# Finding the Shared Key In DHKE

I am trying to work out the shared key between Alice and Bob in DHKE equation. The generator is 7 and the Prime P = 71. Bobs private choice of number is 28 and he also receives g ^ A = 70 from Alice. Alice does not seem to have a private choice of number in this equation, all I know is that g ^ A = 70. So how can I calculate the value of the shared key?

I am just having difficulty working it all out. The modulus arithmetic involved is also confusing me. If someone could try to explain it to me.

I know that it must be: A = 7^x (Mod 71) = 70; and: B = 7^28 (Mod 71);

Then Alice Computes: gB ^ A (Mod 71); and Bob Computes: (gA)^28 (Mod 71);

You are confusing your notation. You are saying that $g^A\equiv 70 \bmod{71}$, but then you are also saying that $A\equiv 7^x\bmod{71}$. That is not true. So let's clarify the notation.
Let $A$ be Alice's private exponent and $B$ be Bob's private exponent.
You know that $B=28$ and you know that $g^A\equiv 70\bmod{71}$. You do not know $A$, however, but you do not need to know $A$.
The shared key between Alice and Bob is ${g^A}^B\equiv {g^B}^A \bmod{p}$. So given what you know, you can compute the shared secret.