Security of Diffie Hellman in specific cyclic group For some $k$, let's say $p = 1+ \prod_{j=1}^k q( j)$, where $q(1)=2$, $q(2)=3$,
if $p$ is prime, the diffie-hellman key exchange is not secure in cyclic group $Z^*_p$. Why?
If $p$ is a prime of the specified structure then the order of the multiplicative group $\mathbb Z_p^*$ is $p-1=\prod_{j=1}^k q( j)$. As you may notice this is a product only consisting of small factors and thereby called smooth.

We actually know an attack on groups with smooth order, called Pohlig-Hellman-Algorithm and the specified structure yields the optimal performance for that algorithm to compute discrete logarithms.