DSA is based on the difficulty of computing discrete logarithm (see Question 52) and is based on schemes proposed by Schnorr [Sch90] and ElGamal (see Question 29). The algorithm is generally considered secure when the key size is large enough. DSS was originally proposed by NIST with a fixed 512-bit key size. After much criticism that this is not secure enough especially for long-term security, NIST revised DSS to allow key sizes up to 1024 bits.
The particular form of the discrete logarithm problem used in DSA is to compute discrete logarithms in certain subgroups in the finite field GF(p) for some prime p. The problem was first proposed for cryptographic use in 1989 by Schnorr [Sch90]. Even though no attacks have been reported on this form of the discrete logarithm problem, further analysis is necessary to fully understand the difficulty of the problem.
Some researchers warned about the existence of "trapdoor" primes in DSA, which could enable a key to be easily broken. These trapdoor primes are relatively rare, however, and are easily avoided if proper key generation procedures are followed [SB93].