Quelle  ZhouGollmann.thy   Sprache: Isabelle

(*  Title:      HOL/Auth/ZhouGollmann.thy
    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 imports Public begin

abbreviation
  TTP :: agent where "TTP == Server"

abbreviation f_sub :: nat where "f_sub == 5"
abbreviation f_nro :: nat where "f_nro == 2"
abbreviation f_nrr :: nat where "f_nrr == 3"
abbreviation f_con :: nat where "f_con == 4"


definition broken :: "agent set" where    
    ― ‹the compromised honest agents; TTP is included as it's not allowed to
 use the protocol›
   "broken == bad - {Spy}"

declare broken_def [simp]

inductive_set zg :: "event list set"
  where

  Nil:  "[] ∈ zg"

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

| Reception:  "[evsr ∈ zg; Says A B X ∈ set evsr] ==> Gets B X # evsr ∈ 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 ∈ zg; Nonce L ∉ used evs1; C = Crypt K (Number m);
           K ∈ 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 ∈ zg"

  (*B must check that NRO is A's signature to learn the sender's name*)
| ZG2: "[evs2 ∈ zg;
           Gets B {Number f_nro, Agent B, Nonce L, C, NRO} ∈ 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 ∈ zg"

  (*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 ∈ zg; C = Crypt K M; K ∈ symKeys;
           Says A B {Number f_nro, Agent B, Nonce L, C, NRO} ∈ set evs3;
           Gets A {Number f_nrr, Agent A, Nonce L, NRR} ∈ 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 ∈ 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.  We also
   give con_K to the Spy. This makes the threat model more dangerous, while 
   also allowing lemma @{text Crypt_used_imp_spies} to omit the condition
   @{term "K \<noteq> priK TTP"}. *)
| ZG4: "[evs4 ∈ zg; K ∈ symKeys;
           Gets TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K}
             ∈ 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}]
       ==> Says TTP Spy con_K
           #
           Notes TTP {Number f_con, Agent A, Agent B, Nonce L, Key K, con_K}
           # evs4 ∈ 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]


text‹A "possibility property": there are traces that reach the end›
lemma "[A ≠ B; TTP ≠ A; TTP ≠ B; K ∈ symKeys] ==>
     ∃L. ∃evs ∈ zg.
           Notes TTP {Number f_con, Agent A, Agent B, Nonce L, Key K,
               Crypt (priK TTP) {Number f_con, Agent A, Agent B, Nonce L, Key K}}
               ∈ set evs"
apply (intro exI bexI)
apply (rule_tac [2] zg.Nil
                    [THEN zg.ZG1, THEN zg.Reception [of _ A B],
                     THEN zg.ZG2, THEN zg.Reception [of _ B A],
                     THEN zg.ZG3, THEN zg.Reception [of _ A TTP], 
                     THEN zg.ZG4])
apply (basic_possibility, auto)
done

subsection ‹Basic Lemmas›

lemma Gets_imp_Says:
     "[Gets B X ∈ set evs; evs ∈ zg] ==> ∃A. Says A B X ∈ set evs"
apply (erule rev_mp)
apply (erule zg.induct, auto)
done

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


text‹Lets us replace proofs about term‹used evs› by simpler proofs
  term‹parts (spies evs)›.›
lemma Crypt_used_imp_spies:
     "[Crypt K X ∈ used evs; evs ∈ zg]
      ==> Crypt K X ∈ 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}
           ∈ set evs;
        sub_K = Crypt (priK A) {Number f_sub, Agent B, Nonce L, Key K};
        evs ∈ zg]
    ==> Gets TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ 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} ∈ set evs; evs ∈ zg]
      ==> C ∈ parts (spies evs)"
by (blast dest: Gets_imp_Says)

(*classical regularity lemma on priK*)
lemma Spy_see_priK [simp]:
     "evs ∈ zg ==> (Key (priK A) ∈ parts (spies evs)) = (A ∈ 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 ∈ zg ==> (Key (priK A) ∈ analz (spies evs)) = (A ∈ bad)"
by auto 


subsection‹About NRO: Validity for term‹B››

text‹Below we prove that if term‹NRO› exists then term‹A› definitely
  it, provided term‹A› is not broken.›

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

lemma NRO_sender:
     "[Says A' B {n, b, l, C, Crypt (priK A) X} ∈ set evs; evs ∈ zg]
    ==> A' ∈ {A,Spy}"
apply (erule rev_mp)  
apply (erule zg.induct, simp_all)
done

text‹Holds also for term‹A = Spy›!›
theorem NRO_validity:
     "[Gets B {Number f_nro, Agent B, Nonce L, C, NRO} ∈ set evs;
        NRO = Crypt (priK A) {Number f_nro, Agent B, Nonce L, C};
        A ∉ broken; evs ∈ zg]
     ==> Says A B {Number f_nro, Agent B, Nonce L, C, NRO} ∈ set evs"
apply (drule Gets_imp_Says, assumption) 
apply clarify 
apply (frule NRO_sender, auto)
txt‹We are left with the case where the sender is term‹Spy› and not
 equal to term‹A›, because term‹A ∉ bad›.
 Thus theorem ‹NRO_validity_good› applies.›
apply (blast dest: NRO_validity_good [OF refl])
done


subsection‹About NRR: Validity for term‹A››

text‹Below we prove that if term‹NRR› exists then term‹B› definitely
  it, provided term‹B› is not broken.›

