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We des ribe eÆ ient onstru tions for various ryptographi primitives in private-key as well as publi -key ryptography. Our major results are two new onstru tions of pseudorandom fun tions. We prove the pseudo-randomness of one onstru tion under the assumption that fa toring (Blum integers) is hard while the other onstru tion is pseudo-random if the de isional version of the DiÆe-Hellman assumption holds. Computing the value of our fun tions at any given point involves two subset produ ts. This is mu h more eÆ ient than previous proposals. Furthermore, these fun tions have the advantage of being in TC 0 (the lass of fun tions omputable by onstant depth ir uits onsisting of a polynomial number of threshold gates). This fa t has several interesting appli ations. The simple algebrai stru ture of the fun tions implies additional features su h as a zero-knowledge proof for statements of the form \y = f s (x)" and \y 6= f s (x)" given a ommitment to a key s of a pseudo-random fun tion f s .