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Vertical federated learning is considered, where an active party, having access to true class labels, wishes to build a classification model by utilizing more features from a passive party, which has no access to the labels, to improve the model accuracy. In the prediction phase, with logistic regression as the classification model, several inference attack techniques are proposed that the adversary, i.e., the active party, can employ to reconstruct the passive party's features, regarded as sensitive information. These attacks, which are mainly based on a classical notion of the center of a set, i.e., the Chebyshev center, are shown to be superior to those proposed in the literature. Moreover, several theoretical performance guarantees are provided for the aforementioned attacks. Subsequently, we consider the minimum amount of information that the adversary needs to fully reconstruct the passive party's features. In particular, it is shown that when the passive party holds one feature, and the adversary is only aware of the signs of the parameters involved, it can perfectly reconstruct that feature when the number of predictions is large enough. Next, as a defense mechanism, a privacy-preserving scheme is proposed that worsen the adversary's reconstruction attacks, while preserving the full benefits that VFL brings to the active party. Finally, experimental results demonstrate the effectiveness of the proposed attacks and the privacy-preserving scheme.