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Radio Frequency Identification (RFID) technique is gaining increasing popularity in supply chain for the product management. By attaching a tag to each product, a reader can employ an authentication protocol to interrogate the tag's information for verification, which facilitates the automatic processing and monitoring of products in many applications. However, most current solutions cannot be directly used as they cannot balance the tradeoff between the privacy and efficiency for individual parties. In this paper, we design a bidirectional efficiency-privacy transferable (BEST) authentication protocol to address this issue. In a relatively secure domain, BEST works in an efficient manner to authenticate batches of tags with less privacy guarantee. Once the tags flow into open environment, BEST can migrate to provide stronger privacy protection to the tags with moderate efficiency degradation. The analytic result shows that BEST can well adapt to the RFID-enabled supply chain.