learn.equalit.ie

Latest revision as of 20:40, 28 July 2014

Procedure Chatroom Init
Input:  $newRoomName$ ,  $participantNick$ 
  global  $myId:=1$ 
  global  $Nick_{{myId}}:=participantNick$ 
  global  $roomName:=newRoomName$ 
  global  $x_{{myId}},y_{{myId}}:=$  Generate Initial Paramters( $myId$ )
  global  $signatureKey_{{myId}}:=(x_{{myId}},y_{{myId}})$ 
   $participantList:=[Nick_{{myId}}]$ 
   $ephemeralPublicPointList:=[y_{{myId}},y_{{other}}]$

Procedure Verify Verifier Generate Init Key
Input:  $schnorrRandomPoint_{{other}}$ ,  $Hv_{{other}}$ ,  $v_{{other}}$ ,  $y_{{other}}$ ,  $Nick_{{other}}$ 
  Verify Verifiers()
  global  $sessionKey:=Hash(x_{{myId}}y_{{other}},sessionId)$ 
   $toBeSigned:=Hash(Hash(sessionId||Hash(y_{1},v_{1})||Hash(y_{2},v_{2})))$ 
  Sign Session and Send( $toBeSigned$ )

Join

Procedure Join
Input:  $newRoomName$ ,  $Nickname_{{myId}}$ ,  $participantId$ 
  global  $myId:=participantId$ 
  global  $roomName:=newRoomName$ 
   $x_{{myId}},y_{{myId}}:=$ Generate Initial Paramters( $myId$ )
  global  $signatureKey_{{myId}}:=(x_{{myId}},y_{{myId}})$ 
  Broadcast(":3mpCat:3Join:3",  $myId$ ,  $Nickname_{{myId}}$ ,  $y_{{myId}}$ )
  global  $participantList,ephemeralPublicPointList:=$  Receive()
  global  $sessionId:=$  Compute Session Id( $roomName$ ,  $participantList$ ,  $ephemeralPublicPointList$ )
  Sign and Send Key Confirmation and Shares()
  Wait On Receive(":3mpCat:3KeyConfirmationShare:3")
  global  $keyShareList,keyConfirmationList,signatureList:=$  Receive()
  Verify Key Confirmations and Signatures( $keyConfirmationList$ ,  $signatureList$ )
  Update Session Key()

Procedure Receive Session Digest
Input:  $currentSessionHistoryDigest$ 
  global  $sessionDigest:=currentSessionHistoryDigest$

Protocol for other participants already in the chat to accept the newcomer

Procedure Accept
Input:  $newParticipant$ 
  Broadcast(":3mpCat:3Join:3",  $myId$ ,  $Nickname_{{myId}}$ ,  $y_{{myId}}$ )
  Wait On Receive(":3mpCat:3Join:3")
  global  $nick_{{NewParticipant}}$ ,  $ephemeralPublicPoint_{{NewParticipant}}:=$  Receive()
  Update Lists( $nick_{{NewParticipant}}$ ,  $ephemeralPublicPoint_{{NewParticipant}}$ )
  global  $sessionId:=$  Compute Session Id( $roomName$ ,  $participantList$ ,  $ephemeralPublicPointList$ )
  Sign and Send Key Confirmation and Shares()
  Wait On Receive(":3mpCat:3KeyConfirmationShare:3")
  global  $keyShareList,keyConfirmationList,signatureList:=$  Receive()
  Verify Key Confirmations and Signatures( $keyConfirmationList$ ,  $signatureList$ )
  Update Session Key()
  Send( $sessionDigest$ )

Farewell

Procedure Shrink on Leave
Input:  $leaverId$ 
  remove  $leaverId$  from  $participantIdList$ 
  global  $sessionId:=$  Compute Session Id()
  if  $|participantList|>1$ , then
    Sign and Send Key Shares()
    Wait On Receive(":3mpCat:3KeyShare:3")
     $keyShareList$  := Receive()
    Update Session Key( $keyShareList$ )

Procedure Sign and Send Key Shares
Input: 
  global  $z_{{myId-1,myId}}:=Hash(k_{{myId,myId-1}},sessionId)$ 
  global  $z_{{myId,myId+1}}:=Hash(k_{{myId,myId+1}},sessionId)$ 
   $keyShare_{{myId}}:=z_{{myId-1,myId}}\oplus z_{{myId,myId+1}}$ 
   $originAuthSignature:=$  ED25519Sign( $SignatureKey$ ,  $sessionId$  ||  $z_{{myId}}$ )
  Broadcast(":3mpCat:3KeyShare:3",  $myId$ ,  $keyShare_{{myId}}$ ,  $originAuthSignature$ )  # we can send this encrypted but leaving person can read it, hence theoretically it is the same as sending it unencrypted.

