We introduce a new information-theoretic primitive called programmable hash functions(PHFs). PHFs can be used to programthe output of a hash function such that it contains solved or unsolved discrete logarithm instances with a certain probability. This is a technique originally used for security proofs in the random oracle model. We give a variety of standard modelrealizations of PHFs (with different parameters). The programmability of PHFs make them a suitable tool to obtain black-box proofs of cryptographic protocols when considering adaptive attacks. We propose generic digital signature schemes from the strong RSA problem and from some hardness assumption on bilinear maps that can be instantiated with any PHF. Our schemes offer various improvements over known constructions. In particular, for a reasonable choice of parameters, we obtain short standard model digital signatures over bilinear maps.