src/HOL/Auth/Yahalom_Bad.thy
author paulson
Sat Aug 17 14:55:08 2002 +0200 (2002-08-17)
changeset 13507 febb8e5d2a9d
parent 11655 923e4d0d36d5
child 13926 6e62e5357a10
permissions -rw-r--r--
tidying of Isar scripts
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(*  Title:      HOL/Auth/Yahalom
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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Inductive relation "yahalom" for the Yahalom protocol.
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Demonstrates of why Oops is necessary.  This protocol can be attacked because
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it doesn't keep NB secret, but without Oops it can be "verified" anyway.
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The issues are discussed in lcp's LICS 2000 invited lecture.
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*)
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theory Yahalom_Bad = Shared:
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consts  yahalom   :: "event list set"
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inductive "yahalom"
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  intros
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         (*Initial trace is empty*)
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   Nil:  "[] \<in> yahalom"
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         (*The spy MAY say anything he CAN say.  We do not expect him to
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           invent new nonces here, but he can also use NS1.  Common to
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           all similar protocols.*)
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   Fake: "[| evsf \<in> yahalom;  X \<in> synth (analz (knows Spy evsf)) |]
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          ==> Says Spy B X  # evsf \<in> yahalom"
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         (*A message that has been sent can be received by the
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           intended recipient.*)
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   Reception: "[| evsr \<in> yahalom;  Says A B X \<in> set evsr |]
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               ==> Gets B X # evsr \<in> yahalom"
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         (*Alice initiates a protocol run*)
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   YM1:  "[| evs1 \<in> yahalom;  Nonce NA \<notin> used evs1 |]
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          ==> Says A B {|Agent A, Nonce NA|} # evs1 \<in> yahalom"
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         (*Bob's response to Alice's message.*)
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   YM2:  "[| evs2 \<in> yahalom;  Nonce NB \<notin> used evs2;
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             Gets B {|Agent A, Nonce NA|} \<in> set evs2 |]
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          ==> Says B Server
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                  {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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                # evs2 \<in> yahalom"
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         (*The Server receives Bob's message.  He responds by sending a
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            new session key to Alice, with a packet for forwarding to Bob.*)
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   YM3:  "[| evs3 \<in> yahalom;  Key KAB \<notin> used evs3;
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             Gets Server
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                  {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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               \<in> set evs3 |]
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          ==> Says Server A
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                   {|Crypt (shrK A) {|Agent B, Key KAB, Nonce NA, Nonce NB|},
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                     Crypt (shrK B) {|Agent A, Key KAB|}|}
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                # evs3 \<in> yahalom"
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         (*Alice receives the Server's (?) message, checks her Nonce, and
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           uses the new session key to send Bob his Nonce.  The premise
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           A \<noteq> Server is needed to prove Says_Server_not_range.*)
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   YM4:  "[| evs4 \<in> yahalom;  A \<noteq> Server;
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             Gets A {|Crypt(shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|}, X|}
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                \<in> set evs4;
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             Says A B {|Agent A, Nonce NA|} \<in> set evs4 |]
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          ==> Says A B {|X, Crypt K (Nonce NB)|} # evs4 \<in> yahalom"
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declare Says_imp_knows_Spy [THEN analz.Inj, dest]
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declare parts.Body  [dest]
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declare Fake_parts_insert_in_Un  [dest]
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declare analz_into_parts [dest]
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(*A "possibility property": there are traces that reach the end*)
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lemma "A \<noteq> Server
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      ==> \<exists>X NB K. \<exists>evs \<in> yahalom.
