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Pseudorandom Functions in Almost Constant Depth from Low-Noise LPN

  • Yu YuEmail author
  • John Steinberger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9666)

Abstract

Pseudorandom functions (PRFs) play a central role in symmetric cryptography. While in principle they can be built from any one-way functions by going through the generic HILL (SICOMP 1999) and GGM (JACM 1986) transforms, some of these steps are inherently sequential and far from practical. Naor, Reingold (FOCS 1997) and Rosen (SICOMP 2002) gave parallelizable constructions of PRFs in NC\(^2\) and TC\(^0\) based on concrete number-theoretic assumptions such as DDH, RSA, and factoring. Banerjee, Peikert, and Rosen (Eurocrypt 2012) constructed relatively more efficient PRFs in NC\(^1\) and TC\(^0\) based on “learning with errors” (LWE) for certain range of parameters. It remains an open problem whether parallelizable PRFs can be based on the “learning parity with noise” (LPN) problem for both theoretical interests and efficiency reasons (as the many modular multiplications and additions in LWE would then be simplified to AND and XOR operations under LPN).

In this paper, we give more efficient and parallelizable constructions of randomized PRFs from LPN under noise rate \(n^{-c}\) (for any constant \(0<c<1)\) and they can be implemented with a family of polynomial-size circuits with unbounded fan-in AND, OR and XOR gates of depth \(\omega (1)\), where \(\omega (1)\) can be any small super-constant (e.g., \(\log \log \log {n}\) or even less). Our work complements the lower bound results by Razborov and Rudich (STOC 1994) that PRFs of beyond quasi-polynomial security are not contained in AC\(^0\)(MOD\(_2\)), i.e., the class of polynomial-size, constant-depth circuit families with unbounded fan-in AND, OR, and XOR gates.

Furthermore, our constructions are security-lifting by exploiting the redundancy of low-noise LPN. We show that in addition to parallelizability (in almost constant depth) the PRF enjoys either of (or any tradeoff between) the following:
  • A PRF on a weak key of sublinear entropy (or equivalently, a uniform key that leaks any \((1 - o(1))\)-fraction) has comparable security to the underlying LPN on a linear size secret.

  • A PRF with key length \(\lambda \) can have security up to \(2^{O(\lambda /\log \lambda )}\), which goes much beyond the security level of the underlying low-noise LPN.

where adversary makes up to certain super-polynomial amount of queries.

Notes

Acknowledgments

Yu Yu is more than grateful to Alon Rosen for motivating this work and many helpful suggestions, and he also thanks Siyao Guo for useful comments. The authors thank Ilan Komargodski for pointing out that the domain extension technique from [10] can also be applied to our constructions with improved efficiency. Yu Yu was supported by the National Basic Research Program of China Grant number 2013CB338004, the National Natural Science Foundation of China Grant (Nos. 61472249, 61572192). John Steinberger was funded by National Basic Research Program of China Grant 2011CBA00300, 2011CBA00301, the National Natural Science Foundation of China Grant 61361136003, and by the China Ministry of Education grant number 20121088050.

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Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringShanghai Jiao Tong UniversityShanghaiChina
  2. 2.State Key Laboratory of Information SecurityInstitute of Information Engineering, Chinese Academy of SciencesBeijingChina
  3. 3.State Key Laboratory of CryptologyBeijingChina
  4. 4.Institute for Interdisciplinary Information SciencesTsinghua UniversityBeijingChina

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