A key building block for collaborative filtering recommender systems is finding users with similar consumption patterns. Given access to the full data regarding the items consumed by each user, one can directly compute the similarity between any two users. However, for massive recommender systems such a naive approach requires a high running time and may be intractable in terms of the space required to store the full data. One way to overcome this is using sketching, a technique that represents massive datasets concisely, while still allowing calculating properties of these datasets. Sketching methods maintain very short fingerprints of the item sets of users, which allow approximately computing the similarity between sets of different users. The state of the art sketch has a very low space complexity, and a recent technique shows how to exponentially speed up the computation time involved in building the fingerprints. Unfortunately, these methods are incompatible, forcing a choice between low running time or a small sketch size. We propose an alternative sketching approach, which achieves both a low space complexity and a low time complexity. We empirically evaluate our algorithm using the Netflix dataset. We analyze the running time and the sketch size of our approach and compare them to alternatives. Further, we show that in practice the accuracy achieved by our approach is even better than the accuracy guaranteed by the theoretical bounds, so it suffices to use even shorter fingerprints to obtain high quality results.