Theory Recur

(*  Title:      HOL/Auth/Recur.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge
*)

sectionThe Otway-Bull Recursive Authentication Protocol

theory Recur imports Public begin

textEnd marker for message bundles
abbreviation
  END :: "msg" where
  "END == Number 0"

(*Two session keys are distributed to each agent except for the initiator,
        who receives one.
  Perhaps the two session keys could be bundled into a single message.
*)
inductive_set (*Server's response to the nested message*)
  respond :: "event list  (msg*msg*key)set"
  for evs :: "event list"
  where
   One:  "Key KAB  used evs
          ==> (Hash[Key(shrK A)] Agent A, Agent B, Nonce NA, END,
               Crypt (shrK A) Key KAB, Agent B, Nonce NA, END,
               KAB)    respond evs"

    (*The most recent session key is passed up to the caller*)
 | Cons: "[| (PA, RA, KAB)  respond evs;
             Key KBC  used evs;  Key KBC  parts {RA};
             PA = Hash[Key(shrK A)] Agent A, Agent B, Nonce NA, P |]
          ==> (Hash[Key(shrK B)] Agent B, Agent C, Nonce NB, PA,
               Crypt (shrK B) Key KBC, Agent C, Nonce NB,
                 Crypt (shrK B) Key KAB, Agent A, Nonce NB,
                 RA,
               KBC)
               respond evs"


(*Induction over "respond" can be difficult due to the complexity of the
  subgoals.  Set "responses" captures the general form of certificates.
*)
inductive_set
  responses :: "event list => msg set"
  for evs :: "event list"
  where
    (*Server terminates lists*)
   Nil:  "END  responses evs"

 | Cons: "[| RA  responses evs;  Key KAB  used evs |]
          ==> Crypt (shrK B) Key KAB, Agent A, Nonce NB,
                RA   responses evs"


inductive_set recur :: "event list set"
  where
         (*Initial trace is empty*)
   Nil:  "[]  recur"

         (*The spy MAY say anything he CAN say.  Common to
           all similar protocols.*)
 | Fake: "[| evsf  recur;  X  synth (analz (knows Spy evsf)) |]
          ==> Says Spy B X  # evsf  recur"

         (*Alice initiates a protocol run.
           END is a placeholder to terminate the nesting.*)
 | RA1:  "[| evs1  recur;  Nonce NA  used evs1 |]
          ==> Says A B (Hash[Key(shrK A)] Agent A, Agent B, Nonce NA, END)
              # evs1  recur"

         (*Bob's response to Alice's message.  C might be the Server.
           We omit PA = ⦃XA, Agent A, Agent B, Nonce NA, P⦄ because
           it complicates proofs, so B may respond to any message at all!*)
 | RA2:  "[| evs2  recur;  Nonce NB  used evs2;
             Says A' B PA  set evs2 |]
          ==> Says B C (Hash[Key(shrK B)] Agent B, Agent C, Nonce NB, PA)
              # evs2  recur"

         (*The Server receives Bob's message and prepares a response.*)
 | RA3:  "[| evs3  recur;  Says B' Server PB  set evs3;
             (PB,RB,K)  respond evs3 |]
          ==> Says Server B RB # evs3  recur"

         (*Bob receives the returned message and compares the Nonces with
           those in the message he previously sent the Server.*)
 | RA4:  "[| evs4  recur;
             Says B  C XH, Agent B, Agent C, Nonce NB,
                         XA, Agent A, Agent B, Nonce NA, P  set evs4;
             Says C' B Crypt (shrK B) Key KBC, Agent C, Nonce NB,
                         Crypt (shrK B) Key KAB, Agent A, Nonce NB,
                         RA  set evs4 |]
          ==> Says B A RA # evs4  recur"

   (*No "oops" message can easily be expressed.  Each session key is
     associated--in two separate messages--with two nonces.  This is
     one try, but it isn't that useful.  Re domino attack, note that
     Recur.thy proves that each session key is secure provided the two
     peers are, even if there are compromised agents elsewhere in
     the chain.  Oops cases proved using parts_cut, Key_in_keysFor_parts,
     etc.

