@MISC{GR_CY_3.MIZ,
  AUTHOR = {Arai, Kenichi and Okazaki, Hiroyuki},
  TITLE = {Properties of Primes and Multiplicative Group of a Field},
  SECTION1 = {Properties of Primes},
  SECTION2 = {Multiplicative Group of a Field},
  DAY = {07},
  MONTH = {April},
  YEAR = {2009},
  ADDRESS1 = {Shinshu University\\ Nagano, Japan},
  ADDRESS2 = {Shinshu University\\ Nagano, Japan},
  SUMMARY = {In the \cite{INT_7.ABS} has been proven that the 
  multiplicative group ${\mathbb Z}/p{{\mathbb Z}^\ast}$ is a cyclic 
  group. Likewise, finite subgroup of the multiplicative group of 
  a field is a cyclic group. However, finite subgroup of the 
  multiplicative group of a field being a cyclic group has not 
  yet been proven. Therefore, it is of importance to prove 
  that finite subgroup of the multiplicative group of a field is 
  a cyclic group. \par
  Meanwhile, in cryptographic system like RSA, in which security 
  basis depends upon the difficulty of factorization of 
  given numbers into prime factors, it is important to employ 
  integers that are difficult to be factorized into prime 
  factors. If both $p$ and $2p+1$ are prime numbers, 
  we call $p$ as Sophie Germain prime, and $2p+1$ as 
  safe prime. It is known that the product of two safe 
  primes is a composite number that is difficult for some 
  factoring algorithms to factorize into prime factors. 
  In addition, safe primes are also important in cryptography system 
  because of their use in discrete logarithm based techniques 
  like Diffie-Hellman key exchange. If $p$ is a safe prime, 
  the multiplicative group of numbers modulo $p$ has a subgroup 
  of large prime order. However, no definitions have not been 
  established yet with the safe prime and Sophie Germain prime. 
  So it is important to give definitions of the Sophie Germain prime and 
  safe prime. \par
  In this article, we prove finite subgroup of the multiplicative 
  group of a field is a cyclic group, and, further, define the 
  safe prime and Sophie Germain prime, and prove several facts 
  about them. In addition, we define Mersenne number (${M}_{n}$), 
  and some facts about Mersenne numbers and prime numbers are proven. }}