Send

Procedure Send
Input:  $metaMessage$ ,  $message$ 
   $keyShareMessage$  = NewKeyShareMessage( $metaMessage$ )
   $cryptMessage$  := AES CTR Encrypt( $sessionKey$ , $message|keyShareMessage$ )
   $originAuthSignature$  := ED25519Sign( $SignatureKey$ ,  $sessionId$  ||  $cryptMetatMessage$ )
   $sessionDigest$  := Compute Session Digest( $lastMessage$ )
  Broadcast(":3mpCat:3",  $sessionId$ ,  $cryptMessage$ ,  $sessionDigest$ ,  $originAuthSignature$ ,":3")

Recieve

Procedure Receive
Input:  $sender$ ,  $encryptedMessage$ ,  $originAuthSignature$ ,  $sessionDigest$ 
   $v:=$  ED25519VerifySignature( $ephemeralPublicKeyList[Sender]$ ,  $sessionId||encryptedMessage$ ,  $originAuthSignature$ )
  Assert( $v$ ) or return Reject
   $message,keyShareMessage:=$  AES CTR Decrypt( $sessionKey$ ,  $encryptedMessage$ ){}
   $isMetaMessage=$ UpdateNewKeyStatus( $keyShareMessage$ )
  Verify Digests( $sessionDiges$ )
  return{ $isMetaMessage,message$ }  # isMetaMessage is true if the message is purely meta message and there is nothing to display

Common functions used by other procedures in different stages

Procedure Generate Initial Paramters
Input:  $myId$ 
   $signaturePrivateKey:=$  RandomBits(256)
   $x_{{myId}}:=$  Ed25519 Scalar( $signaturePrivateKey$ )) #{This is both Diffie-Hellman secret and ephemeral signature private key}
   $y_{{myId}}:=x_{{myId}}P$ 
  return  $x,y$

Procedure Verify Key Confirmation and Signatures
Input:  $signatureList$ ,  $keyConfirmationList$ 
  for each  $participant\in participantList$ , do
    if  $keyConfirmationList[participant][myId]\neq Hash(k_{{myId,participant}},U_{{myId}})$ , then
      Halt()
    else if ED25519VerifySignature( $ephemeralPublicKeyList[particicpant]$ ,  $sessionId|keyShares[myId]$ ,  $originAuthSignature$ ) = Fail , then
      Halt()

Procedure Compute Session Id
Input:  $participantList$ ,  $ephemeralPublicPointList$ 
  return  $Hash(roomeName,zip(participantList,ephemeralPublicPointList))$   #  $zip([a,b],[c,d]):=[(a,c),(b,d)]$

Procedure Verify Signatures
Input:  $longPublicList$ , $schnorrRandomPointList$ , 
# standard signature verification

Procedure Sign and Send Key Confirmation and Share
Input:  $schnorrRandomPointList$ 
  for each  $participant\in participantList$ , do
     $k_{{myId,participant}}:=Hash(x_{{myId}}LP_{{participant}}|lp_{{myId}}y_{{participant}}|x_{{myId}}y_{{participant}})$   # Triple DH
     $kc_{{myId}}:=kc_{{myId}}|Hash(k_{{myId,participant}},U_{{participant}})$ 
  global  $z_{{myId-1,myId}}:=Hash(k_{{myId,myId-1}},sessionId)$ 
  global  $z_{{myId,myId+1}}:=Hash(k_{{myId,myId+1}},sessionId)$ 
   $keyShare_{{myId}}:=z_{{myId-1,myId}}\oplus z_{{myId,myId+1}}$ 
   $originAuthSignature:=$  ED25519Sign( $SignatureKey$ ,  $sessionId$  ||  $z_{{myId}}$ )
  Broadcast(":3mpCat:3KeyConfirmationAndShare:3",  $myId$ ,  $keyShare_{{myId}}$ ,  $originAuthSignature$ ,  $kc_{{myId}}$ )

