src/HOL/Auth/Yahalom.thy
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 14207 f20fbb141673
child 16417 9bc16273c2d4
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
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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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*)
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header{*The Yahalom Protocol*}
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theory Yahalom = Public:
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text{*From page 257 of
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  Burrows, Abadi and Needham (1989).  A Logic of Authentication.
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  Proc. Royal Soc. 426
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This theory has the prototypical example of a secrecy relation, KeyCryptNonce.
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*}
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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, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
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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;  KAB \<in> symKeys;
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             Gets Server 
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                  {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
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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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   YM4:  
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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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           @{term "A \<noteq> Server"} is needed for @{text Says_Server_not_range}.
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           Alice can check that K is symmetric by its length.*}
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	 "[| evs4 \<in> yahalom;  A \<noteq> Server;  K \<in> symKeys;
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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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         (*This message models possible leaks of session keys.  The Nonces
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           identify the protocol run.  Quoting Server here ensures they are
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           correct.*)
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   Oops: "[| evso \<in> yahalom;  
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             Says Server A {|Crypt (shrK A)
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                                   {|Agent B, Key K, Nonce NA, Nonce NB|},
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                             X|}  \<in> set evso |]
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          ==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \<in> yahalom"
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constdefs 
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  KeyWithNonce :: "[key, nat, event list] => bool"
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  "KeyWithNonce K NB evs ==
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     \<exists>A B na X. 
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       Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|} 
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         \<in> set evs"
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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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text{*A "possibility property": there are traces that reach the end*}
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lemma "[| A \<noteq> Server; K \<in> symKeys; Key K \<notin> used [] |]
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      ==> \<exists>X NB. \<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])
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apply (possibility, simp add: used_Cons)
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done
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subsection{*Regularity Lemmas for Yahalom*}
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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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text{*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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text{*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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text{*For Oops*}
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lemma YM4_Key_parts_knows_Spy:
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     "Says Server A {|Crypt (shrK A) {|B,K,NA,NB|}, X|} \<in> set evs
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      ==> K \<in> parts (knows Spy evs)"
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by (blast dest!: parts.Body Says_imp_knows_Spy [THEN parts.Inj])
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text{*Theorems of the form @{term "X \<notin> parts (knows Spy evs)"} imply 
