@MISC{ZMODUL02.MIZ,
  AUTHOR = {Futa, Yuichi and Okazaki, Hiroyuki and Shidama, Yasunari},
  TITLE = {Quotient Module of $\mathbb Z$-module},
  NOTE = {This work was supported by JSPS KAKENHI 22300285.}
  SECTION1 = {Quotient Module of $\mathbb Z$-module and Vector Space},
  SECTION2 = {Linear Combination of $\mathbb Z$-module},
  SECTION3 = {Linearly Independent Subset of $\mathbb Z$-module},
  SECTION4 = {Theorems Related to Submodule},
  DAY = {7},
  MONTH = {May},
  YEAR = {2012},
  ADDRESS1 = {Shinshu University, Nagano, JAPAN},
  ADDRESS2 = {Shinshu University, Nagano, JAPAN},
  ADDRESS3 = {Shinshu University, Nagano, JAPAN},
  SUMMARY = {In this article, we formalize
 a quotient module of $\mathbb Z$-module and a vector space constructed by
 the quotient module.
 We formally prove that for a $\mathbb Z$-module $V$ and a prime number $p$,
 a quotient module $V / p V$ has the structure of a vector space over
 $\mathbb{F}_p$.
 $\mathbb Z$-module is necessary for lattice problems,
 LLL (Lenstra, Lenstra and Lov\'asz) base reduction algorithm
 and cryptographic systems with lattices \cite{LAT}.
 Some theorems in this article are described by translating
 theorems in \cite{RLVECT_2.ABS} and \cite{RLVECT_3.ABS}
 into theorems of $\mathbb Z$-module.}}

@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}