Procedure Update Session Key
Input:  $keyShareList$ 
   $i:=myId$ 
  for each  ${j\in [i,...,i+n-1]}$ , do
      $z_{{j,j+1}}:=z_{{j-1,j}}\oplus keyShareListe[j+1]$ 
    # recovered  $z_{{i-1,i}}$  should be equal to its original value
   global  $sessionKey:=Hash(z_{{j,j+1}}|j\in [1...n])$

Procedure Sign Params Update Session Key
Input:  $toBeSigned$ ,  $signatureList$ , $keyShareList$ 
  Update Session Key()
   $toBeSigned:=Hash(sessionId,||Hash(verifierList,ephemeralPublicPointList,keyShareList)))$ 
   $signature_{{myId}}:=$ Sign Session and Send( $toBeSigned$ )
  Broadcast(":3mpCat:3SignedSessionParameters:3", $signature_{{myId}}$ )

Procedure ComputeSessionDigest
Input:  $lastMessage$ 
  for each  $message$  in Messages Received from  $lastDigestedMessage$ +1 till  $lastMessage$ , do
      $sesionDigest:=Hass(sessionDigest,message)$  
     LRU Cache Store Digest( $sessionDigest$ ,  $message$ )
  return  $sessionDigest$ , $lastMessageId$

Procedure NewKeyShareMessage
Input:  $metaMessage$ 
# Based on metaMessage Determines what type of keyshare needs to be send (Ephemeral point or Group key share) and returen it.

Procedure UpdateNewKeyStatus
Input:  $keyShareMessage$ 
# Update the table of which participant has sent its new ephemeral point or its new group key share

Procedure Hash
Input:  $message$ 
   return SHA-512( $message$ )

learn.equalit.ie

Difference between revisions of "MpOTR/algorithm dump"