text‹Strong conclusion for a good agent›
lemma NRR_validity_good:
     "[NRR = Crypt (priK B) {Number f_nrr, Agent A, Nonce L, C};
        NRR ∈ parts (spies evs);
        B ∉ bad; evs ∈ zg]
     ==> Says B A {Number f_nrr, Agent A, Nonce L, NRR} ∈ 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_sender:
     "[Says B' A {n, a, l, Crypt (priK B) X} ∈ set evs; evs ∈ zg]
    ==> B' ∈ {B,Spy}"
apply (erule rev_mp)  
apply (erule zg.induct, simp_all)
done

text‹Holds also for term‹B = Spy›!›
theorem NRR_validity:
     "[Says B' A {Number f_nrr, Agent A, Nonce L, NRR} ∈ set evs;
        NRR = Crypt (priK B) {Number f_nrr, Agent A, Nonce L, C};
        B ∉ broken; evs ∈ zg]
    ==> Says B A {Number f_nrr, Agent A, Nonce L, NRR} ∈ set evs"
apply clarify 
apply (frule NRR_sender, auto)
txt‹We are left with the case where term‹B' = Spy› and term‹B' ≠ B›,
 i.e. term‹B ∉ bad›, when we can apply ‹NRR_validity_good›.›
 apply (blast dest: NRR_validity_good [OF refl])
done


subsection‹Proofs About term‹sub_K››

text‹Below we prove that if term‹sub_K› exists then term‹A› definitely
  it, provided term‹A› is not broken.›

text‹Strong conclusion for a good agent›
lemma sub_K_validity_good:
     "[sub_K = Crypt (priK A) {Number f_sub, Agent B, Nonce L, Key K};
        sub_K ∈ parts (spies evs);
        A ∉ bad; evs ∈ zg]
     ==> Says A TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ 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)
done

lemma sub_K_sender:
     "[Says A' TTP {n, b, l, k, Crypt (priK A) X} ∈ set evs; evs ∈ zg]
    ==> A' ∈ {A,Spy}"
apply (erule rev_mp)  
apply (erule zg.induct, simp_all)
done

text‹Holds also for term‹A = Spy›!›
theorem sub_K_validity:
     "[Gets TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ set evs;
        sub_K = Crypt (priK A) {Number f_sub, Agent B, Nonce L, Key K};
        A ∉ broken; evs ∈ zg]
     ==> Says A TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ set evs"
apply (drule Gets_imp_Says, assumption) 
apply clarify 
apply (frule sub_K_sender, auto)
txt‹We are left with the case where the sender is term‹Spy› and not
 equal to term‹A›, because term‹A ∉ bad›.
 Thus theorem ‹sub_K_validity_good› applies.›
apply (blast dest: sub_K_validity_good [OF refl])
done



subsection‹Proofs About term‹con_K››

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

lemma con_K_validity:
     "[con_K ∈ used evs;
        con_K = Crypt (priK TTP)
                  {Number f_con, Agent A, Agent B, Nonce L, Key K};
        evs ∈ zg]
    ==> Notes TTP {Number f_con, Agent A, Agent B, Nonce L, Key K, con_K}
          ∈ 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. Importantly, 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}
           ∈ set evs;
        sub_K = Crypt (priK A) {Number f_sub, Agent B, Nonce L, Key K};
        A ∉ broken; evs ∈ zg]
     ==> Says A TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ set evs"
apply clarify
apply (erule rev_mp)
apply (erule zg.induct)
apply (frule_tac [5] ZG2_msg_in_parts_spies, simp_all)
txt‹ZG4›
apply clarify 
apply (rule sub_K_validity, auto) 
done

text‹If term‹con_K› exists, then term‹A› sent term‹sub_K›. We again
 assume that term‹A› is not broken.›
theorem B_sub_K_validity:
     "[con_K ∈ 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 ∉ broken; evs ∈ zg]
     ==> Says A TTP {Number f_sub, Agent B, Nonce L, Key K, sub_K} ∈ set evs"
by (blast dest: con_K_validity Notes_TTP_imp_Says_A)


subsection‹Proving fairness›

text‹Cannot prove that, if term‹B› has NRO, then term‹A› has her NRR.
  would appear that term‹B› has a small advantage, though it is
  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 ∈ parts (spies evs);
        Says A B {Number f_nro, Agent B, Nonce L, Crypt K M', NRO'}
          ∈ set evs;
        A ∉ bad; evs ∈ 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
 . Relies on unicity of labels.›
lemma sub_K_implies_NRR:
     "[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};
         sub_K ∈ parts (spies evs);
         NRO ∈ parts (spies evs);
         sub_K = Crypt (priK A) {Number f_sub, Agent B, Nonce L, Key K};
         A ∉ bad; evs ∈ zg]
     ==> Gets A {Number f_nrr, Agent A, Nonce L, NRR} ∈ set evs"
apply clarify
apply hypsubst_thin
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} ∈ used evs; evs ∈ zg]
      ==> L ∈ 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,
  term‹A› holds NRR. term‹A› must be uncompromised, but there is no
  about term‹B›.›
theorem A_fairness_NRO:
     "[con_K ∈ used evs;
        NRO ∈ 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 ∉ bad; evs ∈ zg]
    ==> Gets A {Number f_nrr, Agent A, Nonce L, NRR} ∈ 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 ∈ 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 ∉ bad; evs ∈ zg]
    ==> Gets B {Number f_nro, Agent B, Nonce L, C, NRO} ∈ 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 ‹con_K_validity›. Cannot conclude that also NRO is available to term‹B›,
  if term‹A› were unfair, term‹A› could build message 3 without
  message 1, which contains NRO.›

end