and Applications

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Book cover Topics in Cryptology – CT-RSA 2018 (CT-RSA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10808))

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Abstract

In STOC 2008, Peikert and Waters introduced a powerful primitive called lossy trapdoor functions (LTFs). In a nutshell, LTFs are functions that behave in one of two modes. In the normal mode, functions are injective and invertible with a trapdoor. In the lossy mode, functions statistically lose information about their inputs. Moreover, the two modes are computationally indistinguishable. In this work, we put forward a relaxation of LTFs, namely, regularly lossy functions (RLFs). Compared to LTFs, the functions in the normal mode are not required to be efficiently invertible or even unnecessary to be injective. Instead, they could also be lossy, but in a regular manner. We also put forward richer abstractions of RLFs, namely all-but-one regularly lossy functions (ABO-RLFs).

We show that (ABO)-RLFs admit efficient constructions from both a variety of number-theoretic assumptions and hash proof system (HPS) for subset membership problems satisfying natural algebraic properties. Thanks to the relaxations on functionality, the constructions enjoy shorter key size and better computational efficiency than that of (ABO)-LTFs.

We demonstrate the applications of (ABO)-RLFs in leakage-resilient cryptography.

  • As a special case of RLFs, lossy functions imply leakage-resilient injective one-way functions with optimal leakage rate \(1-o(1)\).

  • ABO-RLFs immediately imply leakage-resilient message authentication code (MAC) with optimal leakage rate \(1-o(1)\), though in a weak sense.

  • ABO-RLFs together with HPS give rise to leakage-resilient chosen-ciphertext (CCA) secure key encapsulation mechanisms (KEM) (this approach extends naturally to the identity-based setting). Combining the construction of ABO-RLFs from HPS, this gives the first leakage-resilient CCA-secure public-key encryption (PKE) with optimal leakage rate based solely on HPS, and thus goes beyond the barrier posed by Dodis et al. (Asiacrypt 2010).

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Notes

  1. 1.

    This is sometimes called second preimage resistant functions.

  2. 2.

    The leakage bound only subjects to the image size of functions in the lossy mode, which will not be used in real construction.

  3. 3.

    Kiltz et al. [KPSY09] showed that CCA-secure PKE can be constructed from a universal\(_2\) HPS with an authenticated one-time secure symmetric encryption, while universal\(_2\) HPS can be generically obtained from universal\(_1\) HPS via 4-wise independent hash function. At a first glance, their construction can be easily augmented to be leakage-resilient CCA-secure by applying randomness extractor to the projective hash. However, such augment could be very subtle in that the adding of a random seed may render the overall ciphertext easily malleable, and thus cannot be CCA-secure.

  4. 4.

    As shown in [QL13], OT-LFs can be build from ABO-LFs and chameleon hash.

  5. 5.

    Following the treatment of [Wee12], we will write \(\varLambda _{sk}(x)\) as \(\varLambda _x(sk)\) occasionally.

  6. 6.

    The regularity of \(\alpha \) gives an upper bound of \(\nu \).

  7. 7.

    Assume each element in X can be uniquely encoded as a binary string in \(\{0,1\}^m\).

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Acknowledgement

We thank the anonymous reviewers of CT-RSA 2018 for their useful comments.

The first author is supported by the National Key Research and Development Plan (Grant No. 2016YFB0800403), the National Natural Science Foundation of China (Grant No. 61772522), Youth Innovation Promotion Association CAS and Key Research Program of Frontier Sciences, CAS (Grant No. QYZDB-SSW-SYS035). The second author is supported by the National Natural Science Foundation of China (Grant No. 61502400). The third author is supported by the National Natural Science Foundation of China (Grant No. 61602473) and the National Cryptography Development Fund (Grant No. MMJJ20170116).

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Authors and Affiliations

  1. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Yu Chen & Haiyang Xue

  2. School of Cyber Security, University of Chinese Academy of Sciences, Beijing, China

    Yu Chen

  3. National Engineering Laboratory for Wireless Security, Xi’an University of Posts and Telecommunications, Xi’an, China

    Baodong Qin

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  1. Yu Chen
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Correspondence to Haiyang Xue .

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  1. KU Leuven , Leuven, Belgium

    Nigel P. Smart

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Chen, Y., Qin, B., Xue, H. (2018). Regularly Lossy Functions and Applications. In: Smart, N. (eds) Topics in Cryptology – CT-RSA 2018. CT-RSA 2018. Lecture Notes in Computer Science(), vol 10808. Springer, Cham. https://doi.org/10.1007/978-3-319-76953-0_26

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