Theory OtwayRees_Bad

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


sectionThe Otway-Rees Protocol: The Faulty BAN Version

theory OtwayRees_Bad imports Public begin

textThe 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 termB  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]

textA "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


subsectionFor 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

textForwarding lemma: see comments in OtwayRees.thy
lemmas OR2_parts_knows_Spy =
    OR2_analz_knows_Spy [THEN analz_into_parts]


textTheorems of the form termX  parts (spies evs) imply that
NOBODY sends messages containing X!

textSpy 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)


subsectionProofs involving analz

textDescribes 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.
****)


textSession keys are not used to encrypt other session keys

textThe 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)


textThe 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


textCrucial 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 termA=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)


subsectionAttempting to prove stronger properties

textOnly OR1 can have caused such a part of a message to appear. The premise
  termA  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+)


textCrucial property: If the encrypted message appears, and A has used NA
  to start a run, then it originated with the Server!
  The premise termA  B allows use of Crypt_imp_OR1›
textOnly 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.
txtOR3 and OR4
apply (simp_all add: ex_disj_distrib)
 prefer 2 apply (blast intro!: Crypt_imp_OR1)  ― ‹OR4
txtOR3
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