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Vertical Federated Learning (FL) is a new paradigm that enables users with non-overlapping attributes of the same data samples to jointly train a model without directly sharing the raw data. Nevertheless, recent works show that it's still not sufficient to prevent privacy leakage from the training process or the trained model. This paper focuses on studying the privacy-preserving tree boosting algorithms under the vertical FL. The existing solutions based on cryptography involve heavy computation and communication overhead and are vulnerable to inference attacks. Although the solution based on Local Differential Privacy (LDP) addresses the above problems, it leads to the low accuracy of the trained model. This paper explores to improve the accuracy of the widely deployed tree boosting algorithms satisfying differential privacy under vertical FL. Specifically, we introduce a framework called OpBoost. Three order-preserving desensitization algorithms satisfying a variant of LDP called distance-based LDP (dLDP) are designed to desensitize the training data. In particular, we optimize the dLDP definition and study efficient sampling distributions to further improve the accuracy and efficiency of the proposed algorithms. The proposed algorithms provide a trade-off between the privacy of pairs with large distance and the utility of desensitized values. Comprehensive evaluations show that OpBoost has a better performance on prediction accuracy of trained models compared with existing LDP approaches on reasonable settings. Our code is open source. 

We initiate the study of witness authenticating NIZK proof systems (waNIZKs), in which one can use a witness w of a statement x to identify whether a valid proof for x is indeed generated using w. Such a new identification functionality enables more diverse applications, and it also puts new requirements on soundness that: (1) no adversary can generate a valid proof that will not be identified by any witness; (2) or forge a proof using her valid witness to frame others. To work around the obvious obstacle towards conventional zero-knowledgeness, we define entropic zero-knowledgeness that requires the proof to leak no partial information, if the witness has sufficient computational entropy. We give a formal treatment of this new primitive. The modeling turns out to be quite involved and multiple subtle points arise and particular cares are required. We present general constructions from standard assumptions. We also demonstrate three applications in non-malleable (perfectly one-way) hash, group signatures with verifier-local revocations and plaintext-checkable public-key encryption. Our waNIZK provides a new tool to advance the state of the art in all these applications.