src/HOL/Auth/Yahalom_Bad.ML
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(*  Title:      HOL/Auth/Yahalom
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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Inductive relation "yahalom" for the Yahalom protocol.
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From page 257 of
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  Burrows, Abadi and Needham.  A Logic of Authentication.
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  Proc. Royal Soc. 426 (1989)
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*)
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(*A "possibility property": there are traces that reach the end*)
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Goal "A ~= Server \
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\     ==> EX X NB K. EX evs: yahalom.          \
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\            Says A B {|X, Crypt K (Nonce NB)|} : set evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (yahalom.Nil RS 
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          yahalom.YM1 RS yahalom.Reception RS
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          yahalom.YM2 RS yahalom.Reception RS 
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          yahalom.YM3 RS yahalom.Reception RS yahalom.YM4) 2);
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by possibility_tac;
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result();
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Goal "[| Gets B X : set evs; evs : yahalom |] ==> EX A. Says A B X : set evs";
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by (etac rev_mp 1);
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by (etac yahalom.induct 1);
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by Auto_tac;
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qed "Gets_imp_Says";
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(*Must be proved separately for each protocol*)
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Goal "[| Gets B X : set evs; evs : yahalom |]  ==> X : knows Spy evs";
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by (blast_tac (claset() addSDs [Gets_imp_Says, Says_imp_knows_Spy]) 1);
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qed"Gets_imp_knows_Spy";
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AddDs [Gets_imp_knows_Spy RS parts.Inj];
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fun g_not_bad_tac s = 
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  ftac Gets_imp_Says THEN' assume_tac THEN' not_bad_tac s;
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(**** Inductive proofs about yahalom ****)
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(** For reasoning about the encrypted portion of messages **)
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(*Lets us treat YM4 using a similar argument as for the Fake case.*)
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Goal "[| Gets A {|Crypt (shrK A) Y, X|} : set evs;  evs : yahalom |]  \
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\     ==> X : analz (knows Spy evs)";
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by (blast_tac (claset() addSDs [Gets_imp_knows_Spy RS analz.Inj]) 1);
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qed "YM4_analz_knows_Spy";
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bind_thm ("YM4_parts_knows_Spy",
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          YM4_analz_knows_Spy RS (impOfSubs analz_subset_parts));
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(*For proving the easier theorems about X ~: parts (knows Spy evs).*)
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fun parts_knows_Spy_tac i = 
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  EVERY
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   [ftac YM4_parts_knows_Spy (i+6), assume_tac (i+6),
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    prove_simple_subgoals_tac i];
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(*Induction for regularity theorems.  If induction formula has the form
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   X ~: analz (knows Spy evs) --> ... then it shortens the proof by discarding
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   needless information about analz (insert X (knows Spy evs))  *)
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fun parts_induct_tac i = 
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    etac yahalom.induct i
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    THEN 
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    REPEAT (FIRSTGOAL analz_mono_contra_tac)
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    THEN  parts_knows_Spy_tac i;
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(** Theorems of the form X ~: parts (knows Spy evs) imply that NOBODY
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    sends messages containing X! **)
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(*Spy never sees another agent's shared key! (unless it's bad at start)*)
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Goal "evs : yahalom ==> (Key (shrK A) : parts (knows Spy evs)) = (A : bad)";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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by (ALLGOALS Blast_tac);
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qed "Spy_see_shrK";
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Addsimps [Spy_see_shrK];
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Goal "evs : yahalom ==> (Key (shrK A) : analz (knows Spy evs)) = (A : bad)";
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by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset()));
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qed "Spy_analz_shrK";
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Addsimps [Spy_analz_shrK];
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AddSDs [Spy_see_shrK RSN (2, rev_iffD1), 
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	Spy_analz_shrK RSN (2, rev_iffD1)];
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(*Nobody can have used non-existent keys!  Needed to apply analz_insert_Key*)
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Goal "evs : yahalom ==>          \
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\      Key K ~: used evs --> K ~: keysFor (parts (knows Spy evs))";
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by (parts_induct_tac 1);
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(*Fake*)
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by (blast_tac (claset() addSDs [keysFor_parts_insert]) 1);
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(*YM2-4: Because Key K is not fresh, etc.*)
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by (REPEAT (blast_tac (claset() addSEs knows_Spy_partsEs) 1));
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qed_spec_mp "new_keys_not_used";
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bind_thm ("new_keys_not_analzd",
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          [analz_subset_parts RS keysFor_mono,
