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