theory DAA_PnC_Unlinkability_Charge_Data_Records
begin

/*
  Protocol:	DAA_PnC
  Properties:	PR3 - Unlinkable CDRs

This Tamarin model is used to verify the privacy of the charge data authentication process
for the Direct Anonymous Authentication (DAA) based privacy extentsion of the
Plug and Charge (PnC) authentication system. The extension is described in the
paper "Integrating Privacy into the Electric Vehicle Charging Architecture".

It is based on the model from the paper "Formal Analysis and Implementation of a TPM 2.0-based Direct Anonymous Attestation Scheme" accepted to ASIACCS 2020 by
Original Authors:
	Liqun Chen, Surrey Centre for Cyber Security, University of Surrey
	Christoper J.P. Newton, Surrey Centre for Cyber Security, University of Surrey
	Ralf Sasse, Department of Computer Science, ETH Zurich
	Helen Treharne, Surrey Centre for Cyber Security, University of Surrey
	Stephan Wesemeyer, Surrey Centre for Cyber Security, University of Surrey
	Jorden Whitefield, Ericsson AB, Finland
cf. https://github.com/tamarin-prover/tamarin-prover/tree/dddaccbe981343dde1a321ce0c908585d4525918/examples/asiaccs20-eccDAA


time tamarin-prover interactive daa_pnc_anonymity_cdrs.spthy\
 --quit-on-warning --diff --heuristic=O\
 --oraclename=ObsEquOracle_cdrs.py +RTS -N8 -RTS

time tamarin-prover daa_pnc_anonymity_cdrs.spthy\
 --quit-on-warning --diff --heuristic=O\
 --oraclename=ObsEquOracle_cdrs.py\
 --prove=diff_signatures_no_verify +RTS -N8 -RTS


==============================================================================
summary of summaries:

analyzed: daa_pnc_anonymity_cdrs.spthy

  RHS :  diff_signatures_no_verify (exists-trace): verified (6 steps)
  LHS :  diff_signatures_no_verify (exists-trace): verified (6 steps)
  DiffLemma:  Observational_equivalence : verified (6331 steps)

==============================================================================

real	12m37,807s
user	33m49,625s
sys	18m24,725s

*/

builtins:   asymmetric-encryption, symmetric-encryption, signing, hashing//, diffie-hellman//, multiset

functions:  MAC/2, KDF_EK/1,KDF_a/3, KDF_e/4, multp/2, plus/2, //len16/1, 
             H_SHA256/1, H_n_8/8, curlyK/1, RB/2, RD/2, PkX/2, PkY/2,
			  E_S/2, H_k_7/7, 
			 H_n_2/2, H_k_2/2, Nonce/1, H_6/1
			 

// Protocol Restrictions (Axioms)

restriction equality: 	     "All #i    x y    .  Eq( x, y ) @ i ==> x = y"

//the 'Issuer' should only be initialised once
restriction single_issuer_single_init:
	"All #i #j . (Issuer_Init() @ i & Issuer_Init() @ j) ==> (#i=#j)"


// Initialisation of the eMSP (the DAA Issuer) and the CCH (acting as CPS)
// we do not allow key reveals for the issuer
rule Issuer_Init:
		let 
			I=$Iss
			pkX=PkX(~x,'P2')
			pkY=PkY(~y,'P2')
		in
		[ Fr(~x)
			, Fr(~y)
			, Fr(~cps)
		]
		--[Issuer_Init()
			, OnlyOnce('Issuer_Init')]->
		[ !Ltk(I,~x, ~y)
			, !Pk(I, pkX,pkY)
			, Out(<pkX,pkY>)
			, !LtkCPS($CPS_I,~cps)
			, !PkCPS($CPS_I, pk(~cps))
			, Out(pk(~cps))
		]

/*
In this model, we install DAA credentials on two EVs. One with TPM1 and one with TPM2. We then obtain a charge
data from the adversary (acting as CP). It is either authenticated by TPM1 or TPM2 (diff property).
The question is: Can the adversary decide whether the CDR was authenticated by TPM1 or TPM2?
*/