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           new_keys_not_used] MRS contra_subsetD);
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Addsimps [new_keys_not_used, new_keys_not_analzd];
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(*For proofs involving analz.*)
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val analz_knows_Spy_tac = 
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    ftac YM4_analz_knows_Spy 7 THEN assume_tac 7;
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(****
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 The following is to prove theorems of the form
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  Key K : analz (insert (Key KAB) (knows Spy evs)) ==>
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  Key K : analz (knows Spy evs)
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 A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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Goal "evs : yahalom ==>                              \
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\  ALL K KK. KK <= - (range shrK) -->                \
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\         (Key K : analz (Key``KK Un (knows Spy evs))) = \
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\         (K : KK | Key K : analz (knows Spy evs))";
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by (etac yahalom.induct 1);
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by analz_knows_Spy_tac;
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by (REPEAT_FIRST (resolve_tac [allI, impI]));
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by (REPEAT_FIRST (rtac analz_image_freshK_lemma));
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by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
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(*Fake*) 
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by (spy_analz_tac 1);
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qed_spec_mp "analz_image_freshK";
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Goal "[| evs : yahalom;  KAB ~: range shrK |]                  \
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\      ==> Key K : analz (insert (Key KAB) (knows Spy evs)) =  \
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\          (K = KAB | Key K : analz (knows Spy evs))";
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by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
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qed "analz_insert_freshK";
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(*** The Key K uniquely identifies the Server's  message. **)
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Goal "evs : yahalom ==>                                     \
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\   EX A' B' na' nb' X'. ALL A B na nb X.                   \
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\       Says Server A                                       \
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\        {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|}   \
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\       : set evs --> A=A' & B=B' & na=na' & nb=nb' & X=X'";
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by (etac yahalom.induct 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib])));
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by (ALLGOALS Clarify_tac);
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by (ex_strip_tac 2);
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by (Blast_tac 2);
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(*Remaining case: YM3*)
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by (expand_case_tac "K = ?y" 1);
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by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
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(*...we assume X is a recent message and handle this case by contradiction*)
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by (blast_tac (claset() addSEs knows_Spy_partsEs
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                        delrules [conjI]    (*no split-up to 4 subgoals*)) 1);
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val lemma = result();
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Goal "[| Says Server A                                                 \
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\         {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} : set evs;  \
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\       Says Server A'                                                 \
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\         {|Crypt (shrK A') {|Agent B', Key K, na', nb'|}, X'|} : set evs; \
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\       evs : yahalom |]                                    \
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\    ==> A=A' & B=B' & na=na' & nb=nb'";
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by (prove_unique_tac lemma 1);
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qed "unique_session_keys";
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(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
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Goal "[| A ~: bad;  B ~: bad;  evs : 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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\          : set evs -->                                       \
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\         Key K ~: analz (knows Spy evs)";
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by (etac yahalom.induct 1);
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by analz_knows_Spy_tac;
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by (ALLGOALS
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    (asm_simp_tac 
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     (simpset() addsimps split_ifs @ pushes @
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                         [analz_insert_eq, analz_insert_freshK])));
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(*YM3*)
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by (blast_tac (claset() delrules [impCE]
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                        addSEs knows_Spy_partsEs
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                        addIs [impOfSubs analz_subset_parts]) 2);
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(*Fake*) 
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by (spy_analz_tac 1);
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val lemma = result() RS mp RSN(2,rev_notE);
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(*Final version*)
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Goal "[| 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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\          : set evs;                                          \
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\        A ~: bad;  B ~: bad;  evs : yahalom |]                \