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that NOBODY sends messages containing X! *}
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text{*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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by (erule yahalom.induct, force,
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    drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)
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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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text{*Nobody can have used non-existent keys!
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    Needed to apply @{text analz_insert_Key}*}
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lemma new_keys_not_used [simp]:
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    "[|Key K \<notin> used evs; K \<in> symKeys; evs \<in> yahalom|]
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     ==> K \<notin> keysFor (parts (spies 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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txt{*Fake*}
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apply (force dest!: keysFor_parts_insert, auto)
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done
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text{*Earlier, all protocol proofs declared this theorem.
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  But only a few proofs need it, e.g. Yahalom and Kerberos IV.*}
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lemma new_keys_not_analzd:
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 "[|K \<in> symKeys; evs \<in> yahalom; Key K \<notin> used evs|]
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  ==> K \<notin> keysFor (analz (knows Spy evs))"
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by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD])
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text{*Describes the form of K when the Server sends this message.  Useful for
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  Oops as well as main secrecy property.*}
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lemma Says_Server_not_range [simp]:
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     "[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|}
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           \<in> set evs;   evs \<in> yahalom |]
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      ==> K \<notin> range shrK"
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by (erule rev_mp, erule yahalom.induct, simp_all, blast)
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subsection{*Secrecy Theorems*}
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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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text{* 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,
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       drule_tac [7] YM4_analz_knows_Spy, analz_freshK, spy_analz, blast)
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apply (simp only: Says_Server_not_range analz_image_freshK_simps)
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done
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lemma analz_insert_freshK:
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     "[| 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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text{*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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txt{*YM3, by freshness, and YM4*}
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apply blast+
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done
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text{*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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          Notes Spy {|na, nb, Key K|} \<notin> set evs -->
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          Key K \<notin> analz (knows Spy evs)"
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apply (erule yahalom.induct, force,
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       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, Oops*}
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done
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text{*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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         Notes Spy {|na, nb, Key K|} \<notin> 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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subsubsection{* Security Guarantee for A upon receiving YM3 *}
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text{*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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txt{*Fake, YM3*}
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apply blast+
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done
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text{*The obvious combination of @{text A_trusts_YM3} with
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  @{text 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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         Notes Spy {|na, nb, Key K|} \<notin> 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!: A_trusts_YM3 Spy_not_see_encrypted_key)