// We generate two credential requests, one for TPM1 and one for TPM2
rule EV_Generate_Credential_Requests:
	let
		//inputs from Issuer PK
		pkX=PkX(x,'P2')
		pkY=PkY(y,'P2')
		
		//TPM1 details		
		e1=KDF_EK(~TPM_EK_Seed1)
		pke1=pk(e1)
		E_PD1=<'EK_public_data',pke1>
		PC_PD1=<'PC_public_data',pk(~pc1)>
		Q1=multp(~f1, 'P1')
		Q_PD1=<'DAA_public_data', Q1>

		m1=<pke1,pk(~pc1), Q_PD1, ~res_n1, 'join_Issuer_1'>
		signed_m1=H_SHA256(<m1, pk(cps), n1>) // In(n)
		sig_over_m1=sign(signed_m1,~pc1)

		m_out1=aenc(<sig_over_m1,m1>,pk(cps))

		//TPM2 details		
		e2=KDF_EK(~TPM_EK_Seed2)
		pke2=pk(e2)
		E_PD2=<'EK_public_data',pke2>
		PC_PD2=<'PC_public_data',pk(~pc2)>
		Q2=multp(~f2, 'P1')
		Q_PD2=<'DAA_public_data', Q2>

		m2=<pke2,pk(~pc2), Q_PD2, ~res_n2, 'join_Issuer_1'>
		signed_m2=H_SHA256(<m2, pk(cps), n2>) // In(n)
		sig_over_m2=sign(signed_m2,~pc2)

		m_out2=aenc(<sig_over_m2,m2>,pk(cps))

  in
        [	//Issuer details
			!Pk(I,pkX,pkY)		//the issuer's public key
			, !PkCPS(CPS_I, pk(cps))		//the issuer's public key

			, In(n1)
			, In(n2)
			
			, Fr(~TPM_EK_Seed1)
			, Fr(~pc1)
			, Fr(~f1)
			, Fr(~res_n1)

			, Fr(~TPM_EK_Seed2)
			, Fr(~pc2)
			, Fr(~f2)
			, Fr(~res_n2)
      ]
    --[	Generate_TPM_Keys()
			, OnlyOnce( 'Generate_TPM_Keys' )
	]->	
	 [
		CertReq('req1', m_out1, n1)
		, CertReq('req2', m_out2, n2)
		, TPM_EK_QPD('req1', <pke1, PC_PD1, Q_PD1>)
		, TPM_EK_QPD('req2', <pke2, PC_PD2, Q_PD2>) 
	 ]

// This rule combines the role of the CPS and eMSP in the credential issuing process
// First, the CPS decrypts and validates the request and then the eMSP generates the
// DAA credential for the request
rule Issuer_Issue_Credentials:
	let 
		//inputs
		Q=multp(f, 'P1')
		Q_PD=<'DAA_public_data', Q>
		m=<pke,pk(pc), Q_PD, res_n,'join_Issuer_1'>

		signed_m=H_SHA256(<m, pk(~cps), n>)
		m_in=aenc(<sig,m>,pk(~cps))

		//inputs from Issuer PK
		pkX=PkX(~x,'P2')
		pkY=PkY(~y,'P2')
				
		//new values to be calculated
		A=multp(~r,'P1')
		B=multp(~y,A)
		C=plus(multp(~x,A),multp(multp(multp(~r,~x),~y),Q))
		D=multp(multp(~r,~y),Q)
		
		R_B=RB(~l,'P1')
		R_D=RD(~l,Q)
		
		u=H_n_8('P1', Q, R_B, R_D, A, B, C, D)
		j=plus(~l,multp(multp(~y,~r),u))
		
