This work was supported by National Natural Science Foundation of China (Grant Nos. 61472310, U1536202, 61672413, 61672415, 61601107, U1708262) and China 111 Project (Grant No. B16037).
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Figure 2
Protocol CAV.
Figure 5
(Color online) FAR under different $O$ in protocol ${\rm~CAV}_{\delta=1}$.
Figure 6
(Color online) FAR and FRR under different $\delta$.
| Notation | Description |
| $V_i$ | The $i$th verifier located at $v_i$ |
| $P_j$ | The $j$th prover located at $p_j$ |
| $A_k$ | The $k$th adversary located at $a_k$ |
| Area$(O,R)$ | The circular area with center $O$ and radius $R$ |
| $X_i$ | The $i$th BRM message |
| $n_i$ | The $i$th random number |
| $C$ | The traveling speed of messages |
| $d(p,q)$ | The distance between position $p$ and position $q$ |
| $F(\cdot)$ | A secure BSM pseudorandom generator |
| $g(\cdot)$ | A secure MAC function |
| Protocol | RB | DB | SP | SPreg | Ours |
| Area verification | $\checkmark$ | $\times$ | $\times$ | $\times$ | $\checkmark$ |
| Batch verification | $\checkmark$ | $\checkmark$ | $\times$ | $\times$ | $\checkmark$ |
| Resist colluding attacks | $\times$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |
| Without pre-shared key | $\checkmark$ | $\times$ | $\checkmark$ | $\times$ | $\checkmark$ |
| Composition security | $-$ | $-$ | $-$ | $\checkmark$ | $\checkmark$ |
| SP | SPreg | Ours | ||||
| Verifier | Prover | Verifier | Prover | Verifier | Prover | |
| Communication overhead | $|X|+|n|$ | $|n|$ | $|X|+2|n|$ | $2|n|$ | $2|X|+|n|$ | $|n|$ |
| Computation overhead | $3F$ | $3F$ | $3F+g$ | $6F+3g$ | $6F+g$ | $6F+g$ |
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