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356 lines (356 loc) · 194 KB
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<mxCell id="v15FcyH-daBZKtvoUnET-8" value="<div style="text-wrap-mode: nowrap;"><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; text-wrap-mode: wrap;">Choose current timestamp \(T_{cur}^2\) and check if \(|VT_3-T_{cur}^2|&lt;\triangle t \) ?&nbsp;</span></div><div style="text-wrap-mode: nowrap;"><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; text-wrap-mode: wrap;">If yes, then calculate:</span></div><div style=""><span style="text-wrap-mode: wrap; font-family: &quot;Times New Roman&quot;; font-size: 18px;">\(wt_j^*=H(PID_j||\sigma_j||B_j^*||Ch_1||...Ch_K||VT_0||RID_i||VT_3)||H(T_{\varrho^{a_i}}(PK_j))\), \(D_i=T_{\varrho^{c_i}}(B_j^*)\ mod\ q\).&nbsp;&nbsp;</span></div><div style=""><span style="background-color: transparent; font-size: 18px;"><font face="Times New Roman">Verify the authenticity of each drone's identity using the following formula:</font></span></div><div style=""><font face="Times New Roman"><span style="font-size: 18px;">\(H(T_{\varrho^{c_i+wt_1^*}}(VP_1))||T_{\varrho^{c_i+wt_2^*}}(VP_2)||...||T_{\varrho^{c_i+wt_m^*}}(VP_m) \stackrel{?}{=}H(D_1||D_2||...||D_m)\)</span></font></div>" style="rounded=0;whiteSpace=wrap;html=1;labelBorderColor=none;fillColor=#d5e8d4;strokeColor=#82b366;strokeWidth=2;" parent="CkJQmK-AO9A7YE2EVgkL-2" vertex="1">
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<mxCell id="V3z4afdnI8j49kzJQuYH-2" value="<div style=""><font face="Times New Roman"><span style="font-size: 18px;">&nbsp;\(ECV_i\) then calculates the session keys with these \(m\) drones:</span></font></div><div style="text-wrap-mode: nowrap;"><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; text-wrap-mode: wrap;">\(SK_i^j=H(C_i||D_i||wt_j^*)\),&nbsp;</span></div><div style=""><font face="Times New Roman"><span style="font-size: 18px;">Select partial session key \(psk \in \mathbb{Z}_q^*\) and compute:</span></font></div><div style=""><span style="background-color: transparent; font-size: 18px;"><font face="Times New Roman">\(GSK=T_{\varrho^{H(D_1||D_2||...||D_m||psk)}}(\omega) \ mod\ q\), \(VP_i=T_{\varrho^{c_i-wt_j^*}}(\omega)\ mod\ q\)</font></span></div><div style=""><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">Retrieve the current timestamp \(VT_4\) and&nbsp;</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">broadcast the message to all drones: \(M_4=\{VP_i,VT_4\}\)</span></div>" style="rounded=0;whiteSpace=wrap;html=1;labelBorderColor=none;fillColor=#d5e8d4;strokeColor=#82b366;strokeWidth=2;" parent="V3z4afdnI8j49kzJQuYH-1" vertex="1">
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<mxCell id="V3z4afdnI8j49kzJQuYH-27" value="<div style=""><font face="Times New Roman"><span style="font-size: 18px;">After receiving \(M_4\), each drone selects current timestamp \(T_{cur}^3\) to verify&nbsp;</span></font><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">if \(|VT_4-T_{cur}^3|&lt;\triangle t \) ?&nbsp;</span><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">If yes, then</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">&nbsp;</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">verify the legitimacy of \(ECV_i\)</span><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">:</span></div><div style=""><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">\(H(T_{\varrho^{b_j+wt_j}}(VP_i))\stackrel{?}{=}H(T_{\varrho^{b_j}}(C_i))\), if yes, then calculate:</span></div><div style=""><span style="font-family: &quot;Times New Roman&quot;; font-size: 18px; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">\(D_i^*=T_{\varrho^{b_j}}(C_i)\), \(SK_j^i=H(C_i^*||D_i^*||wt_j)\)</span></div>" style="rounded=0;whiteSpace=wrap;html=1;labelBorderColor=none;fillColor=#d5e8d4;strokeColor=#82b366;strokeWidth=2;" parent="V3z4afdnI8j49kzJQuYH-26" vertex="1">
