src/HOL/Auth/Yahalom.thy
author haftmann
Fri Jan 02 08:12:46 2009 +0100 (2009-01-02)
changeset 29332 edc1e2a56398
parent 23746 a455e69c31cc
child 32367 a508148f7c25
permissions -rw-r--r--
named code theorem for Fract_norm
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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 imports Public begin
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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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inductive_set yahalom :: "event list set"
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  where
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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_analz_Spy [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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lemmas Gets_imp_analz_Spy = Gets_imp_knows_Spy [THEN analz.Inj]
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declare Gets_imp_analz_Spy [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)
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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
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  distributed the key quoting nonce NB.  This part says nothing about
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  agent names.  Secrecy of NB is crucial.  Note that @{term "Nonce NB
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   303
  \<notin> analz(knows Spy evs)"} must be the FIRST antecedent of the
wenzelm@17411
   304
  induction formula.*}
wenzelm@17411
   305
paulson@14207
   306
lemma B_trusts_YM4_newK [rule_format]:
paulson@11251
   307
     "[|Crypt K (Nonce NB) \<in> parts (knows Spy evs);
paulson@11251
   308
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
paulson@14207
   309
      ==> \<exists>A B NA. Says Server A
paulson@11251
   310
                      {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|},
paulson@14207
   311
                        Crypt (shrK B) {|Agent A, Key K|}|}
paulson@11251
   312
                     \<in> set evs"
paulson@11251
   313
apply (erule rev_mp, erule rev_mp)
paulson@14207
   314
apply (erule yahalom.induct, force,
paulson@11251
   315
       frule_tac [6] YM4_parts_knows_Spy)
paulson@11251
   316
apply (analz_mono_contra, simp_all)
paulson@14207
   317
txt{*Fake, YM3*}
paulson@11251
   318
apply blast
paulson@11251
   319
apply blast
paulson@14207
   320
txt{*YM4.  A is uncompromised because NB is secure
paulson@14207
   321
  A's certificate guarantees the existence of the Server message*}
paulson@14207
   322
apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
paulson@14207
   323
             dest: Says_imp_spies
paulson@11251
   324
                   parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])
paulson@11251
   325
done
paulson@11251
   326
paulson@11251
   327
paulson@14207
   328
subsubsection{* Towards proving secrecy of Nonce NB *}
paulson@11251
   329
paulson@14207
   330
text{*Lemmas about the predicate KeyWithNonce*}
paulson@11251
   331
paulson@14207
   332
lemma KeyWithNonceI:
paulson@14207
   333
 "Says Server A
paulson@14207
   334
          {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}
paulson@11251
   335
        \<in> set evs ==> KeyWithNonce K NB evs"
paulson@11251
   336
by (unfold KeyWithNonce_def, blast)
paulson@11251
   337
paulson@14207
   338
lemma KeyWithNonce_Says [simp]:
paulson@14207
   339
   "KeyWithNonce K NB (Says S A X # evs) =
paulson@11251
   340
      (Server = S &
paulson@14207
   341
       (\<exists>B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|})
paulson@11251
   342
      | KeyWithNonce K NB evs)"
paulson@11251
   343
by (simp add: KeyWithNonce_def, blast)
paulson@11251
   344
paulson@11251
   345
paulson@14207
   346
lemma KeyWithNonce_Notes [simp]:
paulson@11251
   347
   "KeyWithNonce K NB (Notes A X # evs) = KeyWithNonce K NB evs"
paulson@11251
   348
by (simp add: KeyWithNonce_def)
paulson@11251
   349
paulson@14207
   350
lemma KeyWithNonce_Gets [simp]:
paulson@11251
   351
   "KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs"
paulson@11251
   352
by (simp add: KeyWithNonce_def)
paulson@11251
   353
paulson@14207
   354
text{*A fresh key cannot be associated with any nonce
paulson@14207
   355
  (with respect to a given trace). *}
paulson@14207
   356
lemma fresh_not_KeyWithNonce:
paulson@14207
   357
     "Key K \<notin> used evs ==> ~ KeyWithNonce K NB evs"
paulson@11251
   358
by (unfold KeyWithNonce_def, blast)
paulson@11251
   359
paulson@14207
   360
text{*The Server message associates K with NB' and therefore not with any
paulson@14207
   361
  other nonce NB.*}
paulson@14207
   362
lemma Says_Server_KeyWithNonce:
paulson@14207
   363
