**Quantum Algorithms for Variants of Average-Case Lattice Problems via Filtering**

*Yilei Chen and Qipeng Liu and Mark Zhandry*

**Abstract: **We show polynomial-time quantum algorithms for the following problems:

(*) Short integer solution (SIS) problem under the infinity norm, where the public matrix is very wide, the modulus is a polynomially large prime, and the bound of infinity norm is set to be half of the modulus minus a constant.

(*) Learning with errors (LWE) problem given LWE-like quantum states with polynomially large moduli and certain error distributions, including bounded uniform distributions and Laplace distributions.

(*) Extrapolated dihedral coset problem (EDCP) with certain parameters.

The SIS, LWE, and EDCP problems in their standard forms are as hard as solving lattice problems in the worst case. However, the variants that we can solve are not in the parameter regimes known to be as hard as solving worst-case lattice problems. Still, no classical or quantum polynomial-time algorithms were known for the variants of SIS and LWE we consider. For EDCP, our quantum algorithm slightly extends the result of Ivanyos et al. (2018).

Our algorithms for variants of SIS and EDCP use the existing quantum reductions from those problems to LWE, or more precisely, to the problem of solving LWE given LWE-like quantum states. Our main contribution is solving LWE given LWE-like quantum states with interesting parameters using a filtering technique.

**Category / Keywords: **public-key cryptography / Lattices, SIS, LWE, Quantum Algorithms

**Date: **received 24 Aug 2021, last revised 6 Oct 2021

**Contact author: **qipengliu0 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20211006:062508 (All versions of this report)

**Short URL: **ia.cr/2021/1093

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