Privacy-preserving summation (PPS) is a competent node positioning technique with privacy protection capability. However, the traditional PPS requires all participating nodes to generate and transmit a set of random interference matrices, which results in excessive network traffic. To address this issue, we propose the Privacy-preserving summation with k (PPS-k). The PPS-k randomly designates k nodes to generate and transmit random interference matrices. The generation process of the interference matrices can be changed by adjusting the value of k, which makes it more flexible than PPS. The node positioning network is composed of several static anchors that know their own positions. The anchors can communicate with each other and send measurements to the target, to help the target positioning. We define different scenarios according to where the measurements are stored and design PPS-k-based node localization protocols for different scenarios. We also propose a notion that uses the ratio of the number of extra equations to the number of unknown scalars as an indicator to evaluate the privacy protection capability of PPS based technique. Compared with the traditional evaluation criteria, the privacy protection rate eliminates the influence of the dimension of privacy information on the evaluation result when evaluating algorithms privacy protection performance. The simulation results validate the efficiency of the proposed methods with PPS-k in adjusting traffic and privacy protection capability.