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 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]

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 termX  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 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)


subsection‹Attempting to prove stronger properties›

text‹Only 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+)


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 termA  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