Can you help me understand the Common Modulus Attack in a Lucas Group?
Just find the modular multiplicative inverse $x_C$ of $x_A$ using the extended Euclidean algorithm:

$$(x_A)^{-r}\equiv(x_C)^{|r|}\mod{R}$$

and

$$x_Ax_C \equiv 1\mod{R}$$

Then, once you have $x_C$, you'll be able to compute:

$$(x_C)^{|r|}(x_B)^{s}\mod{R} \equiv M$$

http://en.wikipedia.org/wiki/Modular_exponentiation