src/HOL/Auth/ZhouGollmann.thy

ZhouGollmann: new example (fair non-repudiation protocol)

(*  Title:      HOL/Auth/ZhouGollmann
    ID:         $Id$
    Author:     Giampaolo Bella and L C Paulson, Cambridge Univ Computer Lab
    Copyright   2003  University of Cambridge

The protocol of
  Jianying Zhou and Dieter Gollmann,
  A Fair Non-Repudiation Protocol,
  Security and Privacy 1996 (Oakland)
  55-61
*)

theory ZhouGollmann = Public:

syntax
  TTP :: agent

translations
  "TTP" == "Server"

syntax
  f_sub :: nat
  f_nro :: nat
  f_nrr :: nat
  f_con :: nat

translations
  "f_sub" == "5"
  "f_nro" == "2"
  "f_nrr" == "3"
  "f_con" == "4"


constdefs
  broken :: "agent set"    
    --{*the compromised honest agents; TTP is included as it's not allowed to
        use the protocol*}
   "broken == insert TTP (bad - {Spy})"

declare broken_def [simp]

consts  zg  :: "event list set"

inductive zg
  intros

  Nil:  "[] \<in> zg"

  Fake: "[| evsf \<in> zg;  X \<in> synth (analz (spies evsf)) |]
	 ==> Says Spy B X  # evsf \<in> zg"

Reception:  "[| evsr \<in> zg; A \<noteq> B; Says A B X \<in> set evsr |]
	     ==> Gets B X # evsr \<in> zg"

  (*L is fresh for honest agents.
    We don't require K to be fresh because we don't bother to prove secrecy!
    We just assume that the protocol's objective is to deliver K fairly,
    rather than to keep M secret.*)
  ZG1: "[| evs1 \<in> zg;  Nonce L \<notin> used evs1; C = Crypt K (Number m);
	   K \<in> symKeys;
	   NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|}|]
       ==> Says A B {|Number f_nro, Agent B, Nonce L, C, NRO|} # evs1 \<in> zg"

  (*B must check that NRO is A's signature to learn the sender's name*)
  ZG2: "[| evs2 \<in> zg;
	   Gets B {|Number f_nro, Agent B, Nonce L, C, NRO|} \<in> set evs2;
	   NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|};
	   NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|}|]
       ==> Says B A {|Number f_nrr, Agent A, Nonce L, NRR|} # evs2  \<in>  zg"

  (*K is symmetric must be repeated IF there's spy*)
  (*A must check that NRR is B's signature to learn the sender's name*)
  (*without spy, the matching label would be enough*)
  ZG3: "[| evs3 \<in> zg; C = Crypt K M; K \<in> symKeys;
	   Says A B {|Number f_nro, Agent B, Nonce L, C, NRO|} \<in> set evs3;
	   Gets A {|Number f_nrr, Agent A, Nonce L, NRR|} \<in> set evs3;
	   NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|};
	   sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|}|]
       ==> Says A TTP {|Number f_sub, Agent B, Nonce L, Key K, sub_K|}
	     # evs3 \<in> zg"

 (*TTP checks that sub_K is A's signature to learn who issued K, then
   gives credentials to A and B.  The Notes event models the availability of
   the credentials, but the act of fetching them is not modelled.*)
 (*Also said TTP_prepare_ftp*)
  ZG4: "[| evs4 \<in> zg; K \<in> symKeys;
	   Gets TTP {|Number f_sub, Agent B, Nonce L, Key K, sub_K|}
	     \<in> set evs4;
	   sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
	   con_K = Crypt (priK TTP) {|Number f_con, Agent A, Agent B,
				      Nonce L, Key K|}|]
       ==> Notes TTP {|Number f_con, Agent A, Agent B, Nonce L, Key K, con_K|}
	     # evs4 \<in> zg"


declare Says_imp_knows_Spy [THEN analz.Inj, dest]
declare Fake_parts_insert_in_Un  [dest]
declare analz_into_parts [dest]

declare symKey_neq_priEK [simp]
declare symKey_neq_priEK [THEN not_sym, simp]


subsection {*Basic Lemmas*}

lemma Gets_imp_Says:
     "[| Gets B X \<in> set evs; evs \<in> zg |] ==> \<exists>A. Says A B X \<in> set evs"
apply (erule rev_mp)
apply (erule zg.induct, auto)
done

