@MISC{ZMODUL03.MIZ,
  AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {Free $\mathbb Z$-module},
  SECTION1 = {Free $\mathbb Z$-module},
  SECTION2 = {Finite-rank Free $\mathbb Z$-module},
  SECTION3 = {Rank of a Finite-rank Free $\mathbb Z$-module},
  DAY = {6},
  MONTH = {August},
  YEAR = {2012},
  ADDRESS1 = {Osaka, JAPAN},
  ADDRESS2 = {Shinshu University, Nagano, JAPAN},
  ADDRESS3 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize a free $\mathbb Z$-module
  and its rank.
  We formally prove that for a free finite-rank $\mathbb Z$-module $V$,
  the number of elements in its basis, that is a rank of
  the $\mathbb Z$-module, is constant regardless of selection of its basis.
  $\mathbb Z$-module is necessary 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_3.ABS} and \cite{RLVECT_5.ABS} into theorems of
  $\mathbb Z$-module.},
  NOTE = {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)
