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Collision Resistant Hashing from Sub-exponential Learning Parity with Noise

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Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11922))

Abstract

The Learning Parity with Noise (LPN) problem has recently found many cryptographic applications such as authentication protocols, pseudorandom generators/functions and even asymmetric tasks including public-key encryption (PKE) schemes and oblivious transfer (OT) protocols. It however remains a long-standing open problem whether LPN implies collision resistant hash (CRH) functions. Inspired by the recent work of Applebaum et al. (ITCS 2017), we introduce a general construction of CRH from LPN for various parameter choices. We show that, just to mention a few notable ones, under any of the following hardness assumptions (for the two most common variants of LPN)

  1. 1.

    constant-noise LPN is \(2^{n^{0.5+\varepsilon }}\)-hard for any constant \(\varepsilon >0\);

  2. 2.

    constant-noise LPN is \(2^{\varOmega (n/\log n)}\)-hard given \(q=\mathsf {poly}(n)\) samples;

  3. 3.

    low-noise LPN (of noise rate \(1/\sqrt{n}\)) is \(2^{\varOmega (\sqrt{n}/\log n)}\)-hard given \(q=\mathsf {poly}(n)\) samples.

there exists CRH functions with constant (or even poly-logarithmic) shrinkage, which can be implemented using polynomial-size depth-3 circuits with NOT, (unbounded fan-in) AND and XOR gates. Our technical route LPN \(\rightarrow \) bSVP \(\rightarrow \) CRH is reminiscent of the known reductions for the large-modulus analogue, i.e., LWE \(\rightarrow \) SIS \(\rightarrow \) CRH, where the binary Shortest Vector Problem (bSVP) was recently introduced by Applebaum et al. (ITCS 2017) that enables CRH in a similar manner to Ajtai’s CRH functions based on the Short Integer Solution (SIS) problem.

Furthermore, under additional (arguably minimal) idealized assumptions such as small-domain random functions or random permutations (that trivially imply collision resistance), we still salvage a simple and elegant collision-resistance-preserving domain extender combining the best of the two worlds, namely, maximized (depth one) parallelizability and polynomial shrinkage. In particular, assume \(2^{n^{0.5+\varepsilon }}\)-hard constant-noise LPN or \(2^{n^{0.25+\varepsilon }}\)-hard low-noise LPN, we obtain a collision resistant hash function that evaluates in parallel only a single layer of small-domain random functions (or random permutations) and shrinks polynomially.

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Notes

  1. 1.

    More precisely, [5] obtains a win-win result that either constant-noise LPN implies CRH or one achieves arbitrary polynomial speedup over the BKW algorithm [15].

  2. 2.

    Unlikely collision resistance whose definition is unique and unambiguous, there are several variants of (second) preimage resistance for which people strive to find a compromise that facilitates security proofs yet captures the needs of most applications. Some variants of (second) preimage resistance are implied by collision resistance in the conventional or provisional sense [60].

  3. 3.

    However, our results do not immediately constitute security proofs for the FSB-style hash functions as there remains a substantial gap between the security proved and security level claimed by the FSB instantiation.

  4. 4.

    \(\mathbf M\) in our consideration has dimension \(n\times q\) instead of \({\alpha n\times n}\) considered by [5].

  5. 5.

    The circuit falls into the class \(\mathsf {AC}^0(\mathsf {MOD 2})\). See Sect. 2 for a formal definition.

  6. 6.

    More strictly speaking, the resulting CRH is either \(h_{\mathbf M}\) itself or its domain-extended version (by a parallel repetition).

  7. 7.

    By switching to a new security parameter, we eventually obtain a CRH function with polynomial running time and super-polynomial security for which the \(n^{\varepsilon }\) gap factor plays a vital role.

  8. 8.

    Recall that a non-uniform attacker can obtain polynomial-size non-uniform advice. Thus, if every security parameter corresponds to only a single function h then the attacker can include a pair of x and \(x'\) with \(h(x)=h(x')\) as part of the advice.

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Acknowledgments

Yu Yu was supported by the National Natural Science Foundation of China (Grant Nos. 61872236 and 61572192) and the National Cryptography Development Fund (Grant No. MMJJ20170209). Jiang Zhang is supported by the National Key Research and Development Program of China (Grant No. 2017YFB0802005, 2018YFB0804105), the National Natural Science Foundation of China (Grant Nos. 6160204661932019), and the Young Elite Scientists Sponsorship Program by CAST (2016QNRC001). Jian Weng was partially supported by National Natural Science Foundation of China (Grant Nos. 61825203, U1736203, 61732021). Chun Guo was supported by the Program of Qilu Young Scholars of Shandong University, and also partly funded by Francois-Xavier Standaert via the ERC project SWORD (724725). Xiangxue Li was supported by the National Cryptography Development Fund (Grant No. MMJJ20180106) and the National Natural Science Foundation of China (Grant Nos. 61572192, 61971192). This research is funded in part by the Anhui Initiative in Quantum Information Technologies (Grant No. AHY150100) and Sichuan Science and Technology Program (Grant No. 2017GZDZX0002).

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China

    Yu Yu

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Yu Yu & Jiang Zhang

  3. Jinan University, Guangzhou, 510632, China

    Jian Weng

  4. Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Qingdao, 266237, China

    Chun Guo

  5. School of Cyber Science and Technology, Shandong University, Qingdao, 266237, China

    Chun Guo

  6. School of Software Engineering, East China Normal University, Shanghai, 200062, China

    Xiangxue Li

  7. Westone Cryptologic Research Center, Beijing, 100070, China

    Yu Yu & Xiangxue Li

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Correspondence to Yu Yu , Jiang Zhang , Chun Guo or Xiangxue Li .

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  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Yu, Y., Zhang, J., Weng, J., Guo, C., Li, X. (2019). Collision Resistant Hashing from Sub-exponential Learning Parity with Noise. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11922. Springer, Cham. https://doi.org/10.1007/978-3-030-34621-8_1

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