@MISC{ZMODUL04.MIZ,
  AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {Submodule of free $\mathbb Z$-module},
  SECTION1 = {Vector space of rational field generated by a free $\mathbb$-module},
  SECTION2 = {Submodule of free $\mathbb Z$-module},
  DAY = {31},
  MONTH = {December},
  YEAR = {2013},
  ADDRESS1 = {Japan Advanced Institute of Science and Technology, Ishikawa, JAPAN},
  ADDRESS2 = {Shinshu University, Nagano, JAPAN},
  ADDRESS3 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize a free $\mathbb Z$-module and
  its property.  Specially, we formalize a vector space of rational field
  corresponding to a free $\mathbb Z$-module and prove formally that
  submodules of a free $\mathbb Z$-module are free.
  $\mathbb Z$-module is necassary for lattice problems,
  LLL(Lenstra, Lenstra and Lov\'asz) base reduction algorithm
  and cryptographic systems with lattice \cite{LAT}.
  Some theorems in this article are described by translating
  theorems in \cite{RLVECT_5.ABS} into theorems of
  Z-module, however their proofs are different.},
  ACKNOWLEDGEMENT = {This work was supported by JSPS KAKENHI
  21240001 and 22300285.}}

@ARTICLE(LAT,
      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)