		//s_2_hat='g'^~s_2_dh //pub ecdhe key
		//s_2_temp=pke^~s_2_dh //Z
		s_2_hat=aenc(~s_2_dh, pke) 
		s_2_temp=~s_2_dh

		s_2=KDF_e(s_2_temp,'IDENTITY',s_2_hat,pke)		
		Q_N=<'SHA256',H_SHA256(Q_PD)>			//the name of the DAA key
		k_e=KDF_a(s_2,'STORAGE',Q_N)				
		k_h=KDF_a(s_2,'INTEGRITY','NULL')
		curlyK_2=curlyK(~K_2)
		curlyK_2_hat=senc(curlyK_2,k_e)
		//curlyH=MAC(<len16(curlyK_2_hat),curlyK_2_hat, Q_N>,k_h)
		curlyH=MAC(<curlyK_2_hat, Q_N>,k_h)
		C_hat=senc(<A,B,C,D,u,j>,curlyK_2)

		seed_3_enc=aenc(~seed_3_dh, pke) 
		seed_3_temp=~seed_3_dh

		seed_3=KDF_e(seed_3_temp,'DUPLICATE',seed_3_enc,pke)		
		sk_SENSITIVE=<'TPM_ALG_KEYEDHASH', 'NULL', ~obfuscationValue, ~sk_emaid>
		sk_unique=H_SHA256(<~obfuscationValue, ~sk_emaid>)
		sk_PD=<'SK_EMAID_public_data', sk_unique>
		sk_N=<'SHA256',H_SHA256(sk_PD)>
		sk_k_e=KDF_a(seed_3,'STORAGE',sk_N)				
		sk_k_h=KDF_a(seed_3,'INTEGRITY','NULL')
		sk_SENSITIVE_enc=senc(sk_SENSITIVE,sk_k_e)
		sk_SENSITIVE_hmac=MAC(<sk_SENSITIVE_enc, sk_N>,sk_k_h)
		sk_DUP=<sk_PD, sk_SENSITIVE_hmac, sk_SENSITIVE_enc, seed_3_enc>

		EMSP_Cert=<I,pkX,pkY>

		m_out=<EMSP_Cert, curlyH, curlyK_2_hat, s_2_hat, C_hat, sk_DUP, res_n, 'Host_CompleteJoin'>
		sig_m=sign(H_SHA256(m_out),~cps)	
	in
     [ CertReq(req, m_in, n)
		, !Pk(I,pkX,pkY)
		, !Ltk(I,~x,~y)
		, Fr(~r)
		, Fr(~l)
		, Fr(~s_2_dh)
		, Fr(~K_2)
		, Fr(~sk_emaid), Fr(~seed_3_dh), Fr(~obfuscationValue) // for import
		, !PkCPS(CPS_I,pk(~cps))
		, !LtkCPS(CPS_I, ~cps)
	 ] 
	 --[ Eq(verify(sig,signed_m,pk(pc)), true)	
	 	, CreateRes(req)
	 	, CreateResSig(sig_m)
		, OnlyOnce(<'Issuer_Verify_Challenge', req>)
		]->
	 [ CertRes(req, m_in, n, m_out, sig_m)
	 ]	

// The CPS receives two credential responses from the eMSP
// one from TPM1 and one from TPM2
// The CPS then signs the two responses and forwards one of them
// to the EV (diff property) together with an additional one of TPM2
// and outputs the public data to the adversary
rule Two_Cert_Res:
	let
		m1=<pke1,pk(~pc1), Q_PD1, res_n1, 'join_Issuer_1'>
		m_in1=aenc(<sig_over_m1,m1>,pk(cps))

		m2=<pke2,pk(~pc2), Q_PD2, res_n2, 'join_Issuer_1'>
		m_in2=aenc(<sig_over_m2,m2>,pk(cps))

		sk_DUP1=<sk_PD1, sk_SENSITIVE_hmac1, sk_SENSITIVE_enc1, seed_3_enc1>
		m_out1=<EMSP_Cert1, curlyH1, curlyK_2_hat1, s_2_hat1, C_hat1, sk_DUP1, res_n1, 'Host_CompleteJoin'>
		sig_m1=sign(H_SHA256(m_out1),cps)
		
		sk_DUP2=<sk_PD2, sk_SENSITIVE_hmac2, sk_SENSITIVE_enc2, seed_3_enc2>
		m_out2=<EMSP_Cert2, curlyH2, curlyK_2_hat2, s_2_hat2, C_hat2, sk_DUP2, res_n2, 'Host_CompleteJoin'>
		sig_m2=sign(H_SHA256(m_out2),cps)
		