 "[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB'|}, X|}
paulson@14207
   364
       \<in> set evs;
paulson@14207
   365
     NB \<noteq> NB';  evs \<in> yahalom |]
paulson@11251
   366
  ==> ~ KeyWithNonce K NB evs"
paulson@11251
   367
by (unfold KeyWithNonce_def, blast dest: unique_session_keys)
paulson@11251
   368
paulson@11251
   369
paulson@14207
   370
text{*The only nonces that can be found with the help of session keys are
paulson@11251
   371
  those distributed as nonce NB by the Server.  The form of the theorem
paulson@14207
   372
  recalls @{text analz_image_freshK}, but it is much more complicated.*}
paulson@11251
   373
paulson@11251
   374
paulson@14207
   375
text{*As with @{text analz_image_freshK}, we take some pains to express the 
paulson@14207
   376
  property as a logical equivalence so that the simplifier can apply it.*}
paulson@11251
   377
lemma Nonce_secrecy_lemma:
paulson@14207
   378
     "P --> (X \<in> analz (G Un H)) --> (X \<in> analz H)  ==>
paulson@11251
   379
      P --> (X \<in> analz (G Un H)) = (X \<in> analz H)"
paulson@11251
   380
by (blast intro: analz_mono [THEN subsetD])
paulson@11251
   381
paulson@11251
   382
lemma Nonce_secrecy:
paulson@14207
   383
     "evs \<in> yahalom ==>
paulson@14207
   384
      (\<forall>KK. KK <= - (range shrK) -->
paulson@14207
   385
           (\<forall>K \<in> KK. K \<in> symKeys --> ~ KeyWithNonce K NB evs)   -->
paulson@14207
   386
           (Nonce NB \<in> analz (Key`KK Un (knows Spy evs))) =
paulson@11251
   387
           (Nonce NB \<in> analz (knows Spy evs)))"
paulson@14207
   388
apply (erule yahalom.induct,
paulson@14207
   389
       frule_tac [7] YM4_analz_knows_Spy)
paulson@11251
   390
apply (safe del: allI impI intro!: Nonce_secrecy_lemma [THEN impI, THEN allI])
paulson@14207
   391
apply (simp_all del: image_insert image_Un
paulson@11251
   392
       add: analz_image_freshK_simps split_ifs
paulson@14207
   393
            all_conj_distrib ball_conj_distrib
paulson@11251
   394
            analz_image_freshK fresh_not_KeyWithNonce
paulson@11251
   395
            imp_disj_not1               (*Moves NBa\<noteq>NB to the front*)
paulson@11251
   396
            Says_Server_KeyWithNonce)
wenzelm@17411
   397
txt{*For Oops, simplification proves @{prop "NBa\<noteq>NB"}.  By
wenzelm@17411
   398
  @{term Says_Server_KeyWithNonce}, we get @{prop "~ KeyWithNonce K NB
wenzelm@17411
   399
  evs"}; then simplification can apply the induction hypothesis with
wenzelm@17411
   400
  @{term "KK = {K}"}.*}
paulson@14207
   401
txt{*Fake*}
paulson@11251
   402
apply spy_analz
paulson@14207
   403
txt{*YM2*}
paulson@14207
   404
apply blast
paulson@14207
   405
txt{*YM3*}
paulson@14207
   406
apply blast
paulson@14207
   407
txt{*YM4*}
paulson@13507
   408
apply (erule_tac V = "\<forall>KK. ?P KK" in thin_rl, clarify)
wenzelm@17411
   409
txt{*If @{prop "A \<in> bad"} then @{term NBa} is known, therefore
wenzelm@17411
   410
  @{prop "NBa \<noteq> NB"}.  Previous two steps make the next step
wenzelm@17411
   411
  faster.*}
paulson@11251
   412
apply (blast dest!: Gets_imp_Says Says_imp_spies Crypt_Spy_analz_bad
paulson@11251
   413
         dest: analz.Inj
paulson@11251
   414
           parts.Inj [THEN parts.Fst, THEN A_trusts_YM3, THEN KeyWithNonceI])
paulson@11251
   415
done
paulson@11251
   416
paulson@11251
   417
paulson@14207
   418
text{*Version required below: if NB can be decrypted using a session key then
paulson@14207
   419
   it was distributed with that key.  The more general form above is required
paulson@14207
   420
   for the induction to carry through.*}
paulson@11251
   421
lemma single_Nonce_secrecy:
paulson@14207
   422
     "[| Says Server A
paulson@14207
   423
          {|Crypt (shrK A) {|Agent B, Key KAB, na, Nonce NB'|}, X|}
paulson@14207
   424
         \<in> set evs;
paulson@14207
   425
         NB \<noteq> NB';  KAB \<notin> range shrK;  evs \<in> yahalom |]
paulson@14207
   426
      ==> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =
paulson@11251
   427
          (Nonce NB \<in> analz (knows Spy evs))"
paulson@11251
   428
by (simp_all del: image_insert image_Un imp_disjL
paulson@11251
   429
             add: analz_image_freshK_simps split_ifs
paulson@13507
   430
                  Nonce_secrecy Says_Server_KeyWithNonce)
paulson@11251
   431
paulson@11251
   432
paulson@14207
   433
subsubsection{* The Nonce NB uniquely identifies B's message. *}
paulson@11251
   434
paulson@11251
   435
lemma unique_NB:
paulson@14207
   436
     "[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);
paulson@14207
   437
         Crypt (shrK B') {|Agent A', Nonce NA', nb|} \<in> parts (knows Spy evs);
paulson@14207
   438
        evs \<in> yahalom;  B \<notin> bad;  B' \<notin> bad |]
