International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: Learning with Errors and Extrapolated Dihedral Cosets

Zvika Brakerski
Elena Kirshanova
Damien Stehlé
Weiqiang Wen
DOI: 10.1007/978-3-319-76581-5_24
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Conference: PKC 2018
Abstract: 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.
  title={Learning with Errors and Extrapolated Dihedral Cosets},
  booktitle={Public-Key Cryptography – PKC 2018},
  series={Public-Key Cryptography – PKC 2018},
  author={Zvika Brakerski and Elena Kirshanova and Damien Stehlé and Weiqiang Wen},