Theory OtwayRees_Bad

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


section‹The Otway-Rees Protocol: The Faulty BAN Version›

theory OtwayRees_Bad imports Public begin

text‹The FAULTY version omitting encryption of Nonce NB, as suggested on 
page 247 of
  Burrows, Abadi and Needham (1988).  A Logic of Authentication.
  Proc. Royal Soc. 426

This file illustrates the consequences of such errors.  We can still prove
impressive-looking properties such as ‹Spy_not_see_encrypted_key›, yet
the protocol is open to a middleperson attack.  Attempting to prove some key
lemmas indicates the possibility of this attack.›

inductive_set otway :: "event list set"
  where
   Nil: ― ‹The empty trace›
        "[] ∈ otway"

 | Fake: ― ‹The Spy may say anything he can say.  The sender field is correct,
            but agents don't use that information.›
         "⟦evsf ∈ otway;  X ∈ synth (analz (knows Spy evsf))⟧
          ⟹ Says Spy B X  # evsf ∈ otway"

        
 | Reception: ― ‹A message that has been sent can be received by the
                  intended recipient.›
              "⟦evsr ∈ otway;  Says A B X ∈set evsr⟧
               ⟹ Gets B X # evsr ∈ otway"

 | OR1:  ― ‹Alice initiates a protocol run›
         "⟦evs1 ∈ otway;  Nonce NA ∉ used evs1⟧
          ⟹ Says A B ⦃Nonce NA, Agent A, Agent B,
                         Crypt (shrK A) ⦃Nonce NA, Agent A, Agent B⦄⦄
                 # evs1 ∈ otway"

 | OR2:  ― ‹Bob's response to Alice's message.
             This variant of the protocol does NOT encrypt NB.›
         "⟦evs2 ∈ otway;  Nonce NB ∉ used evs2;
             Gets B ⦃Nonce NA, Agent A, Agent B, X⦄ ∈ set evs2⟧
          ⟹ Says B Server
                  ⦃Nonce NA, Agent A, Agent B, X, Nonce NB,
                    Crypt (shrK B) ⦃Nonce NA, Agent A, Agent B⦄⦄
                 # evs2 ∈ otway"

 | OR3:  ― ‹The Server receives Bob's message and checks that the three NAs
           match.  Then he sends a new session key to Bob with a packet for
           forwarding to Alice.›
         "⟦evs3 ∈ otway;  Key KAB ∉ used evs3;
             Gets Server
                  ⦃Nonce NA, Agent A, Agent B,
                    Crypt (shrK A) ⦃Nonce NA, Agent A, Agent B⦄,
                    Nonce NB,
                    Crypt (shrK B) ⦃Nonce NA, Agent A, Agent B⦄⦄
               ∈ set evs3⟧
          ⟹ Says Server B
                  ⦃Nonce NA,
                    Crypt (shrK A) ⦃Nonce NA, Key KAB⦄,
                    Crypt (shrK B) ⦃Nonce NB, Key KAB⦄⦄
                 # evs3 ∈ otway"

 | OR4:  ― ‹Bob receives the Server's (?) message and compares the Nonces with
             those in the message he previously sent the Server.
             Need \<^term>‹B ≠ Server› because we allow messages to self.›
         "⟦evs4 ∈ otway;  B ≠ Server;
             Says B Server ⦃Nonce NA, Agent A, Agent B, X', Nonce NB,
                             Crypt (shrK B) ⦃Nonce NA, Agent A, Agent B⦄⦄
               ∈ set evs4;
             Gets B ⦃Nonce NA, X, Crypt (shrK B) ⦃Nonce NB, Key K⦄⦄
               ∈ set evs4⟧
          ⟹ Says B A ⦃Nonce NA, X⦄ # evs4 ∈ otway"

 | Oops: ― ‹This message models possible leaks of session keys.  The nonces
             identify the protocol run.›
         "⟦evso ∈ otway;
             Says Server B ⦃Nonce NA, X, Crypt (shrK B) ⦃Nonce NB, Key K⦄⦄
               ∈ set evso⟧
          ⟹ Notes Spy ⦃Nonce NA, Nonce NB, Key K⦄ # evso ∈ otway"