paulson@11251
   439
      ==> NA' = NA & A' = A & B' = B"
paulson@11251
   440
apply (erule rev_mp, erule rev_mp)
paulson@14207
   441
apply (erule yahalom.induct, force,
paulson@11251
   442
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
paulson@14207
   443
txt{*Fake, and YM2 by freshness*}
paulson@11251
   444
apply blast+
paulson@11251
   445
done
paulson@11251
   446
paulson@11251
   447
paulson@14207
   448
text{*Variant useful for proving secrecy of NB.  Because nb is assumed to be
paulson@14207
   449
  secret, we no longer must assume B, B' not bad.*}
paulson@11251
   450
lemma Says_unique_NB:
paulson@14207
   451
     "[| Says C S   {|X,  Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@14207
   452
           \<in> set evs;
paulson@14207
   453
         Gets S' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|}
paulson@14207
   454
           \<in> set evs;
paulson@14207
   455
         nb \<notin> analz (knows Spy evs);  evs \<in> yahalom |]
paulson@11251
   456
      ==> NA' = NA & A' = A & B' = B"
paulson@14207
   457
by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
paulson@11251
   458
          dest: Says_imp_spies unique_NB parts.Inj analz.Inj)
paulson@11251
   459
paulson@11251
   460
paulson@14207
   461
subsubsection{* A nonce value is never used both as NA and as NB *}
paulson@11251
   462
paulson@11251
   463
lemma no_nonce_YM1_YM2:
paulson@11251
   464
     "[|Crypt (shrK B') {|Agent A', Nonce NB, nb'|} \<in> parts(knows Spy evs);
paulson@11251
   465
        Nonce NB \<notin> analz (knows Spy evs);  evs \<in> yahalom|]
paulson@11251
   466
  ==> Crypt (shrK B)  {|Agent A, na, Nonce NB|} \<notin> parts(knows Spy evs)"
paulson@11251
   467
apply (erule rev_mp, erule rev_mp)
paulson@14207
   468
apply (erule yahalom.induct, force,
paulson@11251
   469
       frule_tac [6] YM4_parts_knows_Spy)
paulson@11251
   470
apply (analz_mono_contra, simp_all)
paulson@14207
   471
txt{*Fake, YM2*}
paulson@11251
   472
apply blast+
paulson@11251
   473
done
paulson@11251
   474
paulson@14207
   475
text{*The Server sends YM3 only in response to YM2.*}
paulson@11251
   476
lemma Says_Server_imp_YM2:
paulson@11251
   477
     "[| Says Server A {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} \<in> set evs;
paulson@14207
   478
         evs \<in> yahalom |]
paulson@14207
   479
      ==> Gets Server {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |}
paulson@11251
   480
             \<in> set evs"
paulson@14207
   481
by (erule rev_mp, erule yahalom.induct, auto)
paulson@11251
   482
paulson@14207
   483
text{*A vital theorem for B, that nonce NB remains secure from the Spy.*}
paulson@11251
   484
lemma Spy_not_see_NB :
paulson@14207
   485
     "[| Says B Server
paulson@14207
   486
	        {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@11251
   487
	   \<in> set evs;
paulson@11251
   488
	 (\<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
paulson@14207
   489
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   490
      ==> Nonce NB \<notin> analz (knows Spy evs)"
paulson@11251
   491
apply (erule rev_mp, erule rev_mp)
paulson@14207
   492
apply (erule yahalom.induct, force,
paulson@11251
   493
       frule_tac [6] YM4_analz_knows_Spy)
paulson@11251
   494
apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq
paulson@11251
   495
                     analz_insert_freshK)
paulson@14207
   496
txt{*Fake*}
paulson@11251
   497
apply spy_analz
paulson@14207
   498
txt{*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*}
paulson@11251
   499
apply blast
paulson@14207
   500
txt{*YM2*}
paulson@11251
   501
apply blast
paulson@14207
   502
txt{*Prove YM3 by showing that no NB can also be an NA*}
paulson@11251
   503
apply (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
paulson@14207
   504
txt{*LEVEL 7: YM4 and Oops remain*}
paulson@11251
   505
apply (clarify, simp add: all_conj_distrib)
paulson@14207
   506
txt{*YM4: key K is visible to Spy, contradicting session key secrecy theorem*}
paulson@14207
   507
txt{*Case analysis on Aa:bad; PROOF FAILED problems
wenzelm@17411
   508
  use @{text Says_unique_NB} to identify message components: @{term "Aa=A"}, @{term "Ba=B"}*}
paulson@14207
   509
apply (blast dest!: Says_unique_NB analz_shrK_Decrypt
paulson@14207
   510
                    parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]
paulson@11251
   511
             dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
paulson@11251
   512
                   Spy_not_see_encrypted_key)
paulson@14207
   513
txt{*Oops case: if the nonce is betrayed now, show that the Oops event is
paulson@14207
   514
  covered by the quantified Oops assumption.*}
paulson@11251
   515
apply (clarify, simp add: all_conj_distrib)