Latest revision as of 20:40, 28 July 2014

Contents

Chatroom setup

Join

Protocol for other participants already in the chat to accept the newcomer

Farewell

Send

Recieve

Common functions used by other procedures in different stages

Search

Personal tools

Tools

@@ Line 1: / Line 1: @@
-===Example of an Algorithm in Wiki===
-{{algorithm-begin|name=LargestNumber}}
-   Input: A list of numbers ''L''.
-   Output: The largest number in the list ''L''.
-   ''largest'' ← null
-   '''for each''' ''item'' '''in''' "L"', '''do'''
-     '''if''' ''item'' > ''largest'', '''then'''
-       ''largest'' ← ''item''
-   '''return''' ''largest''
-{{algorithm-end}}
-===mpOTR algorithms===
 ====Chatroom setup====
 <!--alg_chat_setup-->
-{{algorithm-begin|name=Chat Initiator Init}}
+{{algorithm-begin|name=Chatroom Init}}
-  Input: ''newRoomName'', ''participantNick''
+  Input: <math>newRoomName</math>, <math>participantNick</math>
-   Global ''myId := 1''
+   '''global''' <math>myId := 1</math>
-    Global ''Nick_{myId} := participantNick''
+    '''global''' <math>Nick_{myId} := participantNick</math>
-    Global ''roomName := newRoomName''
+    '''global''' <math>roomName := newRoomName</math>
-    Global ''x_{myId}, y_{myId} :='' '''Call''' '''Generate Initial Paramters'''(''myId'')
+    '''global''' <math>x_{myId}, y_{myId} :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Generate Initial Paramters</span>'''(<math>myId</math>)
-    Global ''signatureKey_{myId} := (x_{myId},y_{myId})''
+    '''global''' <math>signatureKey_{myId} := (x_{myId},y_{myId})</math>
-    ''participantList := [Nick_{myId}]''
+    <math>participantList := [Nick_{myId}]</math>
-    ''ephemeralPublicPointList := [y_{myId}, y_{other}]''
+    <math>ephemeralPublicPointList := [y_{myId}, y_{other}]</math>
 {{algorithm-end}}
 {{algorithm-begin|name=Verify Verifier Generate Init Key}}
-  Input: ''schnorrRandomPoint_{other''
+  Input: <math>schnorrRandomPoint_{other}</math>, <math>Hv_{other}</math>, <math>v_{other}</math>, <math>y_{other}</math>, <math>Nick_{other}</math>
-    '''Call''' '''Verify Verifiers'''()
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Verify Verifiers</span>'''()
-    Global ''sessionKey := SHA-512(x_{myId}y_{other}, sessionId)''
+    '''global''' <math>sessionKey := Hash(x_{myId}y_{other}, sessionId)</math>
-    ''toBeSigned := SHA-512(SHA-512(sessionId||SHA-512(y_1, v_1)||SHA-512(y_2,v_2)))''
+    <math>toBeSigned := Hash(Hash(sessionId||Hash(y_1, v_1)||Hash(y_2,v_2)))</math>
-    '''Call''' '''Sign Session and Send'''(''toBeSigned'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Sign Session and Send</span>'''(<math>toBeSigned</math>)
 {{algorithm-end}}
 ====Join====
 <!--alg_join-->
 {{algorithm-begin|name=Join}}
-  Input: ''newRoomName'', ''Nickname_{myId''
+  Input: <math>newRoomName</math>, <math>Nickname_{myId}</math>, <math>participantId</math>
-    Global ''myId := participantId''
+    '''global''' <math>myId := participantId</math>
-    Global ''roomName := newRoomName''
+    '''global''' <math>roomName := newRoomName</math>
-    ''x_{myId}, y_{myId} := '''''Call''' '''Generate Initial Paramters'''(Participant ID ''myId'')
+    <math>x_{myId}, y_{myId} := </math><span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Generate Initial Paramters</span>'''(<math>myId</math>)
-   Global ''signatureKey_{myId} := (x_{myId},y_{myId})''
+   '''global''' <math>signatureKey_{myId} := (x_{myId},y_{myId})</math>
-    '''Call''' '''Broadcast'''(``:3mpCat:3Join:3'', ''myId'', ''Nickname_{myId'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3Join:3", <math>myId</math>, <math>Nickname_{myId}</math>, <math>y_{myId}</math>)
-    Global ''participantList, ephemeralPublicPointList :='' '''Call''' '''Receive'''()
+    '''global''' <math>participantList, ephemeralPublicPointList :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Receive</span>'''()
-    Global ''sessionId := '' '''Call''' '''Compute Session Id'''(''roomName'', ''participantList'', ''ephemeralPublicPointList'')
+    '''global''' <math>sessionId := </math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Compute Session Id</span>'''(<math>roomName</math>, <math>participantList</math>, <math>ephemeralPublicPointList</math>)
-    '''Call''' '''Sign and Send Key Confirmation and Shares'''()