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]

text‹A "possibility property": there are traces that reach the end›
lemma "⟦B ≠ Server; Key K ∉ used []⟧
      ⟹ ∃NA. ∃evs ∈ otway.
            Says B A ⦃Nonce NA, Crypt (shrK A) ⦃Nonce NA, Key K⦄⦄
              ∈ set evs"
apply (intro exI bexI)
apply (rule_tac [2] otway.Nil
                    [THEN otway.OR1, THEN otway.Reception,
                     THEN otway.OR2, THEN otway.Reception,
                     THEN otway.OR3, THEN otway.Reception, THEN otway.OR4])
apply (possibility, simp add: used_Cons) 
done

lemma Gets_imp_Says [dest!]:
     "⟦Gets B X ∈ set evs; evs ∈ otway⟧ ⟹ ∃A. Says A B X ∈ set evs"
apply (erule rev_mp)
apply (erule otway.induct, auto)
done


subsection‹For reasoning about the encrypted portion of messages›

lemma OR2_analz_knows_Spy:
     "⟦Gets B ⦃N, Agent A, Agent B, X⦄ ∈ set evs;  evs ∈ otway⟧
      ⟹ X ∈ analz (knows Spy evs)"
by blast

lemma OR4_analz_knows_Spy:
     "⟦Gets B ⦃N, X, Crypt (shrK B) X'⦄ ∈ set evs;  evs ∈ otway⟧
      ⟹ X ∈ analz (knows Spy evs)"
by blast

lemma Oops_parts_knows_Spy:
     "Says Server B ⦃NA, X, Crypt K' ⦃NB,K⦄⦄ ∈ set evs
      ⟹ K ∈ parts (knows Spy evs)"
by blast

text‹Forwarding lemma: see comments in OtwayRees.thy›
lemmas OR2_parts_knows_Spy =
    OR2_analz_knows_Spy [THEN analz_into_parts]


text‹Theorems of the form \<^term>‹X ∉ parts (spies evs)› imply that
NOBODY sends messages containing X!›

text‹Spy never sees a good agent's shared key!›
lemma Spy_see_shrK [simp]:
     "evs ∈ otway ⟹ (Key (shrK A) ∈ parts (knows Spy evs)) = (A ∈ bad)"
by (erule otway.induct, force,
    drule_tac [4] OR2_parts_knows_Spy, simp_all, blast+)


lemma Spy_analz_shrK [simp]:
     "evs ∈ otway ⟹ (Key (shrK A) ∈ analz (knows Spy evs)) = (A ∈ bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
     "⟦Key (shrK A) ∈ parts (knows Spy evs);  evs ∈ otway⟧ ⟹ A ∈ bad"
by (blast dest: Spy_see_shrK)


subsection‹Proofs involving analz›

text‹Describes the form of K and NA when the Server sends this message.  Also
  for Oops case.›
lemma Says_Server_message_form:
     "⟦Says Server B ⦃NA, X, Crypt (shrK B) ⦃NB, Key K⦄⦄ ∈ set evs;
         evs ∈ otway⟧
      ⟹ K ∉ range shrK ∧ (∃i. NA = Nonce i) ∧ (∃j. NB = Nonce j)"
apply (erule rev_mp)
apply (erule otway.induct, simp_all)
done


(****
 The following is to prove theorems of the form

  Key K ∈ analz (insert (Key KAB) (knows Spy evs)) ⟹
  Key K ∈ analz (knows Spy evs)

 A more general formula must be proved inductively.
****)


text‹Session keys are not used to encrypt other session keys›

text‹The equality makes the induction hypothesis easier to apply›
lemma analz_image_freshK [rule_format]:
 "evs ∈ otway ⟹
   ∀K KK. KK ⊆ -(range shrK) ⟶
          (Key K ∈ analz (Key`KK ∪ (knows Spy evs))) =
          (K ∈ KK | Key K ∈ analz (knows Spy evs))"
apply (erule otway.induct)
apply (frule_tac [8] Says_Server_message_form)
apply (drule_tac [7] OR4_analz_knows_Spy)
apply (drule_tac [5] OR2_analz_knows_Spy, analz_freshK, spy_analz, auto) 
done

lemma analz_insert_freshK:
  "⟦evs ∈ otway;  KAB ∉ range shrK⟧ ⟹
      (Key K ∈ analz (insert (Key KAB) (knows Spy evs))) =
      (K = KAB | Key K ∈ analz (knows Spy evs))"
by (simp only: analz_image_freshK analz_image_freshK_simps)


text‹The Key K uniquely identifies the Server's  message.›
lemma unique_session_keys:
     "⟦Says Server B ⦃NA, X, Crypt (shrK B) ⦃NB, K⦄⦄   ∈ set evs;
         Says Server B' ⦃NA',X',Crypt (shrK B') ⦃NB',K⦄⦄ ∈ set evs;
         evs ∈ otway⟧ ⟹ X=X' ∧ B=B' ∧ NA=NA' ∧ NB=NB'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule otway.induct, simp_all)
apply blast+  ― ‹OR3 and OR4›
done