   Oops:  "[| evso ∈ recur;  Says Server B RB ∈ set evso;
              RB ∈ responses evs';  Key K ∈ parts {RB} |]
           ==> Notes Spy ⦃Key K, RB⦄ # evso ∈ recur"
  *)


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


(** Possibility properties: traces that reach the end
        ONE theorem would be more elegant and faster!
        By induction on a list of agents (no repetitions)
**)


textSimplest case: Alice goes directly to the server
lemma "Key K  used [] 
       ==> NA. evs  recur.
              Says Server A Crypt (shrK A) Key K, Agent Server, Nonce NA,
                    END   set evs"
apply (intro exI bexI)
apply (rule_tac [2] recur.Nil [THEN recur.RA1, 
                             THEN recur.RA3 [OF _ _ respond.One]])
apply (possibility, simp add: used_Cons) 
done


textCase two: Alice, Bob and the server
lemma "[| Key K  used []; Key K'  used []; K  K';
          Nonce NA  used []; Nonce NB  used []; NA < NB |]
       ==> NA. evs  recur.
        Says B A Crypt (shrK A) Key K, Agent B, Nonce NA,
                   END   set evs"
apply (intro exI bexI)
apply (rule_tac [2] 
          recur.Nil
           [THEN recur.RA1 [of _ NA], 
            THEN recur.RA2 [of _ NB],
            THEN recur.RA3 [OF _ _ respond.One 
                                     [THEN respond.Cons [of _ _ K _ K']]],
            THEN recur.RA4], possibility)
apply (auto simp add: used_Cons)
done

(*Case three: Alice, Bob, Charlie and the server Rather slow (5 seconds)*)
lemma "[| Key K  used []; Key K'  used [];  
          Key K''  used []; K  K'; K'  K''; K  K'';
          Nonce NA  used []; Nonce NB  used []; Nonce NC  used []; 
          NA < NB; NB < NC |]
       ==> K. NA. evs  recur.
             Says B A Crypt (shrK A) Key K, Agent B, Nonce NA,
                        END   set evs"
apply (intro exI bexI)
apply (rule_tac [2] 
          recur.Nil [THEN recur.RA1, 
                     THEN recur.RA2, THEN recur.RA2,
                     THEN recur.RA3 
                          [OF _ _ respond.One 
                                  [THEN respond.Cons, THEN respond.Cons]],
                     THEN recur.RA4, THEN recur.RA4])
apply basic_possibility
apply (tactic "DEPTH_SOLVE (swap_res_tac context [refl, conjI, disjCI] 1)")
done


lemma respond_imp_not_used: "(PA,RB,KAB)  respond evs ==> Key KAB  used evs"
by (erule respond.induct, simp_all)

lemma Key_in_parts_respond [rule_format]:
   "[| Key K  parts {RB};  (PB,RB,K')  respond evs |] ==> Key K  used evs"
apply (erule rev_mp, erule respond.induct)
apply (auto dest: Key_not_used respond_imp_not_used)
done

textSimple inductive reasoning about responses
lemma respond_imp_responses:
     "(PA,RB,KAB)  respond evs ==> RB  responses evs"
apply (erule respond.induct)
apply (blast intro!: respond_imp_not_used responses.intros)+
done


(** For reasoning about the encrypted portion of messages **)

lemmas RA2_analz_spies = Says_imp_spies [THEN analz.Inj]

lemma RA4_analz_spies:
     "Says C' B Crypt K X, X', RA  set evs ==> RA  analz (spies evs)"
by blast


(*RA2_analz... and RA4_analz... let us treat those cases using the same
  argument as for the Fake case.  This is possible for most, but not all,
  proofs: Fake does not invent new nonces (as in RA2), and of course Fake
  messages originate from the Spy. *)

lemmas RA2_parts_spies =  RA2_analz_spies [THEN analz_into_parts]
lemmas RA4_parts_spies =  RA4_analz_spies [THEN analz_into_parts]


(** Theorems of the form X ∉ parts (spies evs) imply that NOBODY
    sends messages containing X! **)

(** Spy never sees another agent's shared key! (unless it's bad at start) **)

lemma Spy_see_shrK [simp]:
     "evs  recur ==> (Key (shrK A)  parts (spies evs)) = (A  bad)"
apply (erule recur.induct, auto)
txtRA3.  It's ugly to call auto twice, but it seems necessary.
apply (auto dest: Key_in_parts_respond simp add: parts_insert_spies)
done

lemma Spy_analz_shrK [simp]:
     "evs  recur ==> (Key (shrK A)  analz (spies evs)) = (A  bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
     "[|Key (shrK A)  parts (knows Spy evs);  evs  recur|] ==> A  bad"
by (blast dest: Spy_see_shrK)


(*** Proofs involving analz ***)

(** Session keys are not used to encrypt other session keys **)