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<mxCell id="v15FcyH-daBZKtvoUnET-18" value="<span style="text-align: start; white-space: pre;"><font face="Times New Roman" style="font-size: 18px;">Calculate the share </font></span><font face="Times New Roman"><span style="font-size: 18px; white-space: pre;">based on the procedure outlined in Algorithm 1,</span></font><div><font face="Times New Roman"><span style="font-size: 18px; white-space: pre;">compute \(r(x)=E_0+E_1x+E_2x^2+...+E_tx^t \ mod \ q ,x=Ch_1^j\),</span></font></div><div><font face="Times New Roman"><span style="font-size: 18px; white-space: pre;">broadcast \([g^{share_1^1}h^{r(Ch_1^1)},...,g^{share_1^m}h^{r(Ch_1^m)}]\),</span></font></div><div><span style="color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); background-color: transparent; font-family: &quot;Times New Roman&quot;; font-size: 18px;">Retrieve the current timestamp \(VT_5\) and&nbsp;</span><span style="color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); background-color: transparent; font-family: &quot;Times New Roman&quot;; font-size: 18px;">broadcast the message to all drones (</span><span style="background-color: transparent; font-size: 18px;"><font face="Times New Roman">take \(U_j\) as an example for illustration</font></span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">): \(M_5=\{SEN(SK_i^j)[share_1^j||VT_5]\}\)</span></div>" style="rounded=0;whiteSpace=wrap;html=1;fillColor=#f8cecc;strokeColor=#b85450;strokeWidth=2;" parent="V3z4afdnI8j49kzJQuYH-35" vertex="1">
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<mxCell id="v15FcyH-daBZKtvoUnET-28" value="<font face="Times New Roman"><span style="font-size: 18px;">After receiving \(M_5\), each drone computes&nbsp;</span></font><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">\(SDE(SK_j^i)[M_5]=\{share_1^j, VT_5\}\)&nbsp;</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">&nbsp;</span><font style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));" face="Times New Roman"><span style="font-size: 18px;">Selects current timestamp \(T_{cur}^4\) to verify&nbsp;</span></font><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">if \(|VT_5-T_{cur}^4|&lt;\triangle t \) ?&nbsp;</span><div><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">If yes, then</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-family: &quot;Times New Roman&quot;; font-size: 18px;">&nbsp;</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-size: 18px; font-family: &quot;Times New Roman&quot;;">choose&nbsp;</span><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-size: 18px; font-family: &quot;Times New Roman&quot;; text-align: start; white-space: pre;">a polynomial \((z(.))\) of degree \(m-t-2\) </span></div><div><span style="background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255)); font-size: 18px; font-family: &quot;Times New Roman&quot;; text-align: start; white-space: pre;">Check: </span><span style="font-size: 18px; font-family: &quot;Times New Roman&quot;; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">\(\prod_{j=1}^mv[j]^{z(j) \cdot \lambda_j} \stackrel{?}{=}1_{\mathbb{G}}\),&nbsp;</span><span style="font-size: 18px; font-family: &quot;Times New Roman&quot;; background-color: transparent; color: light-dark(rgb(0, 0, 0), rgb(255, 255, 255));">\(v(j) \stackrel{?}{=}g^{share_1^j}h^{r(Ch_1^j)}\)</span></div>" style="rounded=0;whiteSpace=wrap;html=1;fillColor=#FFE6CC;strokeColor=#d6b656;strokeWidth=2;" parent="V3z4afdnI8j49kzJQuYH-54" vertex="1">
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<mxCell id="wgA1eYtuguO1ax2D-Wl9-3" value="<font style="font-size: 18px;" face="Times New Roman">\(p(x)=\sum_{j=1}^{t+1}[share_1^j]\prod_{k=1,k \neq j}^{t+1}(\frac{x-Ch_1^k}{Ch_1^j-Ch_1^k})] \ mod \ q\)</font><div><font face="Times New Roman"><span style="font-size: 18px;">\(GSK=p(0)=D_0=T_{\varrho^{H(D_1||D_2||...||D-m||psk)}}(\omega)\ mod\ q\)</span></font></div>" style="rounded=0;whiteSpace=wrap;html=1;fillColor=#E5CCFF;strokeColor=#10739e;strokeWidth=2;" parent="V3z4afdnI8j49kzJQuYH-57" vertex="1">
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