# Discrete log problem with modulus prime

I am a bit confused on the hardness of the discrete logarithm problem. Does it become intractale only when it is mod n, where n is a large composite number (Like RSA key). What about if it is mod a prime number p? what is the complexity and method for solving it? Can we reduce the problem of integer factorization to discrete log problem and vice versa?

• DL is hard for prime moduli as long as $n-1$ has certain properties. We typically use primes for Diffie-Hellman or DSA. – CodesInChaos Aug 5 '13 at 8:14

The discrete logarithm problem is usually only considered in prime groups. And there $p-1$ has to have at least one large prime factor.
In general, the discrete logarithm problem could be considered in composite groups, and there you would require the totient $\phi(n)$ to have at least one large prime factor. The reason for this is, that the order of the multiplicative group mod n is $\phi(n)$. In prime order groups this happens to be $p-1$.