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             Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
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apply (intro exI bexI)
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apply (rule_tac [2] yahalom.Nil
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                    [THEN yahalom.YM1, THEN yahalom.Reception,
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                     THEN yahalom.YM2, THEN yahalom.Reception,
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                     THEN yahalom.YM3, THEN yahalom.Reception,
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                     THEN yahalom.YM4], possibility)
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done
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lemma Gets_imp_Says:
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     "[| Gets B X \<in> set evs; evs \<in> yahalom |] ==> \<exists>A. Says A B X \<in> set evs"
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by (erule rev_mp, erule yahalom.induct, auto)
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(*Must be proved separately for each protocol*)
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lemma Gets_imp_knows_Spy:
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     "[| Gets B X \<in> set evs; evs \<in> yahalom |]  ==> X \<in> knows Spy evs"
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by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)
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declare Gets_imp_knows_Spy [THEN analz.Inj, dest]
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(**** Inductive proofs about yahalom ****)
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(** For reasoning about the encrypted portion of messages **)
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(*Lets us treat YM4 using a similar argument as for the Fake case.*)
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lemma YM4_analz_knows_Spy:
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     "[| Gets A {|Crypt (shrK A) Y, X|} \<in> set evs;  evs \<in> yahalom |]
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      ==> X \<in> analz (knows Spy evs)"
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by blast
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lemmas YM4_parts_knows_Spy =
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       YM4_analz_knows_Spy [THEN analz_into_parts, standard]
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(** Theorems of the form X \<notin> parts (knows Spy evs) imply that NOBODY
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    sends messages containing X! **)
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(*Spy never sees a good agent's shared key!*)
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lemma Spy_see_shrK [simp]:
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     "evs \<in> yahalom ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
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apply (erule yahalom.induct, force,
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       drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)
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done
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lemma Spy_analz_shrK [simp]:
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     "evs \<in> yahalom ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
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by auto
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lemma Spy_see_shrK_D [dest!]:
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     "[|Key (shrK A) \<in> parts (knows Spy evs);  evs \<in> yahalom|] ==> A \<in> bad"
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by (blast dest: Spy_see_shrK)
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(*Nobody can have used non-existent keys!  Needed to apply analz_insert_Key*)
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lemma new_keys_not_used [rule_format, simp]:
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 "evs \<in> yahalom ==> Key K \<notin> used evs --> K \<notin> keysFor (parts (knows Spy evs))"
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM3, YM4*)
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apply (blast dest!: keysFor_parts_insert)+
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done
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(****
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 The following is to prove theorems of the form
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  Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
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  Key K \<in> analz (knows Spy evs)
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 A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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lemma analz_image_freshK [rule_format]:
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 "evs \<in> yahalom ==>
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   \<forall>K KK. KK <= - (range shrK) -->
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          (Key K \<in> analz (Key`KK Un (knows Spy evs))) =
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          (K \<in> KK | Key K \<in> analz (knows Spy evs))"
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apply (erule yahalom.induct, force,
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       drule_tac [6] YM4_analz_knows_Spy, analz_freshK, spy_analz)
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done
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lemma analz_insert_freshK: "[| evs \<in> yahalom;  KAB \<notin> range shrK |] ==>
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      (Key K \<in> analz (insert (Key KAB) (knows Spy evs))) =
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      (K = KAB | Key K \<in> analz (knows Spy evs))"
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by (simp only: analz_image_freshK analz_image_freshK_simps)
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(*** The Key K uniquely identifies the Server's  message. **)
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lemma unique_session_keys:
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     "[| Says Server A
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          {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} \<in> set evs;
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        Says Server A'
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          {|Crypt (shrK A') {|Agent B', Key K, na', nb'|}, X'|} \<in> set evs;
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        evs \<in> yahalom |]
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     ==> A=A' & B=B' & na=na' & nb=nb'"
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apply (erule rev_mp, erule rev_mp)
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apply (erule yahalom.induct, simp_all)
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(*YM3, by freshness, and YM4*)
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apply blast+
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done
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(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
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lemma secrecy_lemma:
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     "[| A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Says Server A
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            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},
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              Crypt (shrK B) {|Agent A, Key K|}|}
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           \<in> set evs -->
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          Key K \<notin> analz (knows Spy evs)"
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apply (erule yahalom.induct, force, drule_tac [6] YM4_analz_knows_Spy)
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apply (simp_all add: pushes analz_insert_eq analz_insert_freshK, spy_analz)  (*Fake*)
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apply (blast dest: unique_session_keys)  (*YM3*)
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done
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(*Final version*)
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lemma Spy_not_see_encrypted_key:
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     "[| Says Server A
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            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},
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              Crypt (shrK B) {|Agent A, Key K|}|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest: secrecy_lemma)
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(** Security Guarantee for A upon receiving YM3 **)
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(*If the encrypted message appears then it originated with the Server*)
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lemma A_trusts_YM3:
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     "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} \<in> parts (knows Spy evs);
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         A \<notin> bad;  evs \<in> yahalom |]
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       ==> Says Server A
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            {|Crypt (shrK A) {|Agent B, Key K, na, nb|},
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              Crypt (shrK B) {|Agent A, Key K|}|}
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           \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM3*)
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apply blast+
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done
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(*The obvious combination of A_trusts_YM3 with Spy_not_see_encrypted_key*)
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lemma A_gets_good_key:
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     "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} \<in> parts (knows Spy evs);
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest!: A_trusts_YM3 Spy_not_see_encrypted_key)
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(** Security Guarantees for B upon receiving YM4 **)
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(*B knows, by the first part of A's message, that the Server distributed
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  the key for A and B.  But this part says nothing about nonces.*)
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lemma B_trusts_YM4_shrK:
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     "[| Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs);
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         B \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>NA NB. Says Server A
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                      {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|},
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                        Crypt (shrK B) {|Agent A, Key K|}|}
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                     \<in> set evs"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy, simp_all)
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(*Fake, YM3*)
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apply blast+
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done
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(** Up to now, the reasoning is similar to standard Yahalom.  Now the
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    doubtful reasoning occurs.  We should not be assuming that an unknown
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    key is secure, but the model allows us to: there is no Oops rule to
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    let session keys become compromised.*)
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(*B knows, by the second part of A's message, that the Server distributed
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  the key quoting nonce NB.  This part says nothing about agent names.
