daa_pnc_anonymity_credential_installation.spthy

theory DAA_PnC_Anonymity_Credential_Installation
begin

/*
  Protocol:	DAA_PnC
  Properties:	Weaker version of PR2 - Anonymity of Credential Installation

This Tamarin model is used to verify the privacy of the installation 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_credential_installation.spthy\
 --quit-on-warning --diff --heuristic=O\
 --oraclename=ObsEquOracle_credential_installation.py +RTS -N8 -RTS

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


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

analyzed: daa_pnc_anonymity_credential_installation.spthy

  RHS :  diff_signatures_no_verify (exists-trace): verified (5 steps)
  LHS :  diff_signatures_no_verify (exists-trace): verified (5 steps)
  DiffLemma:  Observational_equivalence : verified (5200 steps)

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

real	37m11,516s
user	102m59,471s
sys	81m59,863s

*/

builtins:   asymmetric-encryption, symmetric-encryption, signing//, 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
			 

// 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_and_CPS_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 generate two EV credential requests. One from EV1/TPM1 with the public endorsement key pke1
and one from EV2/TPM2 with the public endorsement key pke2. The eMSP then issues credentials for one of these requests.
The adversary obtains the credential request mesage, the issued credential, and the public information of TPM1 and TPM2.
The question is: Can the adversary decide whether the credentials have been issued for TPM1 or TPM2?
*/
rule Generate_TPM_DAA_CERTIFY:
	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>)
		sig_over_m2=sign(signed_m2,~pc2)

		m_out2=aenc(<sig_over_m2,m2>,pk(cps))
		
		// Difference property: The adversary cannot distinguish whether the
		// credential request was generated with TPM1 or TPM2
		CERT_REQ_DIFF=diff(<'req1', m_out1, n1>,
						   <'req2', m_out2, n2>)
  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)
      ]
    --[	CreateSigmas(), 
		OnlyOnce( 'SIGN' )
	]->	
	 [
		  CertReq(CERT_REQ_DIFF)
		, Out(<'FirstTPM', pke1, PC_PD1, Q_PD1>)
		, Out(<'SecondTPM', 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))

		CERT_REQ_DIFF=<req, m_in, n>

		//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))
		
		// We use RSA instead of ECDHE keys to keep the model simple
		s_2_hat=aenc(~s_2_dh, pke) //TODO
		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) //TODO len16
		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) //TODO
		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>

		//TODO len16
		//m_out=<EMSP_Cert, curlyH, len16(curlyK_2_hat), curlyK_2_hat, s_2_hat, C_hat, sk_DUP, res_n, 'Host_CompleteJoin'>
		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(CERT_REQ_DIFF)
		, !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, m_out, sig_m)
	 ]	

// The CPS receives the credential responses from the eMSP
// either from TPM1 or TPM2 (diff property)
// The CPS then signs the response and forwards it to the
// CPO (the adversary in this model) together with the original credential request
rule Two_Res:
	let
		m1=<pke1,pk(~pc1), Q_PD1, res_n1, 'join_Issuer_1'>
		m_in1=aenc(<sig_over_m1,m1>,pk(cps))

		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)
	in
	[
		CertRes(req, m_in1, m_out1, sig_m1)
		, !PkCPS(CPS_I,pk(cps))
	]
	--[ 
		Eq(verify(sig_m1,H_SHA256(m_out1),pk(cps)), true)	
		, Diff_Sigs()
		, OnlyOnce('Two_Res')
	]->
	[ Out(<m_in1, m_out1, sig_m1>) ]


lemma diff_signatures_no_verify: exists-trace
"	Ex #t1 #t3 #t6 .
		Issuer_Init() @ t1
		& CreateSigmas() @ t3

		& ( 
			(Ex #k1 . (CreateRes('req1') @k1) )
			|
			(Ex #k1 . (CreateRes('req2') @k1) )
		  ) 

		& Diff_Sigs() @ t6
		& #t1<#t3
		& #t3<#t6
		
		//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