src/HOL/Auth/OtwayRees_Bad.thy

 Inductive relation "otway" for the Otway-Rees protocol.
 
 The FAULTY version omitting encryption of Nonce NB, as suggested on page 247 of
   Burrows, Abadi and Needham.  A Logic of Authentication.
   Proc. Royal Soc. 426 (1989)
+
+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.
 *)
 
-OtwayRees_Bad = Shared + 
-
-consts  otway   :: event list set
-
-inductive otway
-  intrs 
+theory OtwayRees_Bad = Shared:
+
+consts  otway   :: "event list set"
+inductive "otway"
+  intros
          (*Initial trace is empty*)
-    Nil  "[] \\<in> otway"
+   Nil:  "[] \<in> otway"
 
          (*The spy MAY say anything he CAN say.  We do not expect him to
            invent new nonces here, but he can also use NS1.  Common to
            all similar protocols.*)
-    Fake "[| evsf \\<in> otway;  X \\<in> synth (analz (knows Spy evsf)) |]
-          ==> Says Spy B X  # evsf \\<in> otway"
+   Fake: "[| evsf \<in> otway;  X \<in> synth (analz (knows Spy evsf)) |]
+          ==> Says Spy B X  # evsf \<in> otway"
 
          (*A message that has been sent can be received by the
            intended recipient.*)
-    Reception "[| evsr \\<in> otway;  Says A B X \\<in> set evsr |]
-               ==> Gets B X # evsr \\<in> otway"
+   Reception: "[| evsr \<in> otway;  Says A B X \<in>set evsr |]
+               ==> Gets B X # evsr \<in> otway"
 
          (*Alice initiates a protocol run*)
-    OR1  "[| evs1 \\<in> otway;  Nonce NA \\<notin> used evs1 |]
-          ==> Says A B {|Nonce NA, Agent A, Agent B, 
-                         Crypt (shrK A) {|Nonce NA, Agent A, Agent B|} |} 
-                 # evs1 \\<in> otway"
-
-         (*Bob's response to Alice's message. 
+   OR1:  "[| evs1 \<in> otway;  Nonce NA \<notin> used evs1 |]
+          ==> Says A B {|Nonce NA, Agent A, Agent B,
+                         Crypt (shrK A) {|Nonce NA, Agent A, Agent B|} |}
+                 # evs1 \<in> otway"
+
+         (*Bob's response to Alice's message.
            This variant of the protocol does NOT encrypt NB.*)
-    OR2  "[| evs2 \\<in> otway;  Nonce NB \\<notin> used evs2;
-             Gets B {|Nonce NA, Agent A, Agent B, X|} \\<in> set evs2 |]
-          ==> Says B Server 
+   OR2:  "[| evs2 \<in> otway;  Nonce NB \<notin> used evs2;
+             Gets B {|Nonce NA, Agent A, Agent B, X|} \<in> set evs2 |]
+          ==> Says B Server
                   {|Nonce NA, Agent A, Agent B, X, Nonce NB,
                     Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
-                 # evs2 \\<in> otway"
+                 # evs2 \<in> otway"
 
          (*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.*)
-    OR3  "[| evs3 \\<in> otway;  Key KAB \\<notin> used evs3;
-             Gets Server 
-                  {|Nonce NA, Agent A, Agent B, 
-                    Crypt (shrK A) {|Nonce NA, Agent A, Agent B|}, 
-                    Nonce NB, 
+   OR3:  "[| evs3 \<in> otway;  Key KAB \<notin> 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|}|}
-               \\<in> set evs3 |]
-          ==> Says Server B 
-                  {|Nonce NA, 
+               \<in> set evs3 |]
+          ==> Says Server B
+                  {|Nonce NA,
                     Crypt (shrK A) {|Nonce NA, Key KAB|},
                     Crypt (shrK B) {|Nonce NB, Key KAB|}|}
-                 # evs3 \\<in> otway"
+                 # evs3 \<in> otway"
 
