# Can I use computational diffie hellman problem in the following scheme?

Suppose $$X=mnrP , Y=\frac{1}{n}Q, R=e(P,Q)^m$$

$X,Y,P,Q$ are randomly chosen from $G_1$.And only $X,Y$ is given in public.
$m,n,r$ are randomly chosen from $Z_q$.
Anybody who knows $(1/r)$ can easily get $R=e(X,Y)^{\frac{1}{r}}$.
Can I use computational diffie hellman problem by assuming $$X=g^x, Y=g^y$$, then finding $g^yx$.
Because in authenticated identity-based encryption by Ben Lynn, authenticated key $$K= e(H(ID_A),H(ID_B))^s$$ is assumed to be secure if the CDH problem is hard.In this case, $H(ID_A),H(ID_B)$ is declared as public.
Which one is better, guessing problem or computational diffie hellman problem?

Well, the problem "given $X, Y$, recover $R$" is impossible, even to a computationally unbounded adversary. The reason is that any value of the form $e(X, Y)^k$ is possible; for any set of nonzero values for $X, Y, n, m, k$, there exists values for $r, P, Q$ that is consistent with the observed $X, Y$ values, and which $R=e(X,Y)^k$