lemma Gets_imp_knows_Spy:
     "[| Gets B X \<in> set evs; evs \<in> zg |]  ==> X \<in> spies evs"
by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)


text{*Lets us replace proofs about @{term "used evs"} by simpler proofs 
about @{term "parts (spies evs)"}.*}
lemma Crypt_used_imp_spies:
     "[| Crypt K X \<in> used evs;  K \<noteq> priK TTP; evs \<in> zg |]
      ==> Crypt K X \<in> parts (spies evs)"
apply (erule rev_mp)
apply (erule zg.induct)
apply (simp_all add: parts_insert_knows_A) 
done

lemma Notes_TTP_imp_Gets:
     "[|Notes TTP {|Number f_con, Agent A, Agent B, Nonce L, Key K, con_K |}
           \<in> set evs;
        sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
        evs \<in> zg|]
    ==> Gets TTP {|Number f_sub, Agent B, Nonce L, Key K, sub_K|} \<in> set evs"
apply (erule rev_mp)
apply (erule zg.induct, auto)
done

text{*For reasoning about C, which is encrypted in message ZG2*}
lemma ZG2_msg_in_parts_spies:
     "[|Gets B {|F, B', L, C, X|} \<in> set evs; evs \<in> zg|]
      ==> C \<in> parts (spies evs)"
by (blast dest: Gets_imp_Says)

(*classical regularity lemma on priK*)
lemma Spy_see_priK [simp]:
     "evs \<in> zg ==> (Key (priK A) \<in> parts (spies evs)) = (A \<in> bad)"
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, auto)
done

text{*So that blast can use it too*}
declare  Spy_see_priK [THEN [2] rev_iffD1, dest!]

lemma Spy_analz_priK [simp]:
     "evs \<in> zg ==> (Key (priK A) \<in> analz (spies evs)) = (A \<in> bad)"
by auto 


subsection{*About NRO*}

text{*Below we prove that if @{term NRO} exists then @{term A} definitely
sent it, provided @{term A} is not broken.  *}

text{*Strong conclusion for a good agent*}
lemma NRO_authenticity_good:
     "[| NRO \<in> parts (spies evs);
         NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|};
         A \<notin> bad;  evs \<in> zg |]
     ==> Says A B {|Number f_nro, Agent B, Nonce L, C, NRO|} \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, auto)  
done

text{*A compromised agent: we can't be sure whether A or the Spy sends the
message or of the precise form of the message*}
lemma NRO_authenticity_bad:
     "[| NRO \<in> parts (spies evs);
         NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|};
         A \<in> bad;  evs \<in> zg |]
     ==> \<exists>A' \<in> {A,Spy}. \<exists>C Y. Says A' C Y \<in> set evs & NRO \<in> parts {Y}"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt{*ZG3*}
   prefer 4 apply blast
txt{*ZG2*}
   prefer 3 apply blast
txt{*Fake*}
apply (simp add: parts_insert_knows_A, blast) 
txt{*ZG1*}
apply (auto intro!: exI)
done

theorem NRO_authenticity:
     "[| NRO \<in> used evs;
         NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|};
         A \<notin> broken;  evs \<in> zg |]
     ==> \<exists>C Y. Says A C Y \<in> set evs & NRO \<in> parts {Y}"
apply auto
 apply (force dest!: Crypt_used_imp_spies NRO_authenticity_good)
apply (force dest!: Crypt_used_imp_spies NRO_authenticity_bad)
done


subsection{*About NRR*}

text{*Below we prove that if @{term NRR} exists then @{term B} definitely
sent it, provided @{term B} is not broken.*}

text{*Strong conclusion for a good agent*}
lemma NRR_authenticity_good:
     "[| NRR \<in> parts (spies evs);
         NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|};
         B \<notin> bad;  evs \<in> zg |]
     ==> Says B A {|Number f_nrr, Agent A, Nonce L, NRR|} \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, auto)  
done

lemma NRR_authenticity_bad:
     "[| NRR \<in> parts (spies evs);
         NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|};
         B \<in> bad;  evs \<in> zg |]
     ==> \<exists>B' \<in> {B,Spy}. \<exists>C Y. Says B' C Y \<in> set evs & NRR \<in> parts {Y}"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies)
apply (simp_all del: bex_simps)
txt{*ZG3*}
   prefer 4 apply blast
txt{*Fake*}
apply (simp add: parts_insert_knows_A, blast)
txt{*ZG1*}
apply (auto simp del: bex_simps)
txt{*ZG2*}
apply (force intro!: exI)
done