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subsubsection{* Security Guarantees for B upon receiving YM4 *}
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text{*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,
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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, simp_all)
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txt{*Fake, YM3*}
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apply blast+
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done
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text{*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 NB is crucial.  Note that  Nonce NB \<notin> analz(knows Spy evs)  must
paulson@14207
   303
  be the FIRST antecedent of the induction formula.*}
paulson@14207
   304
lemma B_trusts_YM4_newK [rule_format]:
paulson@11251
   305
     "[|Crypt K (Nonce NB) \<in> parts (knows Spy evs);
paulson@11251
   306
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
paulson@14207
   307
      ==> \<exists>A B NA. Says Server A
paulson@11251
   308
                      {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|},
paulson@14207
   309
                        Crypt (shrK B) {|Agent A, Key K|}|}
paulson@11251
   310
                     \<in> set evs"
paulson@11251
   311
apply (erule rev_mp, erule rev_mp)
paulson@14207
   312
apply (erule yahalom.induct, force,
paulson@11251
   313
       frule_tac [6] YM4_parts_knows_Spy)
paulson@11251
   314
apply (analz_mono_contra, simp_all)
paulson@14207
   315
txt{*Fake, YM3*}
paulson@11251
   316
apply blast
paulson@11251
   317
apply blast
paulson@14207
   318
txt{*YM4.  A is uncompromised because NB is secure
paulson@14207
   319
  A's certificate guarantees the existence of the Server message*}
paulson@14207
   320
apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
paulson@14207
   321
             dest: Says_imp_spies
paulson@11251
   322
                   parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])
paulson@11251
   323
done
paulson@11251
   324
paulson@11251
   325
paulson@14207
   326
subsubsection{* Towards proving secrecy of Nonce NB *}
paulson@11251
   327
paulson@14207
   328
text{*Lemmas about the predicate KeyWithNonce*}
paulson@11251
   329
paulson@14207
   330
lemma KeyWithNonceI:
paulson@14207
   331
 "Says Server A
paulson@14207
   332
          {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}
paulson@11251
   333
        \<in> set evs ==> KeyWithNonce K NB evs"
paulson@11251
   334
by (unfold KeyWithNonce_def, blast)
paulson@11251
   335
paulson@14207
   336
lemma KeyWithNonce_Says [simp]:
paulson@14207
   337
   "KeyWithNonce K NB (Says S A X # evs) =
paulson@11251
   338
      (Server = S &
paulson@14207
   339
       (\<exists>B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|})
paulson@11251
   340
      | KeyWithNonce K NB evs)"
paulson@11251
   341
by (simp add: KeyWithNonce_def, blast)
paulson@11251
   342
paulson@11251
   343
paulson@14207
   344
lemma KeyWithNonce_Notes [simp]:
paulson@11251
   345
   "KeyWithNonce K NB (Notes A X # evs) = KeyWithNonce K NB evs"
paulson@11251
   346
by (simp add: KeyWithNonce_def)
paulson@11251
   347
paulson@14207
   348
lemma KeyWithNonce_Gets [simp]:
paulson@11251
   349
   "KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs"
paulson@11251
   350
by (simp add: KeyWithNonce_def)
paulson@11251
   351
paulson@14207
   352
text{*A fresh key cannot be associated with any nonce
paulson@14207
   353
  (with respect to a given trace). *}
paulson@14207
   354
lemma fresh_not_KeyWithNonce:
paulson@14207
   355
     "Key K \<notin> used evs ==> ~ KeyWithNonce K NB evs"
paulson@11251
   356
by (unfold KeyWithNonce_def, blast)
paulson@11251
   357
paulson@14207
   358
text{*The Server message associates K with NB' and therefore not with any
paulson@14207
   359
  other nonce NB.*}
paulson@14207
   360
lemma Says_Server_KeyWithNonce:
paulson@14207
   361
 "[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB'|}, X|}
paulson@14207
   362
       \<in> set evs;
paulson@14207
   363
     NB \<noteq> NB';  evs \<in> yahalom |]
paulson@11251
   364
  ==> ~ KeyWithNonce K NB evs"
paulson@11251
   365
by (unfold KeyWithNonce_def, blast dest: unique_session_keys)
paulson@11251
   366
paulson@11251
   367
paulson@14207
   368
text{*The only nonces that can be found with the help of session keys are
paulson@11251
   369
  those distributed as nonce NB by the Server.  The form of the theorem
paulson@14207
   370
  recalls @{text analz_image_freshK}, but it is much more complicated.*}
paulson@11251
   371
paulson@11251
   372
paulson@14207
   373
text{*As with @{text analz_image_freshK}, we take some pains to express the 
paulson@14207
   374
  property as a logical equivalence so that the simplifier can apply it.*}
paulson@11251
   375
lemma Nonce_secrecy_lemma:
paulson@14207
   376
     "P --> (X \<in> analz (G Un H)) --> (X \<in> analz H)  ==>
paulson@11251
   377
      P --> (X \<in> analz (G Un H)) = (X \<in> analz H)"
paulson@11251
   378
by (blast intro: analz_mono [THEN subsetD])
paulson@11251
   379
paulson@11251
   380
lemma Nonce_secrecy:
