Diffie-Hellman problem

Definition
The Diffie-Hellman problem is a problem asked in the context of an encoding of a group or multi-encoding of a group. $$g$$ is an element of the group and $$x,y$$ are integers. The goal is to find the code-word for $$g^{xy}$$ given the code-words for $$g,g^x,g^y$$ (but without explicitly being given $$x$$ and $$y$$).

Note that $$x,y$$ are not uniquely determined from the knowledge of $$g,g^x,g^y$$. Rather, they are known modulo the order of $$g$$. However, $$g^{xy}$$ is uniquely determined by $$g,g^x,g^y$$.

Relation with cryptography
For certain encodings of groups, the Diffie-Hellman problem is believed to be hard. This belief is encoded using the Diffie-Hellman assumption. Under this assumption, a certain key exchange, called Diffie-Hellman key exchange, can be shown to be secure. The goal of this is for two parties to communicate across an insecure channel (subject to eavesdropping) to establish a common key that is, at the end, known only to both the parties, but not to any eavesdroppers.

Solution

 * Reduction of Diffie-Hellman problem to discrete logarithm problem: The problem reduces to the discrete logarithm problem for the same group.