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Sign and Send Key Confirmation and Shares</span>'''()
-    '''Call''' '''Wait On Receive'''(``:3mpCat:3KeyConfirmationShare:3'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Wait On Receive</span>'''(":3mpCat:3KeyConfirmationShare:3")
-    Global ''keyShareList, keyConfirmationList, signatureList :='' '''Call''' '''Receive'''()
+    '''global''' <math>keyShareList, keyConfirmationList, signatureList :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Receive</span>'''()
-    '''Call''' '''Verify Key Confirmations and Signatures'''(''keyConfirmationList'', ''signatureList'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Verify Key Confirmations and Signatures</span>'''(<math>keyConfirmationList</math>, <math>signatureList</math>)
-    '''Call''' '''Update Session Key'''()
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Update Session Key</span>'''()
 {{algorithm-end}}
 {{algorithm-begin|name=Receive Session Digest}}
-  Input: ''currentSessionHistoryDigest''
+  Input: <math>currentSessionHistoryDigest</math>
-   Global ''sessionDigest := currentSessionHistoryDigest''
+   '''global''' <math>sessionDigest := currentSessionHistoryDigest</math>
 {{algorithm-end}}
+====Protocol for other participants already in the chat to accept the newcomer====
-====Protocol for other participants already in the chat to accept the newcomer====
 <!--alg_accept-->
 {{algorithm-begin|name=Accept}}
-  Input: newParticipant
+  Input: <math>newParticipant</math>
-    '''Call''' '''Broadcast'''(``:3mpCat:3Join:3'', ''myId'', ''Nickname_{myId'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3Join:3", <math>myId</math>, <math>Nickname_{myId}</math>, <math>y_{myId}</math>)
-    '''Call''' '''Wait On Receive'''(``:3mpCat:3Join:3'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Wait On Receive</span>'''(":3mpCat:3Join:3")
-    Global ''nick_{NewParticipant}'', ''ephemeralPublicPoint_{NewParticipant} := '' '''Call''' '''Receive'''()
+    '''global''' <math>nick_{NewParticipant}</math>, <math>ephemeralPublicPoint_{NewParticipant} := </math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Receive</span>'''()
-    '''Call''' '''Update Lists'''(''nick_{NewParticipant'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Update Lists</span>'''(<math>nick_{NewParticipant}</math>, <math>ephemeralPublicPoint_{NewParticipant}</math>)
-    Global ''sessionId := '' '''Call''' '''Compute Session Id'''(''roomName'', ''participantList'', ''ephemeralPublicPointList'')
+    '''global''' <math>sessionId := </math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Compute Session Id</span>'''(<math>roomName</math>, <math>participantList</math>, <math>ephemeralPublicPointList</math>)
-    '''Call''' '''Sign and Send Key Confirmation and Shares'''()
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Sign and Send Key Confirmation and Shares</span>'''()
-    '''Call''' '''Wait On Receive'''(``:3mpCat:3KeyConfirmationShare:3'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Wait On Receive</span>'''(":3mpCat:3KeyConfirmationShare:3")
-    Global ''keyShareList, keyConfirmationList, signatureList :='' '''Call''' '''Receive'''()
+    '''global''' <math>keyShareList, keyConfirmationList, signatureList :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Receive</span>'''()
-    '''Call''' '''Verify Key Confirmations and Signatures'''(''keyConfirmationList'', ''signatureList'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Verify Key Confirmations and Signatures</span>'''(<math>keyConfirmationList</math>, <math>signatureList</math>)
-    '''Call''' '''Update Session Key'''()
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Update Session Key</span>'''()
-    '''Call''' '''Send'''(''sessionDigest'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Send</span>'''(<math>sessionDigest</math>)
 {{algorithm-end}}
@@ Line 85: / Line 67: @@
 <!--alg_farewell-->
 {{algorithm-begin|name=Shrink on Leave}}
-  Input: ''leaverId''
+  Input: <math>leaverId</math>
-    Remove ''leaverId'' from ''participantIdList''
+    '''remove''' <math>leaverId</math> from <math>participantIdList</math>
-    Global ''sessionId :='' '''Call''' '''Compute Session Id'''()
+    '''global''' <math>sessionId :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Compute Session Id</span>'''()
