# Multiplying between generators degree

I have to elements of multiplicative group of finite field with generator g - $$g^x,g^y$$ Can I get? $$g^{xy}$$

• This is Diffie-Hellman problem. Now, what is your actual question? Research on Diffie-Hellman Problem? Or you are given some instance of it? – kelalaka Nov 11 '19 at 15:07
• @kelalaka, Task is next, I have next equation $$g^(x(y + a))$$. And this equation check on recipient, $$g^x,g^y$$ knows everyone (x and y - knows only me and recipient). "a" is index(from 0 to 20 eg) in database . Pipeline is next: I compute $$g^{(x(y + a))}$$ (a - is my index) on my part and send it to recipient. Recipient iterate over all indexes and solve equation, if some result are equals - user is registered. As I understand. There is no way to send msg by intruder without knowing x and y. – Alex Nov 11 '19 at 15:37
• – kelalaka Nov 11 '19 at 17:13
• Hmmmm, if everyone knows $g^x, g^y$, and you transmit $g^{x(y+a)}$, and someone overhears it, and guesses $a$, then he has a guess to $g^{xy}$, which he can use in a later login attempt. Do you have something that protects against this sort of thing? – poncho Nov 12 '19 at 21:57

The problem 'given a group and the elements $$g, g^x, g^y$$, find $$g^{xy}$$' is known as the Diffie-Hellman problem (or, more precisely, the computational Diffie-Hellman problem).
As for how it might be solved, well, in a finite field multiplicative group, the best way of solving it is to solve the discrete log problem; this is, given $$g, g^x$$, recover $$x$$ (and from there, you can compute $$(g^y)^x = g^{xy}$$. If you are in a group where recovering $$x$$ (or $$y$$) directly is infeasible, then solving the DH problem (to the best of our knowledge) is also infeasible.