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A peerreviewed article of this preprint also exists.
This version is not peerreviewed
Algorithm 1 Directing Process 
Input: input departure, $Dp$; destination,$Ds$
Output: reach $Ds=1$

Notations  Definitions 

$GC$  The Global Controller 
$L{C}_{i}$  Local Controller i 
$OL{D}_{i}$  The ith online learning devices 
G  An elliptic curve cycle additive group 
P  A generator of G 
q  The order of G 
p  The size of a field 
${P}_{pub}$  A public key of GC 
s  A private key of GC 
$NL{C}_{i}$  The number of $NL{C}_{i}$ 
$I{D}_{i}$  The identity of $OL{D}_{i}$ 
$PI{D}_{i}$  The pseudo identity of $OL{D}_{i}$ 
$s{k}_{i}$  A private key of $OL{D}_{i}$ 
$P{K}_{i}$  A public key of $OL{D}_{i}$ 
M  A message 
$msg$  A encapsulated message 
$\sigma $  The signature of a message 
${T}_{t}$  The time stamp 
∥  Concatenation operation 
⊕  ExclusiveOR operation 
Operations  Running Time($/\mathit{m}\mathit{s}$) 

${T}_{bp}$  5.086 
${T}_{bm}$  0.694 
${T}_{ba}$  0.0018 
${T}_{mtp}$  0.0992 
${T}_{em}$  0.3218 
${T}_{ea}$  0.0024 
${T}_{h}$  0.001 
Signature  Batch Authentication  Communication Cost  

[35]  $4{T}_{bm}+{T}_{ba}$  $(n+2){T}_{m}+(3n1){T}_{a}+2n{T}_{h}$  296 bytes 
$+{T}_{mtp}+2{T}_{h}$  
[36]  $3{T}_{bm}+2{T}_{h}$  $(2n+1){T}_{ba}+(2n+1){T}_{bm}$  388 bytes 
$+2{T}_{bp}+\left(n\right){T}_{h}$  
[5]  $1{T}_{em}+2{T}_{h}$  $(2n+2){T}_{em}+\left(n\right){T}_{ea}+\left(2n\right){T}_{h}$  144 bytes 
Our  $1{T}_{em}+{T}_{h}$  $2n{T}_{em}+\left(n\right){T}_{ea}+\left(n\right){T}_{h}$  184 bytes 
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. 
Submitted:
02 June 2023
Posted:
05 June 2023
You are already at the latest version
A peerreviewed article of this preprint also exists.
This version is not peerreviewed
Submitted:
02 June 2023
Posted:
05 June 2023
You are already at the latest version
Algorithm 1 Directing Process 
Input: input departure, $Dp$; destination,$Ds$
Output: reach $Ds=1$

Notations  Definitions 

$GC$  The Global Controller 
$L{C}_{i}$  Local Controller i 
$OL{D}_{i}$  The ith online learning devices 
G  An elliptic curve cycle additive group 
P  A generator of G 
q  The order of G 
p  The size of a field 
${P}_{pub}$  A public key of GC 
s  A private key of GC 
$NL{C}_{i}$  The number of $NL{C}_{i}$ 
$I{D}_{i}$  The identity of $OL{D}_{i}$ 
$PI{D}_{i}$  The pseudo identity of $OL{D}_{i}$ 
$s{k}_{i}$  A private key of $OL{D}_{i}$ 
$P{K}_{i}$  A public key of $OL{D}_{i}$ 
M  A message 
$msg$  A encapsulated message 
$\sigma $  The signature of a message 
${T}_{t}$  The time stamp 
∥  Concatenation operation 
⊕  ExclusiveOR operation 
Operations  Running Time($/\mathit{m}\mathit{s}$) 

${T}_{bp}$  5.086 
${T}_{bm}$  0.694 
${T}_{ba}$  0.0018 
${T}_{mtp}$  0.0992 
${T}_{em}$  0.3218 
${T}_{ea}$  0.0024 
${T}_{h}$  0.001 
Signature  Batch Authentication  Communication Cost  

[35]  $4{T}_{bm}+{T}_{ba}$  $(n+2){T}_{m}+(3n1){T}_{a}+2n{T}_{h}$  296 bytes 
$+{T}_{mtp}+2{T}_{h}$  
[36]  $3{T}_{bm}+2{T}_{h}$  $(2n+1){T}_{ba}+(2n+1){T}_{bm}$  388 bytes 
$+2{T}_{bp}+\left(n\right){T}_{h}$  
[5]  $1{T}_{em}+2{T}_{h}$  $(2n+2){T}_{em}+\left(n\right){T}_{ea}+\left(2n\right){T}_{h}$  144 bytes 
Our  $1{T}_{em}+{T}_{h}$  $2n{T}_{em}+\left(n\right){T}_{ea}+\left(n\right){T}_{h}$  184 bytes 
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. 
Jiming Yin
et al.
,
2023
Andreas Plageras
et al.
,
2024
Nurul Fatini Azhar
et al.
,
2020
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