Alice and Bob agree to use a finite cyclic groups practical for cryptography and one of its generator G.Started here, A=M(a,G) is called the (mapped) image within the cyclic group about its generator G of integer a, and a is the preimage of A. Groups commonly used for cryptography include multiplicative group of integers modulo prime p(in which P is the modular multiplication, M is the modular exponentiation) and additive group of points on an elliptic curve over finite fields(in which P is the modular addition of points, M is the modular multiplication between of a point and an integer). The finite cyclic groups practical for cryptography need to satisfy that it is easy to compute M(m,A) from given m and A, but hard to compute an m satisfying M(m,A)=B from given A and B (usually called the discrete logarithm problem of such finite cyclic group). There are generators G (usually more than one) different with the identity element, making M(n,G)=E, and it can be proved that M(0,G)=E, M(1,G), M(2,G)……M(n-1,G) make up all the elements of the group.įor an arbitrary element, and arbitrary integers x, y, it can be proved that M(y,M(x,A))=M(x,M(y,A))=M(xy,A), which is the basis of DH. The inverse element of identity element E is itself.Īpplying m times operation P to the same element A is defined as a new operation M(m,A), with additional definition M(-m,A)=-M(m,A), thus according to these definitions, P(M(x,A),M(y,A))=M(x+y,A), and M(0,A)=E. P(A,E)=A, and there for each element A exists only one inverse element (-A) making P(A,-A)=E. There is an identity element E in the group, with an arbitrary element A, satisfying The basic group operation P is commutative and associative: P(A,B)=P(B,A), P(P(A,B),C)=P(A,P(B,C)). The number of the elements of a finite cyclic group is finite (as its name), and is called the order of the group. Started here, integers are represented with lowercase letters, while elements of theĬyclic group are represented with uppercase letters, and "=" is used to represent mathematical identity. Standardized deployment procedure of using OTR to protect privacy For Debian/Ubuntu users, based on XMPP IM protocol Principle of OTR protocol Diffie–Hellman (DH) key exchangeĭiffie–Hellman key exchange is performed between integers and a finite cyclic group.
0 kommentar(er)