(*Version for "responses" relation.  Handles case RA3 in the theorem below.
  Note that it holds for *any* set H (not just "spies evs")
  satisfying the inductive hypothesis.*)
lemma resp_analz_image_freshK_lemma:
     "[| RB  responses evs;
         K KK. KK  - (range shrK) 
                   (Key K  analz (Key`KK  H)) =
                   (K  KK | Key K  analz H) |]
     ==> K KK. KK  - (range shrK) 
                   (Key K  analz (insert RB (Key`KK  H))) =
                   (K  KK | Key K  analz (insert RB H))"
apply (erule responses.induct)
apply (simp_all del: image_insert
                add: analz_image_freshK_simps, auto)
done 


textVersion for the protocol.  Proof is easy, thanks to the lemma.
lemma raw_analz_image_freshK:
 "evs  recur ==>
   K KK. KK  - (range shrK) 
          (Key K  analz (Key`KK  (spies evs))) =
          (K  KK | Key K  analz (spies evs))"
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
       drule_tac [5] respond_imp_responses,
       drule_tac [6] RA4_analz_spies, analz_freshK, spy_analz)
txtRA3
apply (simp_all add: resp_analz_image_freshK_lemma)
done


(*Instance of the lemma with H replaced by (spies evs):
   [| RB ∈ responses evs;  evs ∈ recur; |]
   ==> KK ⊆ - (range shrK) ⟶
       Key K ∈ analz (insert RB (Key`KK ∪ spies evs)) =
       (K ∈ KK | Key K ∈ analz (insert RB (spies evs)))
*)
lemmas resp_analz_image_freshK =  
       resp_analz_image_freshK_lemma [OF _ raw_analz_image_freshK]

lemma analz_insert_freshK:
     "[| evs  recur;  KAB  range shrK |]
      ==> (Key K  analz (insert (Key KAB) (spies evs))) =
          (K = KAB | Key K  analz (spies evs))"
by (simp del: image_insert
         add: analz_image_freshK_simps raw_analz_image_freshK)


textEverything that's hashed is already in past traffic.
lemma Hash_imp_body:
     "[| Hash Key(shrK A), X  parts (spies evs);
         evs  recur;  A  bad |] ==> X  parts (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct,
       drule_tac [6] RA4_parts_spies,
       drule_tac [5] respond_imp_responses,
       drule_tac [4] RA2_parts_spies)
txtRA3 requires a further induction
apply (erule_tac [5] responses.induct, simp_all)
txtFake
apply (blast intro: parts_insertI)
done


(** The Nonce NA uniquely identifies A's message.
    This theorem applies to steps RA1 and RA2!

  Unicity is not used in other proofs but is desirable in its own right.
**)

lemma unique_NA:
  "[| Hash Key(shrK A), Agent A, B, NA, P  parts (spies evs);
      Hash Key(shrK A), Agent A, B',NA, P'  parts (spies evs);
      evs  recur;  A  bad |]
    ==> B=B'  P=P'"
apply (erule rev_mp, erule rev_mp)
apply (erule recur.induct,
       drule_tac [5] respond_imp_responses)
apply (force, simp_all)
txtFake
apply blast
apply (erule_tac [3] responses.induct)
txtRA1,2: creation of new Nonce
apply simp_all
apply (blast dest!: Hash_imp_body)+
done


(*** Lemmas concerning the Server's response
      (relations "respond" and "responses")
***)

lemma shrK_in_analz_respond [simp]:
     "[| RB  responses evs;  evs  recur |]
  ==> (Key (shrK B)  analz (insert RB (spies evs))) = (Bbad)"
apply (erule responses.induct)
apply (simp_all del: image_insert
                add: analz_image_freshK_simps resp_analz_image_freshK, auto) 
done


lemma resp_analz_insert_lemma:
     "[| Key K  analz (insert RB H);
         K KK. KK  - (range shrK) 
                   (Key K  analz (Key`KK  H)) =
                   (K  KK | Key K  analz H);
         RB  responses evs |]
     ==> (Key K  parts{RB} | Key K  analz H)"
apply (erule rev_mp, erule responses.induct)
apply (simp_all del: image_insert parts_image
             add: analz_image_freshK_simps resp_analz_image_freshK_lemma)
txtSimplification using two distinct treatments of "image"
apply (simp add: parts_insert2, blast)
done

lemmas resp_analz_insert =
       resp_analz_insert_lemma [OF _ raw_analz_image_freshK]

textThe last key returned by respond indeed appears in a certificate
lemma respond_certificate:
     "(Hash[Key(shrK A)] Agent A, B, NA, P, RA, K)  respond evs
      ==> Crypt (shrK A) Key K, B, NA  parts {RA}"
apply (ind_cases "(Hash[Key (shrK A)] Agent A, B, NA, P, RA, K)  respond evs")
apply simp_all
done