          (*Bob receives the Server's (?) message and compares the Nonces with
 	   those in the message he previously sent the Server.
            Need B ~= Server because we allow messages to self.*)
-    OR4  "[| evs4 \\<in> otway;  B ~= Server;
+   OR4:  "[| evs4 \<in> otway;  B ~= Server;
              Says B Server {|Nonce NA, Agent A, Agent B, X', Nonce NB,
                              Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
-               \\<in> set evs4;
+               \<in> set evs4;
              Gets B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
-               \\<in> set evs4 |]
-          ==> Says B A {|Nonce NA, X|} # evs4 \\<in> otway"
+               \<in> set evs4 |]
+          ==> Says B A {|Nonce NA, X|} # evs4 \<in> otway"
 
          (*This message models possible leaks of session keys.  The nonces
            identify the protocol run.*)
-    Oops "[| evso \\<in> otway;  
+   Oops: "[| evso \<in> otway;
              Says Server B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
-               \\<in> set evso |]
-          ==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \\<in> otway"
+               \<in> set evso |]
+          ==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \<in> 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]
+
+(*A "possibility property": there are traces that reach the end*)
+lemma "B \<noteq> Server
+      ==> \<exists>K. \<exists>NA. \<exists>evs \<in> otway.
+            Says B A {|Nonce NA, Crypt (shrK A) {|Nonce NA, Key K|}|}
+              \<in> 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
+done
+
+lemma Gets_imp_Says [dest!]:
+     "[| Gets B X \<in> set evs; evs \<in> otway |] ==> \<exists>A. Says A B X \<in> set evs"
+apply (erule rev_mp)
+apply (erule otway.induct)
+apply auto
+done
+
+
+(**** Inductive proofs about otway ****)
+
+
+(** For reasoning about the encrypted portion of messages **)
+
+lemma OR2_analz_knows_Spy:
+     "[| Gets B {|N, Agent A, Agent B, X|} \<in> set evs;  evs \<in> otway |]
+      ==> X \<in> analz (knows Spy evs)"
+by blast
+
+lemma OR4_analz_knows_Spy:
+     "[| Gets B {|N, X, Crypt (shrK B) X'|} \<in> set evs;  evs \<in> otway |]
+      ==> X \<in> analz (knows Spy evs)"
+by blast
+
+lemma Oops_parts_knows_Spy:
+     "Says Server B {|NA, X, Crypt K' {|NB,K|}|} \<in> set evs
+      ==> K \<in> parts (knows Spy evs)"
+by blast
+
+(*Forwarding lemma: see comments in OtwayRees.thy*)
+lemmas OR2_parts_knows_Spy =
+    OR2_analz_knows_Spy [THEN analz_into_parts, standard]
+
+
+(** Theorems of the form X \<notin> parts (knows Spy evs) imply that NOBODY
+    sends messages containing X! **)
+
+(*Spy never sees a good agent's shared key!*)
+lemma Spy_see_shrK [simp]:
+     "evs \<in> otway ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
+apply (erule otway.induct, force,
+       drule_tac [4] OR2_parts_knows_Spy, simp_all)
+apply blast+
+done
+
+lemma Spy_analz_shrK [simp]:
+     "evs \<in> otway ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
+by auto
+
+lemma Spy_see_shrK_D [dest!]:
+     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> otway|] ==> A \<in> bad"
+by (blast dest: Spy_see_shrK)
+
+
+(*** Proofs involving analz ***)
+
+(*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|}|} \<in> set evs;
+         evs \<in> otway |]
+      ==> K \<notin> range shrK & (\<exists>i. NA = Nonce i) & (\<exists>j. NB = Nonce j)"
+apply (erule rev_mp)
+apply (erule otway.induct, simp_all)
+apply blast
+done
+
+
+(****
+ The following is to prove theorems of the form
+
+  Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
+  Key K \<in> analz (knows Spy evs)
+
+ A more general formula must be proved inductively.
+****)
+
+
+(** Session keys are not used to encrypt other session keys **)
+
+(*The equality makes the induction hypothesis easier to apply*)
