# An Anonymous Offline RFID Grouping-Proof Protocol
## 1 Introduction  
RFID grouping-proof technology is a mechanism that can prove a group of tagged items appeared at the same time and the same place [1]. The grouping-proof protocol can be widely adopted to many applications that need coexistence proof to guarantee the items with RFID tags have been scanned simultaneously, such as supply-chain, health care, and evidence in law [2; 3; 4]. For example, in logistics management, we can generate a proof to guarantee the integrity of the container and the goods in it by scanning their tags simultaneously. In the intelligent health care environment, we can validate the correctness of the medicine taking through scanning the patients and their unit-dose medications at the same time and place [5]. In the manufacturing field, a manufacturer of aircraft equipment can certify that a certain part always leaves its factories with a safety cap attached by scanning their RFID tags simultaneously.  
According to the connection method between the reader and the verifier, there are two different modes: online and offline [4]. The online mode requires a stable connection between the reader and the verifier, such as [6; 7]. In this model, the verifier can send and receive messages from a specific tag(via the reader) during the whole protocol execution. This mode has good real-time performance and high security, but the network condition requirement is relatively high. In some application fields, it is difficult to maintain the network connection between the reader and the background. In addition, the consistent network connection should take the energy efficiency into account [8; 9; 10]. On the other hand, in the offline mode, the stable connection between the reader and the background is unnecessary; the reader can collect tag information and generate multiple grouping-proofs without the participation of the verifier. After these processes, the reader can finally send these proof data to the verifier. In this vein, the verifier in offline mode does not need to communicate with any specific tag (via the reader), it only needs the connection before and after the generation of grouping-proof. The connection requirement is more flexible during the protocol, however, there are many security problems need to be solved in this mode, which has become the research focus in many works proposed in the state of the art [3; 4; 11; 12; 13; 14; 15; 16; 17; 18].  
Figure 1 shows a common offline mode of RFID grouping-proof system. The tags are divided into \(M\) groups: {Group\({}_{1}\), Group\({}_{2}\),..., Group\({}_{M}\)}. Each group represents \(n_{i}\) items with RFID tags. The reader receives group information from the verifier and communicates with tags. If it can simultaneously scan all tags in the \(i\)th group, the reader generates a grouping-proof \(G_{i}^{(n_{i})}\). After all groups are scanned, the reader sends\(\{G_{1}^{(n_{1})},\,G_{2}^{(n_{2})},\,\ldots,\,G_{M}^{(n_{M})}\}\) to the verifier. The verifier checks these proofs and stores them as a record. In the grouping-proof protocol, the simultaneous scan means all tags are scanned by a same reader in a short time interval.