@MISC{ZMODUL01.MIZ,
  AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {$\mathbb Z$-modules},
  SECTION1 = {Definition of $\mathbb Z$-module},
  SECTION2 = {Submodules and Cosets of Submodules in $\mathbb Z$-module},
  SECTION3 = {Operations on Submodules in $\mathbb Z$-module},
  SECTION4 = {Transformation of Abelian Group to $\mathbb Z$-module},
  DAY = {5},
  MONTH = {September},
  YEAR = {2011},
  ADDRESS1 = {Shinshu University, Nagano, JAPAN},
  ADDRESS2 = {Shinshu University, Nagano, JAPAN},
  ADDRESS3 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize
  $\mathbb Z$-module, that is a module over integer ring.
  $\mathbb Z$-module is necassary for lattice problems,
  LLL base reduction algorithm and cryptographic systems
  with lattices \cite{Micci:2002}.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI 22300285.}}

@ARTICLE(Micci:2002,
      AUTHOR  = {Daniele Micciancio and Shafi Goldwasser},
      TITLE   = {Complexity of Lattice Problems:
      A Cryptographic Perspective (The International Series
      in Engineering and Computer Science)},
      PUBLISHER={Springer},
      YEAR    = 2002)
