The Computational Diffie Hellman (CDH) problem is to compute $g^{XY}$ given $g^X$ and $g^Y$ where $g$ generates the group. The Discrete Logarithm (DLOG) problem is to compute $X$ given $g^X$. The latter in $P$ implies the former in $P$.



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> Are there groups where CDH problem is easy and in $P$ while DLOG is difficult and not known to be in $P$?
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Groups in which Computational Diffie Hellman is in $P$ but Discrete Logarithm is not known to be in $P$
Some background for this answer:


1. A computation is *non-uniform* if, in addition to an input $x$, one gets an "advice string" that depends solely on the *length* of $x$. One can view this as precomputation, although this advice string need not be computable (this should not be relevant here though).
In this answer, non-uniformity is used to ensure one has access to parameters of certain elliptic curves, and potentially a factorization of a certain number.
2. The *generic group model* is an abstract model in cryptography where, rather working directly with group elements, one is given oracles for the relevant group operations. Algorithms in the generic group model therefore work for *all* groups with efficiently-computable operations, and do not exploit the particular structure of any single group. There are lower bounds in the generic group model --- solving DLOG in $\Theta(\exp(\sqrt{\log |G|}))$ operations is provably optimal iirc.
The only "mainstream" groups used within cryptography that are plausibly generic are Elliptic Curve groups ($\mathbb{F}\_p^\times$ and related groups admit *non-generic* algorithms that go broadly by the name of "index calculus" algorithms).



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> The paper you quote there requires $|G|$ to be $\mathsf{polylog}|G|$ smooth according to abstract. Are you sure it applies to cyclic multiplicative group mod p a prime? Please post full answer for this group.
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It does not, as $\mathbb{F}\_p^\times$ is not thought to be a "generic" group --- one can speed up DLog via appealing to index calculus methods.
The paper works in the generic group model, so is mainly applicable to groups we think are generic, e.g. elliptic curve groups.


Still, I will include two of the papers results, as your understanding of the paper from the abstract is wrong, and it can handle the case of $|G| = p$ a large prime.


Here is corollary 5 of the paper



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> **Corollary 5**: If the smoothness assumption is true, then there exists a polynomial-time generic algorithm computing discrete logarithms in cyclic groups of order $n$, making calls to a DH oracle for the same group, if and only if all the multiple prime factors of $n$ are of order $(\log n)^{O(1)}$.
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This result is non-uniform, and requires a certain number theoretic assumption.
It clearly handles the case of $|G| = p$ though.


Removing the smoothness assumption (and I believe the non-uniformity assumption, but I have not carefully read), corollary 7 states that



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> **Corollary 7**. Let $P$ be a fixed polynomial, let $G$ be a cyclic group with generator $g$, and let $B := P (\log |G|)$. Then there exists a list of expressions $A(p)$ in $p$ with the following property: if every prime factor $p$ of $|G|$ greater than $B$ is single and if for
> every such prime factor at least one of the expressions $A(p)$ is $B$-smooth, then breaking the DH protocol in $G$ with respect to $g$ is polynomial-time equivalent to computing discrete logarithms in $G$ to the base $g$. The list contains the following expressions:
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> 1. $p − 1 , p + 1 , p + 1 \pm 2a$,
> if $p \equiv 1 (\mod 4)$, where $p = a^2 + b^2$,
> 2. $p + 1 \pm 2a, p + 1 \mp a \pm 2b , p + 1 \pm(a + b)$, if $p \equiv 1 \bmod 3$, where $p = a^2 − ab + b^2, a \equiv 2 \bmod 3$, and $b \equiv 0 \bmod 3$,
> 3. $\frac{(p^k)^l − 1}{p^k − 1} = (p^k)^{l−1} + · · · + p^k + 1$,
> where $k, l = (\log p)^{O(1)}$, and $f(p)$, where $f(x) \in \mathbb{Z}[x]$ is a nonconstant polynomial
> dividing $x^n − 1$ for some $n = O(1)$.
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these technical conditions are related to constructions of what the authors call "auxillary groups", which are the non-uniform information the authors need for their reduction.


This is to say that the result can definitely be improved, for example


1. removing the smoothness assumption for cor. 4, or
2. relaxing the technical conditions with respect to the construction of auxillary groups,


Of main interest to you though, the work is in the generic group model, and the example you have mentioned in the comments is not a generic group.