(*This unicity proof differs from all the others in the HOL/Auth directory.
  The conclusion isn't quite unicity but duplicity, in that there are two
  possibilities.  Also, the presence of two different matching messages in
  the inductive step complicates the case analysis.  Unusually for such proofs,
  the quantifiers appear to be necessary.*)
lemma unique_lemma [rule_format]:
     "(PB,RB,KXY)  respond evs ==>
      A B N. Crypt (shrK A) Key K, Agent B, N  parts {RB} 
      (A' B' N'. Crypt (shrK A') Key K, Agent B', N'  parts {RB} 
      (A'=A  B'=B) | (A'=B  B'=A))"
apply (erule respond.induct)
apply (simp_all add: all_conj_distrib)
apply (blast dest: respond_certificate)
done

lemma unique_session_keys:
     "[| Crypt (shrK A) Key K, Agent B, N  parts {RB};
         Crypt (shrK A') Key K, Agent B', N'  parts {RB};
         (PB,RB,KXY)  respond evs |]
      ==> (A'=A  B'=B) | (A'=B  B'=A)"
by (rule unique_lemma, auto)


(** Crucial secrecy property: Spy does not see the keys sent in msg RA3
    Does not in itself guarantee security: an attack could violate
    the premises, e.g. by having A=Spy **)

lemma respond_Spy_not_see_session_key [rule_format]:
     "[| (PB,RB,KAB)  respond evs;  evs  recur |]
      ==> A A' N. A  bad  A'  bad 
          Crypt (shrK A) Key K, Agent A', N  parts{RB} 
          Key K  analz (insert RB (spies evs))"
apply (erule respond.induct)
apply (frule_tac [2] respond_imp_responses)
apply (frule_tac [2] respond_imp_not_used)
apply (simp_all del: image_insert parts_image
                add: analz_image_freshK_simps split_ifs shrK_in_analz_respond
                     resp_analz_image_freshK parts_insert2)
txtBase case of respond
apply blast
txtInductive step of respond
apply (intro allI conjI impI, simp_all)
txtby unicity, either termB=Aa or termB=A', a contradiction
     if termB  bad   
apply (blast dest: unique_session_keys respond_certificate)
apply (blast dest!: respond_certificate)
apply (blast dest!: resp_analz_insert)
done


lemma Spy_not_see_session_key:
     "[| Crypt (shrK A) Key K, Agent A', N  parts (spies evs);
         A  bad;  A'  bad;  evs  recur |]
      ==> Key K  analz (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
       frule_tac [5] respond_imp_responses,
       drule_tac [6] RA4_analz_spies,
       simp_all add: split_ifs analz_insert_eq analz_insert_freshK)
txtFake
apply spy_analz
txtRA2
apply blast 
txtRA3
apply (simp add: parts_insert_spies)
apply (metis Key_in_parts_respond parts.Body parts.Fst resp_analz_insert 
             respond_Spy_not_see_session_key usedI)
txtRA4
apply blast 
done

(**** Authenticity properties for Agents ****)

textThe response never contains Hashes
lemma Hash_in_parts_respond:
     "[| Hash Key (shrK B), M  parts (insert RB H);
         (PB,RB,K)  respond evs |]
      ==> Hash Key (shrK B), M  parts H"
apply (erule rev_mp)
apply (erule respond_imp_responses [THEN responses.induct], auto)
done

textOnly RA1 or RA2 can have caused such a part of a message to appear.
  This result is of no use to B, who cannot verify the Hash.  Moreover,
  it can say nothing about how recent A's message is.  It might later be
  used to prove B's presence to A at the run's conclusion.
lemma Hash_auth_sender [rule_format]:
     "[| Hash Key(shrK A), Agent A, Agent B, NA, P  parts(spies evs);
         A  bad;  evs  recur |]
      ==> Says A B (Hash[Key(shrK A)] Agent A, Agent B, NA, P)  set evs"
apply (unfold HPair_def)
apply (erule rev_mp)
apply (erule recur.induct,
       drule_tac [6] RA4_parts_spies,
       drule_tac [4] RA2_parts_spies,
       simp_all)
txtFake, RA3
apply (blast dest: Hash_in_parts_respond)+
done

(** These two results subsume (for all agents) the guarantees proved
    separately for A and B in the Otway-Rees protocol.
**)


textCertificates can only originate with the Server.
lemma Cert_imp_Server_msg:
     "[| Crypt (shrK A) Y  parts (spies evs);
         A  bad;  evs  recur |]
      ==> C RC. Says Server C RC  set evs  
                   Crypt (shrK A) Y  parts {RC}"
apply (erule rev_mp, erule recur.induct, simp_all)
txtFake
apply blast
txtRA1
apply blast
txtRA2: it cannot be a new Nonce, contradiction.
apply blast
txtRA3.  Pity that the proof is so brittle: this step requires the rewriting,
       which however would break all other steps.
apply (simp add: parts_insert_spies, blast)
txtRA4
apply blast
done

end