Algorithm 4 Audit (Replay)

Input:  $PP,p{k}_{\mathrm{vrf}},T,Acc,{({\epsilon }_i^{\mathrm{tot}},k_i)}_i,B$ Output:  result, S   1:State$\left[i\right]\leftarrow ({\epsilon }_i^{\mathrm{tot}},re{c}_{\mathrm{init}},0)$ for each i; $BatchQueue\leftarrow ⌀$; $S\leftarrow ⌀$   2:for each $recor{d}_t\in T$ do   3:    $(i,t,o{p}_t,{\theta }_t,y_t,\mathsf{\Delta }{\epsilon }_t,id{x}_t,re{c}_t,met{a}_t,{\pi }_t,r_t,{\pi }_t^{\mathrm{vrf}})\leftarrow recor{d}_t$   4:    $({\epsilon }_{\mathrm{bal}},re{c}_{\mathrm{prev}},cnt)\leftarrow \mathrm{State}\left[i\right]$   5:    if $\mathrm{VRF}.\mathrm{Verify}(p{k}_{\mathrm{vrf}},(i,t),r_t,{\pi }_t^{\mathrm{vrf}})=0$ or $id{x}_t\ne H(i\parallel t\parallel r_t)$ then   6:        return $(\mathrm{fail},recor{d}_t)$   7:    end if   8:    if $re{c}_t\ne H(re{c}_{\mathrm{prev}}\parallel id{x}_t\parallel met{a}_t)$ or $\mathsf{\Delta }{\epsilon }_t\ne \epsilon (o{p}_t,{\theta }_t)$ then   9:        return $(\mathrm{fail},recor{d}_t)$ 10:    end if 11:    ${\epsilon }_{\mathrm{new}}\leftarrow {\epsilon }_{\mathrm{bal}}-\mathsf{\Delta }{\epsilon }_t$ 12:    if ${\epsilon }_{\mathrm{new}}<0$ then 13:        return $(\mathrm{fail},recor{d}_t)$ 14:    end if 15:    $cnt\leftarrow cnt+1$ 16:    if $cnt>k_i$ then 17:        $S\leftarrow S\cup \left\{i\right\}$ 18:    end if 19:    $in{p}_t\leftarrow (i,C_{i,t},o{p}_t,{\theta }_t,y_t,\mathsf{\Delta }{\epsilon }_t,{\epsilon }_t,k_i,id{x}_t,re{c}_t)$ 20:    $\mathrm{Add}(in{p}_t,{\pi }_t)$ to $BatchQueue$; $\mathrm{State}\left[i\right]\leftarrow ({\epsilon }_{\mathrm{new}},re{c}_t,cnt)$ 21:    if $\left|BatchQueue\right|\ge B$ then 22:        $valid\leftarrow \mathrm{ZK}.\mathrm{BatchVerify}(PP,R,BatchQueue)$ 23:        if $valid=0$ then 24:           return $(\mathrm{fail},BatchQueue)$ 25:        end if 26:        $BatchQueue\leftarrow ⌀$ 27:    end if 28: end for 29:if $\mathrm{Size}\left(BatchQueue\right)>0$ and $\mathrm{ZK}.\mathrm{BatchVerify}(PP,R,BatchQueue)=0$ then 30:    return $(\mathrm{fail},BatchQueue)$ 31: end if 32:for each device i do 33:    $({\epsilon }^{*},{\mathsf{\Delta }}^{*})\leftarrow \mathrm{Acc}\left(\mathrm{History}\mathrm{of}i\right)$ 34:    if ${\epsilon }^{*}>{\epsilon }_i^{\mathrm{tot}}$ then 35:        $S\leftarrow S\cup \left\{i\right\}$ 36:    end if 37: end for 38:return $(\mathrm{fail},S)$ if $S\ne ⌀$ else $(\mathrm{pass},⌀)$