paulson@14207
   381
     "evs \<in> yahalom ==>
paulson@14207
   382
      (\<forall>KK. KK <= - (range shrK) -->
paulson@14207
   383
           (\<forall>K \<in> KK. K \<in> symKeys --> ~ KeyWithNonce K NB evs)   -->
paulson@14207
   384
           (Nonce NB \<in> analz (Key`KK Un (knows Spy evs))) =
paulson@11251
   385
           (Nonce NB \<in> analz (knows Spy evs)))"
paulson@14207
   386
apply (erule yahalom.induct,
paulson@14207
   387
       frule_tac [7] YM4_analz_knows_Spy)
paulson@11251
   388
apply (safe del: allI impI intro!: Nonce_secrecy_lemma [THEN impI, THEN allI])
paulson@14207
   389
apply (simp_all del: image_insert image_Un
paulson@11251
   390
       add: analz_image_freshK_simps split_ifs
paulson@14207
   391
            all_conj_distrib ball_conj_distrib
paulson@11251
   392
            analz_image_freshK fresh_not_KeyWithNonce
paulson@11251
   393
            imp_disj_not1               (*Moves NBa\<noteq>NB to the front*)
paulson@11251
   394
            Says_Server_KeyWithNonce)
paulson@14207
   395
txt{*For Oops, simplification proves NBa\<noteq>NB.  By Says_Server_KeyWithNonce,
paulson@11251
   396
  we get (~ KeyWithNonce K NB evs); then simplification can apply the
paulson@14207
   397
  induction hypothesis with KK = {K}.*}
paulson@14207
   398
txt{*Fake*}
paulson@11251
   399
apply spy_analz
paulson@14207
   400
txt{*YM2*}
paulson@14207
   401
apply blast
paulson@14207
   402
txt{*YM3*}
paulson@14207
   403
apply blast
paulson@14207
   404
txt{*YM4*}
paulson@13507
   405
apply (erule_tac V = "\<forall>KK. ?P KK" in thin_rl, clarify)
paulson@14207
   406
txt{*If A \<in> bad then NBa is known, therefore NBa \<noteq> NB.  Previous two steps 
paulson@14207
   407
   make the next step faster.*}
paulson@11251
   408
apply (blast dest!: Gets_imp_Says Says_imp_spies Crypt_Spy_analz_bad
paulson@11251
   409
         dest: analz.Inj
paulson@11251
   410
           parts.Inj [THEN parts.Fst, THEN A_trusts_YM3, THEN KeyWithNonceI])
paulson@11251
   411
done
paulson@11251
   412
paulson@11251
   413
paulson@14207
   414
text{*Version required below: if NB can be decrypted using a session key then
paulson@14207
   415
   it was distributed with that key.  The more general form above is required
paulson@14207
   416
   for the induction to carry through.*}
paulson@11251
   417
lemma single_Nonce_secrecy:
paulson@14207
   418
     "[| Says Server A
paulson@14207
   419
          {|Crypt (shrK A) {|Agent B, Key KAB, na, Nonce NB'|}, X|}
paulson@14207
   420
         \<in> set evs;
paulson@14207
   421
         NB \<noteq> NB';  KAB \<notin> range shrK;  evs \<in> yahalom |]
paulson@14207
   422
      ==> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =
paulson@11251
   423
          (Nonce NB \<in> analz (knows Spy evs))"
paulson@11251
   424
by (simp_all del: image_insert image_Un imp_disjL
paulson@11251
   425
             add: analz_image_freshK_simps split_ifs
paulson@13507
   426
                  Nonce_secrecy Says_Server_KeyWithNonce)
paulson@11251
   427
paulson@11251
   428
paulson@14207
   429
subsubsection{* The Nonce NB uniquely identifies B's message. *}
paulson@11251
   430
paulson@11251
   431
lemma unique_NB:
paulson@14207
   432
     "[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);
paulson@14207
   433
         Crypt (shrK B') {|Agent A', Nonce NA', nb|} \<in> parts (knows Spy evs);
paulson@14207
   434
        evs \<in> yahalom;  B \<notin> bad;  B' \<notin> bad |]
paulson@11251
   435
      ==> NA' = NA & A' = A & B' = B"
paulson@11251
   436
apply (erule rev_mp, erule rev_mp)
paulson@14207
   437
apply (erule yahalom.induct, force,
paulson@11251
   438
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
paulson@14207
   439
txt{*Fake, and YM2 by freshness*}
paulson@11251
   440
apply blast+
paulson@11251
   441
done
paulson@11251
   442
paulson@11251
   443
paulson@14207
   444
text{*Variant useful for proving secrecy of NB.  Because nb is assumed to be
paulson@14207
   445
  secret, we no longer must assume B, B' not bad.*}
paulson@11251
   446
lemma Says_unique_NB:
paulson@14207
   447
     "[| Says C S   {|X,  Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@14207
   448
           \<in> set evs;
paulson@14207
   449
         Gets S' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|}
paulson@14207
   450
           \<in> set evs;
paulson@14207
   451
         nb \<notin> analz (knows Spy evs);  evs \<in> yahalom |]
paulson@11251
   452
      ==> NA' = NA & A' = A & B' = B"
paulson@14207
   453
by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
paulson@11251
   454
          dest: Says_imp_spies unique_NB parts.Inj analz.Inj)
paulson@11251
   455
paulson@11251
   456
paulson@14207
   457
subsubsection{* A nonce value is never used both as NA and as NB *}
paulson@11251
   458
paulson@11251
   459
lemma no_nonce_YM1_YM2:
paulson@11251
   460
     "[|Crypt (shrK B') {|Agent A', Nonce NB, nb'|} \<in> parts(knows Spy evs);
paulson@11251
   461
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
paulson@11251
   462
  ==> Crypt (shrK B)  {|Agent A, na, Nonce NB|} \<notin> parts(knows Spy evs)"
paulson@11251
   463
apply (erule rev_mp, erule rev_mp)
paulson@14207
   464
apply (erule yahalom.induct, force,
paulson@11251
   465
       frule_tac [6] YM4_parts_knows_Spy)
