The hardness of the learning with errors (LWE) problem is one of the most fruitful resources of modern cryptography. In particular, it is one of the most prominent candidates for secure post-quantum cryptography. Understanding its quantum complexity is therefore an important goal. We show that under quantum polynomial time reductions, LWE is equivalent to a relaxed version of the dihedral coset problem (DCP), which we call extrapolated DCP (eDCP). The extent of extrapolation varies with the LWE noise rate. By considering different extents of extrapolation, our result generalizes Regev's famous proof that if DCP is in BQP (quantum poly-time) then so is LWE (FOCS 02). We also discuss a connection between eDCP and Childs and Van Dam's algorithm for generalized hidden shift problems (SODA 07). Our result implies that a BQP solution for LWE might not require the full power of solving DCP, but rather only a solution for its relaxed version, eDCP, which could be easier.

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We prove that the module learning with errors (M-LWE) problem with arbitrary polynomial-sized modulus p is classically at least as hard as standard worst-case lattice problems, as long as the module rank d is not smaller than the number field degree n. Previous publications only showed the hardness under quantum reductions. We achieve this result in an analogous manner as in the case of the learning with errors (LWE) problem. First, we show the classical hardness of M-LWE with an exponential-sized modulus. In a second step, we prove the hardness of M-LWE using a binary secret. And finally, we provide a modulus reduction technique. The complete result applies to the class of powerof-two cyclotomic fields. However, several tools hold for more general classes of number fields and may be of independent interest.