text‹Crucial secrecy property: Spy does not see the keys sent in msg OR3
    Does not in itself guarantee security: an attack could violate
    the premises, e.g. by having \<^term>‹A=Spy››
lemma secrecy_lemma:
 "⟦A ∉ bad;  B ∉ bad;  evs ∈ otway⟧
  ⟹ Says Server B
        ⦃NA, Crypt (shrK A) ⦃NA, Key K⦄,
          Crypt (shrK B) ⦃NB, Key K⦄⦄ ∈ set evs ⟶
      Notes Spy ⦃NA, NB, Key K⦄ ∉ set evs ⟶
      Key K ∉ analz (knows Spy evs)"
apply (erule otway.induct, force)
apply (frule_tac [7] Says_Server_message_form)
apply (drule_tac [6] OR4_analz_knows_Spy)
apply (drule_tac [4] OR2_analz_knows_Spy)
apply (simp_all add: analz_insert_eq analz_insert_freshK pushes)
apply spy_analz  ― ‹Fake›
apply (blast dest: unique_session_keys)+  ― ‹OR3, OR4, Oops›
done


lemma Spy_not_see_encrypted_key:
     "⟦Says Server B
          ⦃NA, Crypt (shrK A) ⦃NA, Key K⦄,
                Crypt (shrK B) ⦃NB, Key K⦄⦄ ∈ set evs;
         Notes Spy ⦃NA, NB, Key K⦄ ∉ set evs;
         A ∉ bad;  B ∉ bad;  evs ∈ otway⟧
      ⟹ Key K ∉ analz (knows Spy evs)"
by (blast dest: Says_Server_message_form secrecy_lemma)


subsection‹Attempting to prove stronger properties›

text‹Only OR1 can have caused such a part of a message to appear. The premise
  \<^term>‹A ≠ B› prevents OR2's similar-looking cryptogram from being picked 
  up. Original Otway-Rees doesn't need it.›
lemma Crypt_imp_OR1 [rule_format]:
     "⟦A ∉ bad;  A ≠ B;  evs ∈ otway⟧
      ⟹ Crypt (shrK A) ⦃NA, Agent A, Agent B⦄ ∈ parts (knows Spy evs) ⟶
          Says A B ⦃NA, Agent A, Agent B,
                     Crypt (shrK A) ⦃NA, Agent A, Agent B⦄⦄  ∈ set evs"
by (erule otway.induct, force,
    drule_tac [4] OR2_parts_knows_Spy, simp_all, blast+)


text‹Crucial property: If the encrypted message appears, and A has used NA
  to start a run, then it originated with the Server!
  The premise \<^term>‹A ≠ B› allows use of ‹Crypt_imp_OR1››
text‹Only it is FALSE.  Somebody could make a fake message to Server
          substituting some other nonce NA' for NB.›
lemma "⟦A ∉ bad;  A ≠ B;  evs ∈ otway⟧
       ⟹ Crypt (shrK A) ⦃NA, Key K⦄ ∈ parts (knows Spy evs) ⟶
           Says A B ⦃NA, Agent A, Agent B,
                      Crypt (shrK A) ⦃NA, Agent A, Agent B⦄⦄
            ∈ set evs ⟶
           (∃B NB. Says Server B
                ⦃NA,
                  Crypt (shrK A) ⦃NA, Key K⦄,
                  Crypt (shrK B) ⦃NB, Key K⦄⦄ ∈ set evs)"
apply (erule otway.induct, force,
       drule_tac [4] OR2_parts_knows_Spy, simp_all)
apply blast  ― ‹Fake›
apply blast  ― ‹OR1: it cannot be a new Nonce, contradiction.›
txt‹OR3 and OR4›
apply (simp_all add: ex_disj_distrib)
 prefer 2 apply (blast intro!: Crypt_imp_OR1)  ― ‹OR4›
txt‹OR3›
apply clarify
(*The hypotheses at this point suggest an attack in which nonce NB is used
  in two different roles:
          Gets Server
           ⦃Nonce NA, Agent Aa, Agent A,
             Crypt (shrK Aa) ⦃Nonce NA, Agent Aa, Agent A⦄, Nonce NB,
             Crypt (shrK A) ⦃Nonce NA, Agent Aa, Agent A⦄⦄
          ∈ set evs3
          Says A B
           ⦃Nonce NB, Agent A, Agent B,
             Crypt (shrK A) ⦃Nonce NB, Agent A, Agent B⦄⦄
          ∈ set evs3;
*)


(*Thus the key property A_can_trust probably fails too.*)
oops

end