+lemma analz_image_freshK [rule_format]:
+ "evs \<in> otway ==>
+   \<forall>K KK. KK <= -(range shrK) -->
+          (Key K \<in> analz (Key`KK Un (knows Spy evs))) =
+          (K \<in> KK | Key K \<in> 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 analz_freshK
+apply spy_analz
+done
+
+lemma analz_insert_freshK:
+  "[| evs \<in> otway;  KAB \<notin> range shrK |] ==>
+      Key K \<in> analz (insert (Key KAB) (knows Spy evs)) =
+      (K = KAB | Key K \<in> analz (knows Spy evs))"
+by (simp only: analz_image_freshK analz_image_freshK_simps)
+
+
+(*** The Key K uniquely identifies the Server's  message. **)
+
+lemma unique_session_keys:
+     "[| Says Server B {|NA, X, Crypt (shrK B) {|NB, K|}|}   \<in> set evs;
+         Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} \<in> set evs;
+         evs \<in> otway |] ==> X=X' & B=B' & NA=NA' & NB=NB'"
+apply (erule rev_mp)
+apply (erule rev_mp)
+apply (erule otway.induct, simp_all)
+(*Remaining cases: OR3 and OR4*)
+apply blast+
+done
+
+
+(** 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 A=Spy **)
+
+lemma secrecy_lemma:
+ "[| A \<notin> bad;  B \<notin> bad;  evs \<in> otway |]
+  ==> Says Server B
+        {|NA, Crypt (shrK A) {|NA, Key K|},
+          Crypt (shrK B) {|NB, Key K|}|} \<in> set evs -->
+      Notes Spy {|NA, NB, Key K|} \<notin> set evs -->
+      Key K \<notin> 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*)
+(*OR3, OR4, Oops*)
+apply (blast dest: unique_session_keys)+
+done
+
+
+lemma Spy_not_see_encrypted_key:
+     "[| Says Server B
+          {|NA, Crypt (shrK A) {|NA, Key K|},
+                Crypt (shrK B) {|NB, Key K|}|} \<in> set evs;
+         Notes Spy {|NA, NB, Key K|} \<notin> set evs;
+         A \<notin> bad;  B \<notin> bad;  evs \<in> otway |]
+      ==> Key K \<notin> analz (knows Spy evs)"
+by (blast dest: Says_Server_message_form secrecy_lemma)
+
+
+(*** Attempting to prove stronger properties ***)
+
+(*Only OR1 can have caused such a part of a message to appear.
+  The premise A \<noteq> 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 \<notin> bad;  A \<noteq> B;  evs \<in> otway |]
+      ==> Crypt (shrK A) {|NA, Agent A, Agent B|} \<in> parts (knows Spy evs) -->
+          Says A B {|NA, Agent A, Agent B,
+                     Crypt (shrK A) {|NA, Agent A, Agent B|}|}  \<in> set evs"
+apply (erule otway.induct, force,
+       drule_tac [4] OR2_parts_knows_Spy, simp_all)
+apply blast+
+done
+
+
+(*Crucial property: If the encrypted message appears, and A has used NA
+  to start a run, then it originated with the Server!
+  The premise A \<noteq> B allows use of Crypt_imp_OR1*)
+(*Only it is FALSE.  Somebody could make a fake message to Server
+          substituting some other nonce NA' for NB.*)
+lemma "[| A \<notin> bad;  A \<noteq> B;  evs \<in> otway |]
+       ==> Crypt (shrK A) {|NA, Key K|} \<in> parts (knows Spy evs) -->
+           Says A B {|NA, Agent A, Agent B,
+                      Crypt (shrK A) {|NA, Agent A, Agent B|}|}
+            \<in> set evs -->
+           (\<exists>B NB. Says Server B
+                {|NA,
+                  Crypt (shrK A) {|NA, Key K|},
+                  Crypt (shrK B) {|NB, Key K|}|}  \<in> set evs)"
+apply (erule otway.induct, force,
+       drule_tac [4] OR2_parts_knows_Spy)
+apply simp_all
+(*Fake*)
+apply blast
+(*OR1: it cannot be a new Nonce, contradiction.*)
+apply blast
+(*OR3 and OR4*)
+apply (simp_all add: ex_disj_distrib)
+(*OR4*)
+prefer 2 apply (blast intro!: Crypt_imp_OR1)
+(*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|}|}
+          \<in> set evs3
+          Says A B
+           {|Nonce NB, Agent A, Agent B,
+             Crypt (shrK A) {|Nonce NB, Agent A, Agent B|}|}
+          \<in> set evs3;
+*)
+
+
+(*Thus the key property A_can_trust probably fails too.*)
+oops
 
 end