theorem NRR_authenticity:
     "[| NRR \<in> used evs;
         NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|};
         B \<notin> broken;  evs \<in> zg |]
     ==> \<exists>C Y. Says B C Y \<in> set evs & NRR \<in> parts {Y}"
apply auto
 apply (force dest!: Crypt_used_imp_spies NRR_authenticity_good)
apply (force dest!: Crypt_used_imp_spies NRR_authenticity_bad)
done


subsection{*Proofs About @{term sub_K}*}

text{*Below we prove that if @{term sub_K} exists then @{term A} definitely
sent it, provided @{term A} is not broken.  *}

text{*Strong conclusion for a good agent*}
lemma sub_K_authenticity_good:
     "[| sub_K \<in> parts (spies evs);
         sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
         A \<notin> bad;  evs \<in> zg |]
     ==> Says A TTP {|Number f_sub, Agent B, Nonce L, Key K, sub_K|} \<in> set evs"
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt{*Fake*} 
apply (blast dest!: Fake_parts_sing_imp_Un)
done

text{*A compromised agent: we can't be sure whether A or the Spy sends the
message or of the precise form of the message*}
lemma sub_K_authenticity_bad:
     "[| sub_K \<in> parts (spies evs);
         sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
         A \<in> bad;  evs \<in> zg |]
     ==> \<exists>A' \<in> {A,Spy}. \<exists>C Y. Says A' C Y \<in> set evs & sub_K \<in> parts {Y}"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies)
apply (simp_all del: bex_simps)
txt{*Fake*}
apply (simp add: parts_insert_knows_A, blast)
txt{*ZG1*}
apply (auto simp del: bex_simps)
txt{*ZG3*}
apply (force intro!: exI)
done

theorem sub_K_authenticity:
     "[| sub_K \<in> used evs;
         sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
         A \<notin> broken;  evs \<in> zg |]
     ==> \<exists>C Y. Says A C Y \<in> set evs & sub_K \<in> parts {Y}"
apply auto
 apply (force dest!: Crypt_used_imp_spies sub_K_authenticity_good)
apply (force dest!: Crypt_used_imp_spies sub_K_authenticity_bad)
done


subsection{*Proofs About @{term con_K}*}

text{*Below we prove that if @{term con_K} exists, then @{term TTP} has it,
and therefore @{term A} and @{term B}) can get it too.  Moreover, we know
that @{term A} sent @{term sub_K}*}

lemma con_K_authenticity:
     "[|con_K \<in> used evs;
        con_K = Crypt (priK TTP)
                  {|Number f_con, Agent A, Agent B, Nonce L, Key K|};
        evs \<in> zg |]
    ==> Notes TTP {|Number f_con, Agent A, Agent B, Nonce L, Key K, con_K|}
          \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt{*Fake*}
apply (blast dest!: Fake_parts_sing_imp_Un)
txt{*ZG2*}
apply (blast dest: parts_cut)
done

text{*If @{term TTP} holds @{term con_K} then @{term A} sent
 @{term sub_K}.  We assume that @{term A} is not broken.  Nothing needs to
 be assumed about the form of @{term con_K}!*}
lemma Notes_TTP_imp_Says_A:
     "[|Notes TTP {|Number f_con, Agent A, Agent B, Nonce L, Key K, con_K|}
           \<in> set evs;
        sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
        A \<notin> broken; evs \<in> zg|]
    ==> \<exists>C Y. Says A C Y \<in> set evs & sub_K \<in> parts {Y}"
by (blast dest!: Notes_TTP_imp_Gets [THEN Gets_imp_knows_Spy, THEN parts.Inj] intro: sub_K_authenticity)

text{*If @{term con_K} exists, then @{term A} sent @{term sub_K}.*}
theorem B_sub_K_authenticity:
     "[|con_K \<in> used evs;
        con_K = Crypt (priK TTP) {|Number f_con, Agent A, Agent B,
                                   Nonce L, Key K|};
        sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
        A \<notin> broken; B \<noteq> TTP; evs \<in> zg|]
    ==> \<exists>C Y. Says A C Y \<in> set evs & sub_K \<in> parts {Y}"
by (blast dest: con_K_authenticity Notes_TTP_imp_Says_A)


