One of the problems traditionally used to create
public key cryptography and
digital signature systems. It is a so called 'one way function', meaning that it is function that is easy to compute in one direction, but hard to reverse. It is distinct from
secure hashes in that it is easy to make
zero-knowledge proofs around it.
The Discrete Logarithms problem is based on determining 'e' in the calculation g^e mod p, given only the value of the function, the value of g, and the value of p.
'g' is a generator (random), p is a large prime, and 'e' is an integer. This is believed to be hard.
Discrete Logarithms are definately harder to break than RSA. RSA is vulnerable to calculation of discrete logarithms, advances in factoring, and specific attacks against RSA. An oracle for solving RSA does not help in solving either the discrete logarithm problem or in factoring.
Diffie-Hellman key exchange is based on the discrete logarithms problem; as is ElGamal and DSS. A desciption of how to make a signature system from Discrete Logarithms is in Discrete Logarithm Signatures.