paulson@11251
   516
apply (frule Says_Server_imp_YM2, assumption)
paulson@11251
   517
apply (case_tac "NB = NBa")
paulson@14207
   518
txt{*If NB=NBa then all other components of the Oops message agree*}
paulson@11251
   519
apply (blast dest: Says_unique_NB)
wenzelm@17411
   520
txt{*case @{prop "NB \<noteq> NBa"}*}
paulson@11251
   521
apply (simp add: single_Nonce_secrecy)
paulson@11251
   522
apply (blast dest!: no_nonce_YM1_YM2 (*to prove NB\<noteq>NAa*))
paulson@11251
   523
done
paulson@11251
   524
paulson@11251
   525
paulson@14207
   526
text{*B's session key guarantee from YM4.  The two certificates contribute to a
paulson@11251
   527
  single conclusion about the Server's message.  Note that the "Notes Spy"
wenzelm@17411
   528
  assumption must quantify over @{text \<forall>} POSSIBLE keys instead of our particular K.
paulson@11251
   529
  If this run is broken and the spy substitutes a certificate containing an
paulson@14207
   530
  old key, B has no means of telling.*}
paulson@11251
   531
lemma B_trusts_YM4:
paulson@14207
   532
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
paulson@14207
   533
                  Crypt K (Nonce NB)|} \<in> set evs;
paulson@14207
   534
         Says B Server
paulson@14207
   535
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@14207
   536
           \<in> set evs;
paulson@14207
   537
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;
paulson@14207
   538
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@14207
   539
       ==> Says Server A
paulson@14207
   540
                   {|Crypt (shrK A) {|Agent B, Key K,
paulson@14207
   541
                             Nonce NA, Nonce NB|},
paulson@14207
   542
                     Crypt (shrK B) {|Agent A, Key K|}|}
paulson@11251
   543
             \<in> set evs"
paulson@14207
   544
by (blast dest: Spy_not_see_NB Says_unique_NB
paulson@11251
   545
                Says_Server_imp_YM2 B_trusts_YM4_newK)
paulson@11251
   546
paulson@11251
   547
paulson@11251
   548
paulson@14207
   549
text{*The obvious combination of @{text B_trusts_YM4} with 
paulson@14207
   550
  @{text Spy_not_see_encrypted_key}*}
paulson@11251
   551
lemma B_gets_good_key:
paulson@11251
   552
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
paulson@11251
   553
                  Crypt K (Nonce NB)|} \<in> set evs;
paulson@14207
   554
         Says B Server
paulson@14207
   555
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
paulson@14207
   556
           \<in> set evs;
paulson@14207
   557
         \<forall>k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs;
paulson@14207
   558
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   559
      ==> Key K \<notin> analz (knows Spy evs)"
paulson@11251
   560
by (blast dest!: B_trusts_YM4 Spy_not_see_encrypted_key)
paulson@11251
   561
paulson@11251
   562
paulson@14207
   563
subsection{*Authenticating B to A*}
paulson@11251
   564
paulson@14207
   565
text{*The encryption in message YM2 tells us it cannot be faked.*}
paulson@11251
   566
lemma B_Said_YM2 [rule_format]:
paulson@11251
   567
     "[|Crypt (shrK B) {|Agent A, Nonce NA, nb|} \<in> parts (knows Spy evs);
paulson@11251
   568
        evs \<in> yahalom|]
paulson@11251
   569
      ==> B \<notin> bad -->
paulson@11251
   570
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@11251
   571
            \<in> set evs"
paulson@14207
   572
apply (erule rev_mp, erule yahalom.induct, force,
paulson@11251
   573
       frule_tac [6] YM4_parts_knows_Spy, simp_all)
paulson@14207
   574
txt{*Fake*}
paulson@11251
   575
apply blast
paulson@11251
   576
done
paulson@11251
   577
paulson@14207
   578
text{*If the server sends YM3 then B sent YM2*}
paulson@11251
   579
lemma YM3_auth_B_to_A_lemma:
paulson@14207
   580
     "[|Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
paulson@11251
   581
       \<in> set evs;  evs \<in> yahalom|]
paulson@14207
   582
      ==> B \<notin> bad -->
paulson@11251
   583
          Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
paulson@11251
   584
            \<in> set evs"
paulson@11251
   585
apply (erule rev_mp, erule yahalom.induct, simp_all)
paulson@14207
   586
txt{*YM3, YM4*}
paulson@11251
   587
apply (blast dest!: B_Said_YM2)+
paulson@11251
   588
done
paulson@11251
   589
paulson@14207
   590
text{*If A receives YM3 then B has used nonce NA (and therefore is alive)*}
paulson@11251
   591
lemma YM3_auth_B_to_A:
paulson@14207
   592
     "[| Gets A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
paulson@14207
   593
           \<in> set evs;
paulson@14207
   594
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@14207
   595
      ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
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   596