-  \If{''|participantList| > 1''}
+   '''if''' <math>|participantList| > 1</math>''', then'''
-   '''Call''' '''Sign and Send Key Shares'''()
+     <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Sign and Send Key Shares</span>'''()
-   '''Call''' '''Wait On Receive'''(``:3mpCat:3KeyShare:3'')
+     <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Wait On Receive</span>'''(":3mpCat:3KeyShare:3")
-   ''keyShareList'' := Receive{}
+     <math>keyShareList</math> := <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Receive</span>'''()
-   '''Call''' '''Update Session Key'''(''keyShareList'')
+     <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Update Session Key</span>'''(<math>keyShareList</math>)
 {{algorithm-end}}
@@ Line 98: / Line 80: @@
 {{algorithm-begin|name=Sign and Send Key Shares}}
   Input:
-    Global ''z_{myId -1, myId} := SHA-512(k_{myId,myId-1}, sessionId)''
+    '''global''' <math>z_{myId -1, myId} := Hash(k_{myId,myId-1}, sessionId)</math>
-    Global ''z_{myId, myId+1} := SHA-512(k_{myId,myId+1}, sessionId)''
+    '''global''' <math>z_{myId, myId+1} := Hash(k_{myId,myId+1}, sessionId)</math>
-    ''keyShare_{myId} := z_{myId -1, myId} \oplus z_{myId, myId+1}''
+    <math>keyShare_{myId} := z_{myId -1, myId} \oplus z_{myId, myId+1}</math>
-    ''originAuthSignature :='' '''Call''' '''ED25519Sign'''(''SignatureKey'', ''sessionId'' || ''z_{myId'')
+    <math>originAuthSignature :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''ED25519Sign</span>'''(<math>SignatureKey</math>, <math>sessionId</math> || <math>z_{myId}</math>)
-    '''Call''' '''Broadcast'''(``:3mpCat:3KeyShare:3'', ''myId'', ''keyShare_{myId'')
+    <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3KeyShare:3", <math>myId</math>, <math>keyShare_{myId}</math>, <math>originAuthSignature</math>)  # we can send this encrypted but leaving person can read it, hence theoretically it is the same as sending it unencrypted.
 {{algorithm-end}}
 ====Send====
 <!--alg_send-->
 {{algorithm-begin|name=Send}}
-  Input: Message
+  Input: <math>metaMessage</math>, <math>message</math>
- ''keyShareMessage'' = '''Call''' '''NewKeyShareMessage'''(MetaMessage)
+   <math>keyShareMessage</math> = <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''NewKeyShareMessage</span>'''(<math>metaMessage</math>)
- ''cryptMessage'' := '''Call''' '''AES CTR Encrypt'''(''sessionKey'',''message | keyShareMessage'')
+   <math>cryptMessage</math> := <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''AES CTR Encrypt</span>'''(<math>sessionKey</math>,<math>message | keyShareMessage</math>)
- ''originAuthSignature'' := '''Call''' '''ED25519Sign'''(''SignatureKey'', ''sessionId'' || ''cryptMetatMessage'')
+   <math>originAuthSignature</math> := <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''ED25519Sign</span>'''(<math>SignatureKey</math>, <math>sessionId</math> || <math>cryptMetatMessage</math>)
- ''sessionDigest'' := '''Call''' '''Compute Session Digest'''(''lastMessage'')
+   <math>sessionDigest</math> := <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Compute Session Digest</span>'''(<math>lastMessage</math>)
- '''Call''' '''Broadcast'''(``:3mpCat:3'', ''sessionId'', ''cryptMessage'', ''sessionDigest'', ''originAuthSignature'',``:3'')
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3", <math>sessionId</math>, <math>cryptMessage</math>, <math>sessionDigest</math>, <math>originAuthSignature</math>,":3")
 {{algorithm-end}}
 ====Recieve====
 <!--alg_recv-->
 {{algorithm-begin|name=Receive}}
-  Input: ''sender'', ''encryptedMessage'', ''originAuthSignature'', ''sessionDigest''
+  Input: <math>sender</math>, <math>encryptedMessage</math>, <math>originAuthSignature</math>, <math>sessionDigest</math>
- ''v := '' '''Call''' '''ED25519VerifySignature'''(''ephemeralPublicKeyList[Sender]'', ''sessionId || encryptedMessage'', ''originAuthSignature'')
+   <math>v := </math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''ED25519VerifySignature</span>'''(<math>ephemeralPublicKeyList[Sender]</math>, <math>sessionId || encryptedMessage</math>, <math>originAuthSignature</math>)
- '''Call''' '''Assert'''(''v'')
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Assert</span>'''(<math>v</math>) or '''return''' Reject
- ''message, keyShareMessage :='' '''Call''' '''AES CTR Decrypt'''(''sessionKey'', ''encryptedMessage'')