paulson@11251
   466
apply (analz_mono_contra, simp_all)
paulson@14207
   467
txt{*Fake, YM2*}
paulson@11251
   468
apply blast+
paulson@11251
   469
done
paulson@11251
   470
paulson@14207
   471
text{*The Server sends YM3 only in response to YM2.*}
paulson@11251
   472
lemma Says_Server_imp_YM2:
paulson@11251
   473
     "[| Says Server A {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} \<in> set evs;
paulson@14207
   474
         evs \<in> yahalom |]
paulson@14207
   475
      ==> Gets Server {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |}
paulson@11251
   476
             \<in> set evs"
paulson@14207
   477
by (erule rev_mp, erule yahalom.induct, auto)
paulson@11251
   478
paulson@14207
   479
text{*A vital theorem for B, that nonce NB remains secure from the Spy.*}
paulson@11251
   480
lemma Spy_not_see_NB :
paulson@14207
   481
     "[| Says B Server
paulson@14207
   482
	        {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@11251
   483
	   \<in> set evs;
paulson@11251
   484
	 (\<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
paulson@14207
   485
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   486
      ==> Nonce NB \<notin> analz (knows Spy evs)"
paulson@11251
   487
apply (erule rev_mp, erule rev_mp)
paulson@14207
   488
apply (erule yahalom.induct, force,
paulson@11251
   489
       frule_tac [6] YM4_analz_knows_Spy)
paulson@11251
   490
apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq
paulson@11251
   491
                     analz_insert_freshK)
paulson@14207
   492
txt{*Fake*}
paulson@11251
   493
apply spy_analz
paulson@14207
   494
txt{*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*}
paulson@11251
   495
apply blast
paulson@14207
   496
txt{*YM2*}
paulson@11251
   497
apply blast
paulson@14207
   498
txt{*Prove YM3 by showing that no NB can also be an NA*}
paulson@11251
   499
apply (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
paulson@14207
   500
txt{*LEVEL 7: YM4 and Oops remain*}
paulson@11251
   501
apply (clarify, simp add: all_conj_distrib)
paulson@14207
   502
txt{*YM4: key K is visible to Spy, contradicting session key secrecy theorem*}
paulson@14207
   503
txt{*Case analysis on Aa:bad; PROOF FAILED problems
paulson@14207
   504
  use Says_unique_NB to identify message components: Aa=A, Ba=B*}
paulson@14207
   505
apply (blast dest!: Says_unique_NB analz_shrK_Decrypt
paulson@14207
   506
                    parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]
paulson@11251
   507
             dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
paulson@11251
   508
                   Spy_not_see_encrypted_key)
paulson@14207
   509
txt{*Oops case: if the nonce is betrayed now, show that the Oops event is
paulson@14207
   510
  covered by the quantified Oops assumption.*}
paulson@11251
   511
apply (clarify, simp add: all_conj_distrib)
paulson@11251
   512
apply (frule Says_Server_imp_YM2, assumption)
paulson@11251
   513
apply (case_tac "NB = NBa")
paulson@14207
   514
txt{*If NB=NBa then all other components of the Oops message agree*}
paulson@11251
   515
apply (blast dest: Says_unique_NB)
paulson@14207
   516
txt{*case NB \<noteq> NBa*}
paulson@11251
   517
apply (simp add: single_Nonce_secrecy)
paulson@11251
   518
apply (blast dest!: no_nonce_YM1_YM2 (*to prove NB\<noteq>NAa*))
paulson@11251
   519
done
paulson@11251
   520
paulson@11251
   521
paulson@14207
   522
text{*B's session key guarantee from YM4.  The two certificates contribute to a
paulson@11251
   523
  single conclusion about the Server's message.  Note that the "Notes Spy"
paulson@11251
   524
  assumption must quantify over \<forall>POSSIBLE keys instead of our particular K.
paulson@11251
   525
  If this run is broken and the spy substitutes a certificate containing an
paulson@14207
   526
  old key, B has no means of telling.*}
paulson@11251
   527
lemma B_trusts_YM4:
paulson@14207
   528
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
paulson@14207
   529
                  Crypt K (Nonce NB)|} \<in> set evs;
paulson@14207
   530
         Says B Server
paulson@14207
   531
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@14207
   532
           \<in> set evs;
paulson@14207
   533
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;
paulson@14207
   534
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@14207
   535
       ==> Says Server A
paulson@14207
   536
                   {|Crypt (shrK A) {|Agent B, Key K,
paulson@14207
   537
                             Nonce NA, Nonce NB|},
paulson@14207
   538
                     Crypt (shrK B) {|Agent A, Key K|}|}
paulson@11251
   539
             \<in> set evs"
paulson@14207
   540
by (blast dest: Spy_not_see_NB Says_unique_NB
paulson@11251
   541
                Says_Server_imp_YM2 B_trusts_YM4_newK)
paulson@11251
   542
paulson@11251
   543
paulson@11251
   544
paulson@14207
   545
text{*The obvious combination of @{text B_trusts_YM4} with 
paulson@14207
   546
  @{text Spy_not_see_encrypted_key}*}
paulson@11251
   547
lemma B_gets_good_key:
paulson@11251
   548
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
paulson@11251
   549
                  Crypt K (Nonce NB)|} \<in> set evs;
paulson@14207
   550
         Says B Server
paulson@14207
   551
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@14207
