@MISC{VSDIFF_1.MIZ,
  AUTHOR = {Arai, Kenichi and Wakabayashi, Ken and Okazaki, Hiroyuki},
  TITLE = {Difference of Function on Vector Space over $\mathbb{F}$},
  DAY = {26},
  MONTH = {September},
  YEAR = {2014},
  ADDRESS1 = {Tokyo University of Science, Chiba, Japan},
  ADDRESS2 = {Shinshu University, Nagano, Japan},
  ADDRESS3 = {Shinshu University, Nagano, Japan},
  SUMMARY = {In \cite{DIFF_1.ABS}, the definitions of forward difference,
    backward difference, and central difference were formalized.
    Those definitions are formalized as difference of a function from
    $\mathbb{R}$ to $\mathbb{R}$. In this article, we formalize the
    definitions of forward difference, backward difference, and central
    difference of a function on a vector space over $\mathbb{F}$.
    In cryptology, these definitions are very important in evaluating
    the security of cryptographic systems \cite{Lai1994}. Moreover, we
    formalize some facts about these definitions.},
  ACKNOWLEDGEMENT = {We sincerely thank Professor Yasunari Shidama
    for his helpful advices.}}

@ARTICLE{Lai1994,
  AUTHOR = {X. Lai},
  TITLE = {Higher Order Derivatives and Differential Cryptoanalysis},
  JOURNAL = {Communications and Cryptography},
  PAGES = {227--233},
  PUBLISHER = {Kluwer Academic Publishers},
  YEAR = {1994}}