subsection{*Proving fairness*}

text{*Cannot prove that, if @{term B} has NRO, then  @{term A} has her NRR.
It would appear that @{term B} has a small advantage, though it is
useless to win disputes: @{term B} needs to present @{term con_K} as well.*}

text{*Strange: unicity of the label protects @{term A}?*}
lemma A_unicity: 
     "[|NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, Crypt K M|};
        NRO \<in> parts (spies evs);
        Says A B {|Number f_nro, Agent B, Nonce L, Crypt K M', NRO'|}
          \<in> set evs;
        A \<notin> bad; evs \<in> zg |]
     ==> M'=M"
apply clarify
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, auto) 
txt{*ZG1: freshness*}
apply (blast dest: parts.Body) 
done


text{*Fairness lemma: if @{term sub_K} exists, then @{term A} holds 
NRR.  Relies on unicity of labels.*}
lemma sub_K_implies_NRR:
     "[| sub_K \<in> parts (spies evs);
         NRO \<in> parts (spies evs);
         sub_K = Crypt (priK A) {|Number f_sub, Agent B, Nonce L, Key K|};
         NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, Crypt K M|};
         NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, Crypt K M|};
         A \<notin> bad;  evs \<in> zg |]
     ==> Gets A {|Number f_nrr, Agent A, Nonce L, NRR|} \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt{*Fake*}
apply blast 
txt{*ZG1: freshness*}
apply (blast dest: parts.Body) 
txt{*ZG3*}
apply (blast dest: A_unicity [OF refl]) 
done


lemma Crypt_used_imp_L_used:
     "[| Crypt (priK TTP) {|F, A, B, L, K|} \<in> used evs; evs \<in> zg |]
      ==> L \<in> used evs"
apply (erule rev_mp)
apply (erule zg.induct, auto)
txt{*Fake*}
apply (blast dest!: Fake_parts_sing_imp_Un)
txt{*ZG2: freshness*}
apply (blast dest: parts.Body) 
done


text{*Fairness for @{term A}: if @{term con_K} and @{term NRO} exist, 
then @{term A} holds NRR.  @{term A} must be uncompromised, but there is no
assumption about @{term B}.*}
theorem A_fairness_NRO:
     "[|con_K \<in> used evs;
        NRO \<in> parts (spies evs);
        con_K = Crypt (priK TTP)
                      {|Number f_con, Agent A, Agent B, Nonce L, Key K|};
        NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, Crypt K M|};
        NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, Crypt K M|};
        A \<notin> bad;  evs \<in> zg |]
    ==> Gets A {|Number f_nrr, Agent A, Nonce L, NRR|} \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
   txt{*Fake*}
   apply (simp add: parts_insert_knows_A) 
   apply (blast dest: Fake_parts_sing_imp_Un) 
  txt{*ZG1*}
  apply (blast dest: Crypt_used_imp_L_used) 
 txt{*ZG2*}
 apply (blast dest: parts_cut)
txt{*ZG4*}
apply (blast intro: sub_K_implies_NRR [OF _ _ refl] 
             dest: Gets_imp_knows_Spy [THEN parts.Inj])
done

text{*Fairness for @{term B}: NRR exists at all, then @{term B} holds NRO.
@{term B} must be uncompromised, but there is no assumption about @{term
A}. *}
theorem B_fairness_NRR:
     "[|NRR \<in> used evs;
        NRR = Crypt (priK B) {|Number f_nrr, Agent A, Nonce L, C|};
        NRO = Crypt (priK A) {|Number f_nro, Agent B, Nonce L, C|};
        B \<notin> bad; evs \<in> zg |]
    ==> Gets B {|Number f_nro, Agent B, Nonce L, C, NRO|} \<in> set evs"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt{*Fake*}
apply (blast dest!: Fake_parts_sing_imp_Un)
txt{*ZG2*}
apply (blast dest: parts_cut)
done


text{*If @{term con_K} exists at all, then @{term B} can get it, by @{text
con_K_authenticity}.  Cannot conclude that also NRO is available to @{term B},
because if @{term A} were unfair, @{term A} could build message 3 without
building message 1, which contains NRO. *}

end