Cấu trúc tựa đẳng cấu giữa vành đa thức có 2 lớp kề cyclic và trường số

Abstract

The well-known problem of computing discrete logarithms in finite field $GF(p)$ has acquired importance in many studies due to its applicability in cryptography. This is widely thought to be very computationnaly hard if large prime p is selected. Polynomial ring with two cyclotomic cosets $Z_{2}[x]/(x^{n}+1)$ is a special ring with only two idempotent elements. In this paper, a quasi-isomorphic structure between polynomial ring with two cyclotomic cosets $Z,[x]/(x^{n}+1)$ and field $GF(p)$ with only one idempotent element (where $p=2^{n}-1$ is Mersenne primer) is presented. Conclusions on this isomorphism are the result of mathematical analysis and are illustrated by concrete examples. Based on this structure we can construct Discrete logarithm problem over polynomial rings. Discrete logarithm problem over polynomial rings can be used in many cryto-systems (for authentication, digital signatures, encryption.etc..).