We show how to implement cryptographic primitives based on the realistic assumption that quantum storage of qubits is noisy. We thereby consider individual-storage attacks; i.e., the dishonest party attempts to store each incoming qubit separately. Our model is similar to the model of bounded-quantum storage; however, we consider an explicit noise model inspired by present-day technology. To illustrate the power of this new model, we show that a protocol for oblivious transfer is secure for any amount of quantum-storage noise, as long as honest players can perform perfect quantum operations. Our model also allows us to show the security of protocols that cope with noise in the operations of the honest players and achieve more advanced tasks such as secure identification.