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  Secrecy of K is assumed; the valid Yahalom proof uses (and later proves)
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  the secrecy of NB.*)
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lemma B_trusts_YM4_newK [rule_format]:
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     "[|Key K \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
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      ==> Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
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          (\<exists>A B NA. Says Server A
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                      {|Crypt (shrK A) {|Agent B, Key K,
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                                Nonce NA, Nonce NB|},
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                        Crypt (shrK B) {|Agent A, Key K|}|}
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                     \<in> set evs)"
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apply (erule rev_mp)
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy)
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apply (analz_mono_contra, simp_all)
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(*Fake*)
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apply blast
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(*YM3*)
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apply blast
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(*A is uncompromised because NB is secure
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  A's certificate guarantees the existence of the Server message*)
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apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
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             dest: Says_imp_spies
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                   parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])
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done
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(*B's session key guarantee from YM4.  The two certificates contribute to a
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  single conclusion about the Server's message. *)
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lemma B_trusts_YM4:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
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                  Crypt K (Nonce NB)|} \<in> set evs;
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         Says B Server
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           {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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       ==> \<exists>na nb. Says Server A
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                   {|Crypt (shrK A) {|Agent B, Key K, na, nb|},
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                     Crypt (shrK B) {|Agent A, Key K|}|}
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             \<in> set evs"
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by (blast dest: B_trusts_YM4_newK B_trusts_YM4_shrK Spy_not_see_encrypted_key
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                unique_session_keys)
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(*The obvious combination of B_trusts_YM4 with Spy_not_see_encrypted_key*)
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lemma B_gets_good_key:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
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                  Crypt K (Nonce NB)|} \<in> set evs;
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         Says B Server
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           {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> Key K \<notin> analz (knows Spy evs)"
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by (blast dest!: B_trusts_YM4 Spy_not_see_encrypted_key)
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(*** Authenticating B to A: these proofs are not considered.
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     They are irrelevant to showing the need for Oops. ***)
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(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)
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(*Assuming the session key is secure, if both certificates are present then
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  A has said NB.  We can't be sure about the rest of A's message, but only
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  NB matters for freshness.*)
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lemma A_Said_YM3_lemma [rule_format]:
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     "evs \<in> yahalom
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      ==> Key K \<notin> analz (knows Spy evs) -->
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          Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
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          Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs) -->
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          B \<notin> bad -->
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          (\<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs)"
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apply (erule yahalom.induct, force,
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       frule_tac [6] YM4_parts_knows_Spy)
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apply (analz_mono_contra, simp_all)
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(*Fake*)
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apply blast
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(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
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apply (force dest!: Crypt_imp_keysFor)
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(*YM4: was Crypt K (Nonce NB) the very last message?  If not, use ind. hyp.*)
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apply (simp add: ex_disj_distrib)
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(*yes: apply unicity of session keys*)
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apply (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK
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                    Crypt_Spy_analz_bad
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             dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
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done
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(*If B receives YM4 then A has used nonce NB (and therefore is alive).
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  Moreover, A associates K with NB (thus is talking about the same run).
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  Other premises guarantee secrecy of K.*)
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lemma YM4_imp_A_Said_YM3 [rule_format]:
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     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
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                  Crypt K (Nonce NB)|} \<in> set evs;
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         Says B Server
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           {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}
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           \<in> set evs;
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         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
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      ==> \<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
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apply (blast intro!: A_Said_YM3_lemma
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            dest: Spy_not_see_encrypted_key B_trusts_YM4 Gets_imp_Says)
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done
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end