+   <math>message, keyShareMessage :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''AES CTR Decrypt</span>'''(<math>sessionKey</math>, <math>encryptedMessage</math>){}
- ''isMetaMessage = '''''Call''' '''UpdateNewKeyStatus'''(''keyShareMessage'')
+   <math>isMetaMessage = </math><span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''UpdateNewKeyStatus</span>'''(<math>keyShareMessage</math>)
- '''Call''' '''Verify Digests'''(''sessionDiges'')
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Verify Digests</span>'''(<math>sessionDiges</math>)
- '''Return'''{''isMetaMessage, message''}
+   '''return'''{<math>isMetaMessage, message</math>}  # isMetaMessage is true if the message is purely meta message and there is nothing to display
 {{algorithm-end}}
-\subsection{Common functions}
 ====Common functions used by other procedures in different stages====
 <!--alg_comm-->
 {{algorithm-begin|name=Generate Initial Paramters}}
-  Input: ''myId''
+  Input: <math>myId</math>
-    ''signaturePrivateKey := '' '''Call''' '''RandomBits'''(256)
+    <math>signaturePrivateKey := </math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''RandomBits</span>'''(256)
-    ''x_{myId} :='' '''Call''' '''Ed25519 Scalar'''(''signaturePrivateKey''))
+    <math>x_{myId} :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Ed25519 Scalar</span>'''(<math>signaturePrivateKey</math>)) #{This is both Diffie-Hellman secret and ephemeral signature private key}
-    ''y_{myId} := x_{myId}P''
+    <math>y_{myId} := x_{myId}P</math>
-  '''Return''' ''x,y''
+   '''return''' <math>x,y</math>
 {{algorithm-end}}
 {{algorithm-begin|name=Verify Key Confirmation and Signatures}}
-  Input: ''signatureList'', ''keyConfirmationList''
+  Input: <math>signatureList</math>, <math>keyConfirmationList</math>
-    '''For each''' ''participant \in participantList''}, '''do'''
+    '''for each''' <math>participant \in participantList</math>, '''do'''
-      \If{''keyConfirmationList[participant][myId] \neq SHA-512(k_{myId,participant}, U_{myId})''}
+      '''if''' <math>keyConfirmationList[participant][myId] \neq Hash(k_{myId,participant} , U_{myId} )</math>''', then'''
-       '''Call''' '''Halt'''()
+       <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Halt</span>'''()
-      ' \If{'''Call''' '''ED25519VerifySignature'''(''ephemeralPublicKeyList[particicpant]'', ''sessionId ||keyShares[myId]'', ''originAuthSignature'')'
+     '''else''' '''if''' <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''ED25519VerifySignature</span>'''(<math>ephemeralPublicKeyList[particicpant]</math>, <math>sessionId | keyShares[myId]</math>, <math>originAuthSignature</math>) = Fail ''', then'''
-       '''Call''' '''Halt'''()
+       <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Halt</span>'''()
 {{algorithm-end}}
 {{algorithm-begin|name=Compute Session Id}}
-  Input: ''participantList'', ''ephemeralPublicPointList''
+  Input: <math>participantList</math>, <math>ephemeralPublicPointList</math>
-    '''Return''' ''SHA-512(roomeName, zip(participantList, ephemeralPublicPointList))'' # ''zip([a,b],[c,d]):=[(a,c),(b,d)]''
+    '''return''' <math>Hash(roomeName, zip(participantList, ephemeralPublicPointList))</math>  # <math>zip([a,b],[c,d]):=[(a,c),(b,d)]</math>
 {{algorithm-end}}
 {{algorithm-begin|name=Verify Signatures}}
-  Input: ''longPublicList'',''schnorrRandomPointList'',
+  Input: <math>longPublicList</math>,<math>schnorrRandomPointList</math>,
+ # standard signature verification
 {{algorithm-end}}
 {{algorithm-begin|name=Sign and Send Key Confirmation and Share}}
-  Input: ''schnorrRandomPointList''
+  Input: <math>schnorrRandomPointList</math>
- '''For each''' ''participant \in participantList'', '''do'''
+   '''for each''' <math>participant \in participantList</math>, '''do'''
-      ''k_{myId, participant} := H(g^{lp_{myId}}LP_{participant}y_{participant}^{x_{myId}})'' # Triple DH
+      <math>k_{myId, participant} := Hash(x_{myId}LP_{participant} |lp_{myId}y_{participant} | x_{myId}y_{participant})</math>  # Triple DH
-      ''kc_{myId} := kc_{myId} | H(k_{myId,participant}, U_{participant})''
+      <math>kc_{myId} := kc_{myId} | Hash(k_{myId,participant}, U_{participant})</math>
+   '''global''' <math>z_{myId -1, myId} := Hash(k_{myId,myId-1}, sessionId)</math>