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\     ==> Key K ~: analz (knows Spy evs)";
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by (blast_tac (claset() addSEs [lemma]) 1);
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qed "Spy_not_see_encrypted_key";
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(** Security Guarantee for A upon receiving YM3 **)
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(*If the encrypted message appears then it originated with the Server*)
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Goal "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} : parts (knows Spy evs); \
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\        A ~: bad;  evs : 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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\          : set evs";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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qed "A_trusts_YM3";
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(*The obvious combination of A_trusts_YM3 with Spy_not_see_encrypted_key*)
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Goal "[| Crypt (shrK A) {|Agent B, Key K, na, nb|} : parts (knows Spy evs); \
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\        A ~: bad;  B ~: bad;  evs : yahalom |]                \
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\     ==> Key K ~: analz (knows Spy evs)";
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by (blast_tac (claset() addSDs [A_trusts_YM3, Spy_not_see_encrypted_key]) 1);
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qed "A_gets_good_key";
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(** Security Guarantees for B upon receiving YM4 **)
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(*B knows, by the first part of A's message, that the Server distributed 
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  the key for A and B.  But this part says nothing about nonces.*)
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Goal "[| Crypt (shrK B) {|Agent A, Key K|} : parts (knows Spy evs);  \
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\        B ~: bad;  evs : yahalom |]                                 \
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\     ==> EX 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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\                    : set evs";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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(*YM3*)
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by (Blast_tac 1);
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qed "B_trusts_YM4_shrK";
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(** Up to now, the reasoning is similar to standard Yahalom.  Now the
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    doubtful reasoning occurs.  We should not be assuming that an unknown
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    key is secure, but the model allows us to: there is no Oops rule to
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    let session keys go.*)
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(*B knows, by the second part of A's message, that the Server distributed 
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  the key quoting nonce NB.  This part says nothing about agent names. 
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  Secrecy of K is assumed; the valid Yahalom proof uses (and later proves)
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  the secrecy of NB.*)
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Goal "evs : yahalom                                          \
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\     ==> Key K ~: analz (knows Spy evs) -->                 \
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\         Crypt K (Nonce NB) : parts (knows Spy evs) -->     \
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\         (EX A B NA. Says Server A                          \
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\                     {|Crypt (shrK A) {|Agent B, Key K,     \
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\                               Nonce NA, Nonce NB|},        \
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\                       Crypt (shrK B) {|Agent A, Key K|}|}  \
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\                    : set evs)";
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by (parts_induct_tac 1);
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by (ALLGOALS Clarify_tac);
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(*YM3 & Fake*)
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by (Blast_tac 2);
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by (Fake_parts_insert_tac 1);
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(*YM4*)
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(*A is uncompromised because NB is secure*)
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by (g_not_bad_tac "A" 1);
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(*A's certificate guarantees the existence of the Server message*)
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by (blast_tac (claset() addDs [Says_imp_knows_Spy RS parts.Inj RS parts.Fst RS
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			       A_trusts_YM3]) 1);
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bind_thm ("B_trusts_YM4_newK", result() RS mp RSN (2, rev_mp));
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(*B's session key guarantee from YM4.  The two certificates contribute to a
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  single conclusion about the Server's message. *)
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Goal "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},                    \
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\                 Crypt K (Nonce NB)|} : set evs;                       \
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\        Says B Server                                                  \
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\          {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|}  \
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\          : set evs;                                                   \
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\        A ~: bad;  B ~: bad;  evs : yahalom |]                         \
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\      ==> EX na nb. Says Server A                                      \
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\                  {|Crypt (shrK A) {|Agent B, Key K, na, nb|},         \
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\                    Crypt (shrK B) {|Agent A, Key K|}|}                \
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\            : set evs";
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by (etac (Gets_imp_knows_Spy RS parts.Inj RS MPair_parts) 1 THEN
