# Novel Designated Ownership Transfer with Grouping Proof
## 2 Proposed Protocol
### Initialization Phase  
When a cargo shipment with a tag collection \(T^{q}\) is delivered to the recipient \(P^{q}\), the reader requests the verifier to establish a secure multicast connection to ensure that the generated grouping proofs are in accordance with those on the recipient's reader and that the message can be transmitted to \(\delta q\) tags in \(T^{q}\). Accordingly, the verifier generates a k-ary group key with a subtree height difference of \(\leq 1(h_{max}\!=\!\left\lceil\log_{k}\left(\delta q/k\right)\right\rceil)\) by using the secret key \(Kt_{i}^{q}\), where \(Kt_{i}^{q}\) is shared between the verifier and the tags \(TID_{i}^{q}\). In summary, the group keys that can transmit messages to \(\delta\)q tags in \(T^{q}\) are defined as \(GK_{0}^{q}\). Figure 1 shows an example of a 3-ary group key (\(GK_{0}^{q}\) is the starting node) generated for a set of 23 tags; the group key \(GK_{1}^{q}\) is employed to encrypt the multicast messages transmitted to the tags numbered from \(TID_{1}^{q}\) to \(TID_{q}^{q}\), and tags \(TID_{1}^{q}\), \(TID_{2}^{q}\), and \(TID_{3}^{q}\) can decrypt the multicast messagesencrypted with the group key \(GK_{4}^{q}\) by using their own shared keys \(Kt_{1}^{q}\), \(Kt_{2}^{q}\), and \(Kt_{3}^{q}\), respectively. The! details of group key generation can be found in [PERSON] and [PERSON]'s proposed methods [17].  
Moreover, a verification code \(TA^{q}\) is generated for \(\delta q\) tags in \(T^{q}\) according to Equation (1).  
\[TA^{q}=TH_{1}^{q}\parallel\ldots\parallel TH_{\delta q}^{q},\text{where }\forall \text{i}\text{ TH}_{\text{i}}^{q}=\text{H}(\text{TID}_{\text{i}}^{q}\parallel K _{\text{i}}^{q}\parallel\text{ TS}). \tag{1}\]  
After the verifier has generated the verification codes \(TA^{q}\), group key \(GK_{0}^{0}\), and timestamp \(TS_{v}\), they are then transmitted to the transporter's reader to be forwarded to the \(n\) recipients' readers.  
Figure 1: The 3-ary key tree for a group of tags.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(A\) & a transporter who delivers cargo \\ \(T^{q}\) & a cargo shipment with a tag collection \\ \(P^{q}\) & the \(q\)th recipient who receives the cargo \\ \(AID\) & an identification code for \(A\) \\ \(RID_{0}\) & an identification code of the reader used by \(A\) \\ \(PID^{q}\) & an identification code of \(P^{q}\) \\ \(RID^{q}_{i}\) & an identification code of the \(j\)th reader used by \(P^{q}\) \\ \(TID^{q}_{i}\) & an identification code of the \(i\)th tag for \(P^{q}\) \\ \(TH^{q}_{i}\) & a hash value for verifying \(TID^{q}_{i}\) \\ \(Kt^{q}_{i}\) & a secret key shared between \(RID^{q}_{i}\) and \(V\) \\ \(Kt^{q}_{i}\) & a secret key shared between \(TID^{q}_{i}\) and \(V\) \\ \(Ky^{q}_{i}\) & a secret key shared between \(PID^{q}\) and \(TID^{q}_{i}\) \\ \(GK_{4}\) & a secret key shared between \(G^{q}_{i}\) and \(V\) \\ \(SK^{q}_{i}\) & a session key shared among readers \\ \(PK^{q}/PR^{q}\) & a public/private key pair for \(a\) \\ \(PK^{q}/PR^{q}\) & a public/private key pair \(P^{q}\) \\ \(Nr_{i}^{q}\) & a random number generated by \(RID^{q}_{i}\) \\ \(Nt_{i}^{q}\) & a random number generated by \(TID^{q}_{i}\) \\ \(Na\) & a random number generated by \(a\) \\ \(Np^{q}\) & a random number generated by \(P^{q}\) \\ \(TS_{v}\) & a timestamp generated by \(V\) \\ \(E(key,Msg)\) & an encryption function with two inputs: the message ( \(Msg\)) and the symmetric key (\(key\)) \\ \(Sign(key,Msg)\) & a signing function with two inputs \(Msg\) and \(key\) \\ \(MAC(key,Msg)\) & a key-hashing function for generating message authentication codes, \\  & where the inputs are \(Msg\) and \(Key\) \\ \(H(Msg)\) & a hashing function with an input \(Msg\) \\ \(OT\) & a ownership transfer protocol \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbol notations.