I know that there are elliptic curve groups, used in pairing-based cryptography, where the decisional Diffie Hellman problem (ie. given $g$, $g^a$, $g^b$ and $c$, determine if $c = g^{ab}$ is easy but the computational Diffie Hellman problem (given $g$, $g^a$ and $g^b$, find $g^{ab}$) is hard, and I know that if you can solve the discrete log problem (given $g$ and $g^a$ find $a$) you can do both.

But are there groups where you can fairly easily do the Computational Diffie Hellman problem but it's hard to find a discrete log?



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