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Tytuł: Properties of Primes and Multiplicative Group of a Field
Autorzy: Arai, Kenichi
Okazaki, Hiroyuki
Data wydania: 2009
Data dodania: 1-gru-2015
Wydawca: De Gruyter Open
Źródło: Formalized Mathematics, Volume 17, Issue 2, 2009, Pages 151-155
Abstrakt: In the [16] has been proven that the multiplicative group Z/pZ* 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. 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. 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 (Mn), and some facts about Mersenne numbers and prime numbers are proven.
Afiliacja: Arai Kenichi - Shinshu University, Nagano, Japan
Okazaki Hiroyuki - Shinshu University, Nagano, Japan
URI: http://hdl.handle.net/11320/3531
DOI: 10.2478/v10037-009-0017-7
ISSN: 1426-2630
1898-9934
Typ Dokumentu: Article
Występuje w kolekcji(ach):Formalized Mathematics, 2009, Volume 17, Issue 2

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