The round complexity of zero-knowledge protocols is a longstanding open question, yet to be settled under standard assumptions. So far, the question has appeared equally challenging for relaxations such as weak zero-knowledge and witness hiding. Protocols satisfying these relaxed notions under standard assumptions have at least four messages, just like full-fledged zero-knowledge. The difficulty in improving round complexity stems from a fundamental barrier: none of these notions can be achieved in three messages via reductions (or simulators) that treat the verifier as a black box. We introduce a new non-black-box technique and use it to obtain the first protocols that cross this barrier under standard assumptions. We obtain weak zero-knowledge for NP in two messages, assuming the existence of quasipolynomially-secure fully-homomorphic encryption and other standard primitives (known based on the quasipolynomial hardness of Learning with Errors), and subexponentially-secure one-way functions. We also obtain weak zero-knowledge for NP in three messages under standard polynomial assumptions (following for example from fully homomorphic encryption and factoring). We also give, under polynomial assumptions, a two-message witness-hiding protocol for any language L ∈ NP that has a witness encryption scheme. This protocol is publicly verifiable. Our technique is based on a new homomorphic trapdoor paradigm, which can be seen as a non-black-box analog of the classic Feige-Lapidot-Shamir trapdoor paradigm.