		// Difference property: The adversary cannot distinguish whether the
		// charge authorisation request (and the CDR) was generated with TPM1 or TPM2
		Auth_DIFF=diff( <'req1', m_in1, n1, m_out1, sig_m1, <pke1, PC_PD1, Q_PD1>>,
		 				<'req2', m_in2, n2, m_out2, sig_m2, <pke2, PC_PD2, Q_PD2>>)
	in
     [ 
		CertRes('req1', m_in1, n1, m_out1, sig_m1)
		, CertRes('req2', m_in2, n2, m_out2, sig_m2)
		, TPM_EK_QPD('req1',<pke1, PC_PD1, Q_PD1>)
		, TPM_EK_QPD('req2',<pke2, PC_PD2, Q_PD2>) 	
		, !PkCPS(CPS_I,pk(cps))
	] 
	 --[ 
		Eq(verify(sig_m1,H_SHA256(m_out1),pk(cps)), true)	
		, Eq(verify(sig_m2,H_SHA256(m_out2),pk(cps)), true)	
		, Two_Cert_Res()
		, OnlyOnce('Two_Cert_Res')
		]->
	 [
	 	EV_Start_Auth( Auth_DIFF )
		, Out(<'FirstTPM', pke1, PC_PD1, Q_PD1, sk_PD1>)
		, Out(<'SecondTPM', pke2, PC_PD2, Q_PD2, sk_PD2>) 	
	 ]	

// The EV obtains a credential response either for TPM1 or TPM2 (diff property)
// as well as charge data from the adversary (impersonating the CP)
// The EV then uses the obtained credential to authenticate the charge data and sends the
// authenticated data back to the adversary
rule EV_Auth:
	let
		e=KDF_EK(~TPM_EK_Seed)
		//pke1='g'^e1
		pke=pk(e)
		E_PD=<'EK_public_data',pke>
		PC_PD=<'PC_public_data',pk(pc)>
		Q=multp(~f, 'P1')
		Q_PD=<'DAA_public_data', Q>

		i_x=h(<i_x_t, pke>)

		m=<pke,pk(pc), Q_PD, res_n, 'join_Issuer_1'>
		signed_m=H_SHA256(<m, pk(cps), n>) 
		m_in=aenc(<sig_over_m,m>,pk(cps))

		pkX=PkX(x,'P2')
		pkY=PkY(y,'P2')
		EMSP_Cert=<I,pkX,pkY>

		A=multp(r,'P1')
		B=multp(y,A)
		C=plus(multp(x,A),multp(multp(multp(r,x),y),Q))
		D=multp(multp(r,y),Q)	

		curlyK_2_hat=senc(curlyK_2,k_e)
		C_hat=senc(<A,B,C,D,u,j>,curlyK_2)
		sk_SENSITIVE=<'TPM_ALG_KEYEDHASH', 'NULL', obfuscationValue, sk_emaid>
		sk_SENSITIVE_enc=senc(sk_SENSITIVE,sk_k_e)
		sk_DUP=<sk_PD, sk_SENSITIVE_hmac, sk_SENSITIVE_enc, seed_3_enc>
		m_out=<EMSP_Cert, curlyH, curlyK_2_hat, s_2_hat, C_hat, sk_DUP, res_n, 'Host_CompleteJoin'>

		Auth_DIFF=<req, m_in, n, m_out, sig_m, <pke, PC_PD, Q_PD>>

		//Host_Randomise_Credentials
		/*
		bsn='bottom'
		R=multp(~l,A)
		S=multp(~l,B)
		T=multp(~l,C)
		W=multp(~l,D)
		s_2_bar=bsn
		y_2=bsn

