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**2017/1260**

**Title:**

Collision Resistant Hashing from Sub-exponential Learning Parity with Noise

**Authors:**
Yu Yu, Jiang Zhang, Jian Weng, Chun Guo, Xiangxue Li

**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. Based on the recent work of Applebaum et al. (ITCS 2017), we introduce a general framework for constructing 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) constant-noise LPN is 2^{n^{0.5+\epsilon}}-hard for any constant \epsilon>0; 2) constant-noise LPN is 2^{\Omega(n/\log n)}-hard given q=poly(n) samples; 3) low-noise LPN (of noise rate 1/\sqrt{n}) is 2^{\Omega(\sqrt{n}/\log n)}-hard given q=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 that is (asymptotically) more parallel and efficient than previously known. In particular, assume 2^{n^{0.5+\epsilon}}-hard constant-noise LPN or 2^{n^{0.25+\epsilon}}-hard low-noise LPN, we obtain a polynomially shrinking collision resistant hash function that evaluates in parallel only a single layer of small-domain random functions (or random permutations) and produces their XOR sum as output.

**ePrint:**
https://eprint.iacr.org/2017/1260

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