       \<in> set evs"
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   597
by (blast dest!: A_trusts_YM3 YM3_auth_B_to_A_lemma elim: knows_Spy_partsEs)
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   598
paulson@11251
   599
paulson@14207
   600
subsection{*Authenticating A to B using the certificate 
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   601
  @{term "Crypt K (Nonce NB)"}*}
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   602
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   603
text{*Assuming the session key is secure, if both certificates are present then
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   604
  A has said NB.  We can't be sure about the rest of A's message, but only
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   605
  NB matters for freshness.*}
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   606
lemma A_Said_YM3_lemma [rule_format]:
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   607
     "evs \<in> yahalom
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   608
      ==> Key K \<notin> analz (knows Spy evs) -->
paulson@11251
   609
          Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
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   610
          Crypt (shrK B) {|Agent A, Key K|} \<in> parts (knows Spy evs) -->
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   611
          B \<notin> bad -->
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   612
          (\<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs)"
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   613
apply (erule yahalom.induct, force,
paulson@11251
   614
       frule_tac [6] YM4_parts_knows_Spy)
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   615
apply (analz_mono_contra, simp_all)
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   616
txt{*Fake*}
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   617
apply blast
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   618
txt{*YM3: by @{text new_keys_not_used}, the message
paulson@14207
   619
   @{term "Crypt K (Nonce NB)"} could not exist*}
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   620
apply (force dest!: Crypt_imp_keysFor)
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   621
txt{*YM4: was @{term "Crypt K (Nonce NB)"} the very last message?
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   622
    If not, use the induction hypothesis*}
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   623
apply (simp add: ex_disj_distrib)
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   624
txt{*yes: apply unicity of session keys*}
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   625
apply (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK
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   626
                    Crypt_Spy_analz_bad
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   627
             dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
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   628
done
paulson@11251
   629
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   630
text{*If B receives YM4 then A has used nonce NB (and therefore is alive).
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   631
  Moreover, A associates K with NB (thus is talking about the same run).
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   632
  Other premises guarantee secrecy of K.*}
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   633
lemma YM4_imp_A_Said_YM3 [rule_format]:
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   634
     "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
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   635
                  Crypt K (Nonce NB)|} \<in> set evs;
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   636
         Says B Server
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   637
           {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
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   638
           \<in> set evs;
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   639
         (\<forall>NA k. Notes Spy {|Nonce NA, Nonce NB, k|} \<notin> set evs);
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   640
         A \<notin> bad;  B \<notin> bad;  evs \<in> yahalom |]
paulson@11251
   641
      ==> \<exists>X. Says A B {|X, Crypt K (Nonce NB)|} \<in> set evs"
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   642
by (blast intro!: A_Said_YM3_lemma
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   643
          dest: Spy_not_see_encrypted_key B_trusts_YM4 Gets_imp_Says)
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   644
paulson@1985
   645
end