@MISC{ASYMPT_3.MIZ,
  AUTHOR = {Okazaki, Hiroyuki},
  TITLE = {Algebra of Polynomially Bounded Sequences and Negligible Functions},
  SECTION1 = {Preliminaries},
  SECTION2 = {Algebra of Polynomially Bounded Sequences},
  SECTION3 = {Negligible Functions},
  DAY = {14},
  MONTH = {August},
  YEAR = {2015},
  ADDRESS1 = {Shinshu University, Nagano, Japan},
  SUMMARY = {In this article, we formalize negligible functions that play an essential role in cryptology \cite{GOLDREICH,BELLARE}.
              Generally, a cryptosystem is secure if the probability of succeeding any attacks against the
              cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences \cite{ASYMPT_2.ABS}. 
              Next, we formalize negligible functions and prove the set of negligible functions is a subset 
             of the algebra of polynomially bounded sequences. Moreover, we then introduce equivalence relation 
             between polynomially bounded sequences, using negligible functions.},
  ACKNOWLEDGEMENT = {The authors would like to express my gratitude to
    Prof. Yuichi Futa and Prof. Yasunari Shidama for their support and encouragement.}}

@BOOK{GOLDREICH,
      AUTHOR = {Oded Goldreich},
      TITLE = {Foundations of Cryptography: Volume 1, Basic Tools},
      PUBLISHER = {Cambridge University Press},
      YEAR = {2001}}

@TECHREPORT{BELLARE,
 AUTHOR={Mihir Bellare},
 TITLE={A Note on Negligible Functions},
 INSTITUTION={University of California at San Diego},
 YEAR=2002
 }}