+   '''global''' <math>z_{myId, myId+1} := Hash(k_{myId,myId+1}, sessionId)</math>
+   <math>keyShare_{myId} := z_{myId -1, myId} \oplus z_{myId, myId+1}</math>
+   <math>originAuthSignature :=</math> <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''ED25519Sign</span>'''(<math>SignatureKey</math>, <math>sessionId</math> || <math>z_{myId}</math>)
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3KeyConfirmationAndShare:3", <math>myId</math>, <math>keyShare_{myId}</math>, <math>originAuthSignature</math>, <math>kc_{myId}</math>)
+{{algorithm-end}}
-   Global ''z_{myId -1, myId} := SHA-512(k_{myId,myId-1}, sessionId)''
+{{algorithm-begin|name=Update Session Key}}
-   Global ''z_{myId, myId+1} := SHA-512(k_{myId,myId+1}, sessionId)''
+ Input: <math>keyShareList</math>
-   ''keyShare_{myId} := z_{myId -1, myId} \oplus z_{myId, myId+1}''
+   <math>i := myId</math>
+   '''for each''' <math>{j \in [i,...,i+n-1]}</math>, '''do'''
+      <math>z_{j,j+1} := z_{j-1,j} \oplus keyShareListe[j+1]</math>
+     # recovered <math>z_{i-1,i}</math> should be equal to its original value
+    '''global''' <math>sessionKey := Hash(z_{j,j+1} | j \in [1...n])</math>
+{{algorithm-end}}
-     ''originAuthSignature :='' '''Call''' '''ED25519Sign'''(''SignatureKey'', ''sessionId'' || ''z_{myId'')
+{{algorithm-begin|name=Sign Params Update Session Key}}
-    '''Call''' '''Broadcast'''(``:3mpCat:3KeyConfirmationAndShare:3'', ''myId'', ''keyShare_{myId'')
+ Input: <math>toBeSigned</math>, <math>signatureList</math>,<math>keyShareList</math>
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Update Session Key</span>'''()
+   <math>toBeSigned := Hash(sessionId, ||Hash(verifierList, ephemeralPublicPointList, keyShareList)))</math>
+    <math>signature_{myId} := </math><span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Sign Session and Send</span>'''(<math>toBeSigned</math>)
+   <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''Broadcast</span>'''(":3mpCat:3SignedSessionParameters:3",<math>signature_{myId}</math>)
 {{algorithm-end}}
-{{algorithm-begin|name=Update Session Key}}
+{{algorithm-begin|name=ComputeSessionDigest}}
-  Input: ''keyShareList''
+  Input: <math>lastMessage</math>
- ''i := myId''
+   '''for each''' <math>message</math> in Messages Received from <math>lastDigestedMessage</math>+1 till <math>lastMessage</math>, '''do'''
-'''For each''' ''{j \in [i,...,i+n-1]}''}, '''do'''
+      <math>sesionDigest := Hass(sessionDigest, message)</math>
-   ''z_{j,j+1} := z_{j-1,j} \oplus keyShareListe[j+1]''
+      <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''LRU Cache Store Digest</span>'''(<math>sessionDigest</math>, <math>message</math>)
- # recovered ''z_{i-1,i}'' should be equal to its original value
+   '''return''' <math>sessionDigest</math>,<math>lastMessageId</math>
- Global ''sessionKey := SHA-512(z_{j,j+1} | j \in [1...n])''
 {{algorithm-end}}
-{{algorithm-begin|name=Sign Params Update Session Key}}
+{{algorithm-begin|name=NewKeyShareMessage}}
-  Input: ''toBeSigned'', ''signatureList'', ''keyShareList''
+  Input: <math>metaMessage</math>
-  '''Call''' '''Update Session Key'''()
+  # Based on metaMessage Determines what type of keyshare needs to be send (Ephemeral point or Group key share) and returen it.
- ''toBeSigned := SHA-512(sessionId, ||SHA-512(verifierList, ephemeralPublicPointList, keyShareList)))''
- ''signature_{myId} := '''''Call''' '''Sign Session and Send'''(''toBeSigned'')
- '''Call''' '''Broadcast'''(``:3mpCat:3SignedSessionParameters:3'',''signature_{myId'')
 {{algorithm-end}}
-{{algorithm-begin|name=ComputeSessionDigest}}
+{{algorithm-begin|name=UpdateNewKeyStatus}}
-  Input: ''lastMessage''
+  Input: <math>keyShareMessage</math>
-'''For each''' ''message'' in Messages Received from ''lastDigestedMessage''+1 till ''lastMessage''}, '''do'''
+ # Update the table of which participant has sent its new ephemeral point or its new group key share
-  ''sesionDigest :='' '''Call''' '''SHA-512'''(''sessionDigest'', ''message'')
+{{algorithm-end}}
- '''Call''' '''LRU Cache Store Digest'''(''sessionDigest'', ''message'')
- '''Return''' ''sessionDigest'',''lastMessageId''
+{{algorithm-begin|name=Hash}}
+  Input: <math>message</math>
+    '''return''' <span style="font-family: serif; font-size: larger; font-variant:small-caps;">'''SHA-512</span>'''(<math>message</math>)
 {{algorithm-end}}