		//TPM2_Commit
		E=E_S(~r_cv1,S)*/

		//TPM_Create_Session_Key
		Qk=pk(~g)
		/*
		Qk_PD=<'SessionKey_public_data', Qk>
		Qk_n=<'SHA256',H_SHA256(Qk_PD)>
		Qk_SD=senc(~g,aes_key)

		//Host_Load_Qk_For_Ceritfication
		credData='CredentialData'
		c=H_k_7(credData,R,S,T,W,E, sid)*/
		m_buffer=<'00',i_x>

		//TPM2_Load_And_Certify
		/*N1=QName('SHA256',H_SHA256('root'))
		N2=QName('SHA256',H_SHA256(E_PD))
		N3=H_SHA256(<N1, N2>)
		Qk_QualName=H_SHA256(<N3, Qk_n>)*/
		/*
		curlyA=<'certificationData', Qk_n>//, Qk_n, Qk_QualName>
		credData='CredentialData'
		small_c=H_k_7(credData,R,S,T,W,E, sid)
		h1=H_k_2(small_c, H_6(curlyA))
		n_C=Nonce(~rnd_n_C)
		h2=H_n_2(n_C, h1)
		small_s=plus(~r_cv1, multp(h2, ~f))*/

		//TPM2_HMAC1
		M_id=MAC(m_buffer, sk_emaid)
		X1=h(M_id)

		//Host_Receive_Certified_Q_k
		//sigma_K=<Qk_PD, curlyA, bsn, R, S, T, W, h2, small_s, n_C>
		//auth_m1=<EMSP_Cert, X1,  sigma_K, 'TPM_Certificate_Of_Q_K'>

		//Host_Auth
		m_buffer2=<'01',i_x>

		//TPM2_HMAC2
		M_auth=MAC(m_buffer2, sk_emaid)

		//Host_Auth2
		X2=h(h(<M_auth, nonce_ix>))
		//authH=h(<$CP, nonce, X2>) 

		//TPM2_Sign_SessionKey
		//sig_over_auth=sign(authH,~g)

		//Host_Auth3
		//auth_m2=<authH, sig_over_auth, ~i_x, X2, 'AuthorizationReq'>

		// CP_Verify
		auth_m_emsp=<I, X1, nonce_ix, X2, Qk, 'EMSP_Auth'>

		//EV_DataSign
		dataTBS=h(<'charge_data', dataID, M_auth>)
		dataSig=sign(dataTBS,~g)

		//CP_DataRec
		data_m=<I, 'charge_data', dataID, dataSig>
	in
     [ 
	 	EV_Start_Auth(Auth_DIFF)
		, !PkCPS(CPS_I,pk(cps))
		//, Fr(~l)
		//, Fr(~r_cv1)
		, Fr(~g) 
		, In(i_x_t) //In & onlyonce
		, In(<$CP, sid, <nonce, nonce_ix>, <'charge_data', dataID>>)
		, Fr(~rnd_n_C)
				
	] 
	 --[ 
		Eq(verify(sig_m,H_SHA256(m_out),pk(cps)), true)
		, Eq(verify(sig_over_m,signed_m,pk(pc)), true)
		, EV_Auth()
		, OnlyOnce('EV_Auth')
		]->
	 [
		Out(<auth_m_emsp, data_m>)
	 ]	


lemma diff_signatures_no_verify: exists-trace
"	Ex #t1 #t3 #t4 #t5 #t6 #t7 .
		Issuer_Init() @ t1
		& Generate_TPM_Keys() @ t3
		& CreateRes('req1') @ t4
		& CreateRes('req2') @ t5
		& Two_Cert_Res() @ t6
		& EV_Auth() @ t7
		& #t1<#t3
		& #t3<#t4
		& #t4<#t5
		& #t5<#t6
		& #t6<#t7
		
		//we had no key reveal
		//& not( Ex RevealEvent ENTITY #k1 . KeyReveal(RevealEvent, ENTITY)@k1) 
		
		//restrict rules to only run once in a trace
		& (All event #i #j . OnlyOnce(event)@i & OnlyOnce(event)@j ==> #i=#j)
"

end