   552
           \<in> set evs;
paulson@14207
   553
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;
paulson@14207
   554
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   555
      ==> Key K \<notin> analz (knows Spy evs)"
paulson@11251
   556
by (blast dest!: B_trusts_YM4 Spy_not_see_encrypted_key)
paulson@11251
   557
paulson@11251
   558
paulson@14207
   559
subsection{*Authenticating B to A*}
paulson@11251
   560
paulson@14207
   561
text{*The encryption in message YM2 tells us it cannot be faked.*}
paulson@11251
   562
lemma B_Said_YM2 [rule_format]:
paulson@11251
   563
     "[|Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);
paulson@11251
   564
        evs \<in> yahalom|]
paulson@11251
   565
      ==> B \<notin> bad -->
paulson@11251
   566
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@11251
   567
            \<in> set evs"
paulson@14207
   568
apply (erule rev_mp, erule yahalom.induct, force,
paulson@11251
   569
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
paulson@14207
   570
txt{*Fake*}
paulson@11251
   571
apply blast
paulson@11251
   572
done
paulson@11251
   573
paulson@14207
   574
text{*If the server sends YM3 then B sent YM2*}
paulson@11251
   575
lemma YM3_auth_B_to_A_lemma:
paulson@14207
   576
     "[|Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
paulson@11251
   577
       \<in> set evs;  evs \<in> yahalom|]
paulson@14207
   578
      ==> B \<notin> bad -->
paulson@11251
   579
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@11251
   580
            \<in> set evs"
paulson@11251
   581
apply (erule rev_mp, erule yahalom.induct, simp_all)
paulson@14207
   582
txt{*YM3, YM4*}
paulson@11251
   583
apply (blast dest!: B_Said_YM2)+
paulson@11251
   584
done
paulson@11251
   585
paulson@14207
   586
text{*If A receives YM3 then B has used nonce NA (and therefore is alive)*}
paulson@11251
   587
lemma YM3_auth_B_to_A:
paulson@14207
   588
     "[| Gets A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
paulson@14207
   589
           \<in> set evs;
paulson@14207
   590
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@14207
   591
      ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@11251
   592
       \<in> set evs"
paulson@11251
   593
by (blast dest!: A_trusts_YM3 YM3_auth_B_to_A_lemma elim: knows_Spy_partsEs)
paulson@11251
   594
paulson@11251
   595
paulson@14207
   596
subsection{*Authenticating A to B using the certificate 
paulson@14207
   597
  @{term "Crypt K (Nonce NB)"}*}
paulson@11251
   598
paulson@14207
   599
text{*Assuming the session key is secure, if both certificates are present then
paulson@11251
   600
  A has said NB.  We can't be sure about the rest of A's message, but only
paulson@14207
   601
  NB matters for freshness.*}
paulson@11251
   602
lemma A_Said_YM3_lemma [rule_format]:
paulson@11251
   603
     "evs \<in> yahalom
paulson@11251
   604
      ==> Key K \<notin> analz (knows Spy evs) -->
paulson@11251
   605
          Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
paulson@11251
   606
          Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs) -->
paulson@11251
   607
          B \<notin> bad -->
paulson@11251
   608
          (\<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs)"
paulson@14207
   609
apply (erule yahalom.induct, force,
paulson@11251
   610
       frule_tac [6] YM4_parts_knows_Spy)
paulson@11251
   611
apply (analz_mono_contra, simp_all)
paulson@14207
   612
txt{*Fake*}
paulson@11251
   613
apply blast
paulson@14207
   614
txt{*YM3: by @{text new_keys_not_used}, the message
paulson@14207
   615
   @{term "Crypt K (Nonce NB)"} could not exist*}
paulson@11251
   616
apply (force dest!: Crypt_imp_keysFor)
paulson@14207
   617
txt{*YM4: was @{term "Crypt K (Nonce NB)"} the very last message?
paulson@14207
   618
    If not, use the induction hypothesis*}
paulson@11251
   619
apply (simp add: ex_disj_distrib)
paulson@14207
   620
txt{*yes: apply unicity of session keys*}
paulson@11251
   621
apply (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK
paulson@14207
   622
                    Crypt_Spy_analz_bad
paulson@11251
   623
             dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
paulson@11251
   624
done
paulson@11251
   625
paulson@14207
   626
text{*If B receives YM4 then A has used nonce NB (and therefore is alive).
paulson@11251
   627
  Moreover, A associates K with NB (thus is talking about the same run).
paulson@14207
   628
  Other premises guarantee secrecy of K.*}
paulson@11251
   629
lemma YM4_imp_A_Said_YM3 [rule_format]:
paulson@11251
   630
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
paulson@11251
   631
                  Crypt K (Nonce NB)|} \<in> set evs;
paulson@11251
   632
         Says B Server
paulson@11251
   633
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@11251
   634
           \<in> set evs;
paulson@11251
   635
         (\<forall>NA k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
paulson@11251
   636
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   637
      ==> \<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
paulson@14207
   638
by (blast intro!: A_Said_YM3_lemma
paulson@11251
   639
          dest: Spy_not_see_encrypted_key B_trusts_YM4 Gets_imp_Says)
paulson@3447
   640
paulson@1985
   641
end