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    assume_tac 1 THEN dtac B_trusts_YM4_shrK 1);
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by (dtac B_trusts_YM4_newK 3);
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by (REPEAT_FIRST (eresolve_tac [asm_rl, exE]));
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by (etac Spy_not_see_encrypted_key 1 THEN REPEAT (assume_tac 1));
7499
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by (ftac unique_session_keys 1 THEN REPEAT (assume_tac 1));
6400
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by (blast_tac (claset() addDs []) 1);
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qed "B_trusts_YM4";
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(*The obvious combination of B_trusts_YM4 with Spy_not_see_encrypted_key*)
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Goal "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},                   \
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\                    Crypt K (Nonce NB)|} : set evs;                   \
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\        Says B Server                                                 \
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\          {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|} \
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\          : set evs;                                                  \
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\        A ~: bad;  B ~: bad;  evs : yahalom |]                \
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\     ==> Key K ~: analz (knows Spy evs)";
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by (blast_tac (claset() addSDs [B_trusts_YM4, Spy_not_see_encrypted_key]) 1);
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qed "B_gets_good_key";
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(*** Authenticating B to A: these proofs are not considered.
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     They are irrelevant to showing the need for Oops. ***)
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(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)
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(*Assuming the session key is secure, if both certificates are present then
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paulson
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  A has said NB.  We can't be sure about the rest of A's message, but only
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paulson
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  NB matters for freshness.*)  
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Goal "evs : yahalom                                              \
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paulson
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\     ==> Key K ~: analz (knows Spy evs) -->                     \
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\         Crypt K (Nonce NB) : parts (knows Spy evs) -->         \
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\         Crypt (shrK B) {|Agent A, Key K|} : parts (knows Spy evs) --> \
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\         B ~: bad -->                                           \
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\         (EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs)";
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paulson
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by (parts_induct_tac 1);
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paulson
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(*Fake*)
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by (Fake_parts_insert_tac 1);
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(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
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by (fast_tac (claset() addSDs [Crypt_imp_keysFor] addss (simpset())) 1); 
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(*YM4: was Crypt K (Nonce NB) the very last message?  If not, use ind. hyp.*)
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by (asm_simp_tac (simpset() addsimps [ex_disj_distrib]) 1);
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paulson
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(*yes: apply unicity of session keys*)
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paulson
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by (g_not_bad_tac "Aa" 1);
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paulson
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by (blast_tac (claset() addSEs [MPair_parts]
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                        addSDs [A_trusts_YM3, B_trusts_YM4_shrK]
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		        addDs  [Says_imp_knows_Spy RS parts.Inj,
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				unique_session_keys]) 1);
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qed_spec_mp "A_Said_YM3_lemma";
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(*If B receives YM4 then A has used nonce NB (and therefore is alive).
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  Moreover, A associates K with NB (thus is talking about the same run).
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  Other premises guarantee secrecy of K.*)
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Goal "[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},                   \
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\                 Crypt K (Nonce NB)|} : set evs;                      \
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\        Says B Server                                                 \
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\          {|Agent B, Nonce NB, Crypt (shrK B) {|Agent A, Nonce NA|}|} \
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\          : set evs;                                                  \
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\        A ~: bad;  B ~: bad;  evs : yahalom |]       \
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\     ==> EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs";
7499
23e090051cb8 isatool expandshort;
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by (ftac B_trusts_YM4 1);
6400
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by (REPEAT_FIRST assume_tac);
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by (etac (Gets_imp_knows_Spy RS parts.Inj RS MPair_parts) 1 THEN assume_tac 1);
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paulson
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by (Clarify_tac 1);
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by (rtac A_Said_YM3_lemma 1);
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paulson
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by (rtac Spy_not_see_encrypted_key 2);
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paulson
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by (REPEAT_FIRST assume_tac);
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qed_spec_mp "YM4_imp_A_Said_YM3";