src/HOL/Auth/Yahalom2.ML
author paulson
Fri Jan 17 12:49:31 1997 +0100 (1997-01-17)
changeset 2516 4d68fbe6378b
parent 2451 ce85a2aafc7a
child 2637 e9b203f854ae
permissions -rw-r--r--
Now with Andy Gordon's treatment of freshness to replace newN/K
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(*  Title:      HOL/Auth/Yahalom2
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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, Variant 2.
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This version trades encryption of NB for additional explicitness in YM3.
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From page 259 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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open Yahalom2;
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proof_timing:=true;
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HOL_quantifiers := false;
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(*A "possibility property": there are traces that reach the end*)
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goal thy 
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 "!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
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\        ==> EX X NB K. EX evs: yahalom lost.          \
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\               Says A B {|X, Crypt K (Nonce NB)|} : set_of_list evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS 
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          yahalom.YM4) 2);
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by possibility_tac;
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result();
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(**** Inductive proofs about yahalom ****)
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(*Monotonicity*)
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goal thy "!!evs. lost' <= lost ==> yahalom lost' <= yahalom lost";
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by (rtac subsetI 1);
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by (etac yahalom.induct 1);
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by (REPEAT_FIRST
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    (best_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
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                              :: yahalom.intrs))));
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qed "yahalom_mono";
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(*Nobody sends themselves messages*)
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goal thy "!!evs. evs: yahalom lost ==> ALL A X. Says A A X ~: set_of_list evs";
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by (etac yahalom.induct 1);
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by (Auto_tac());
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qed_spec_mp "not_Says_to_self";
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Addsimps [not_Says_to_self];
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AddSEs   [not_Says_to_self RSN (2, rev_notE)];
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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 thy "!!evs. Says S A {|NB, Crypt (shrK A) Y, X|} : set_of_list evs ==> \
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\                X : analz (sees lost Spy evs)";
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by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
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qed "YM4_analz_sees_Spy";
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bind_thm ("YM4_parts_sees_Spy",
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          YM4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
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(*Relates to both YM4 and Oops*)
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goal thy "!!evs. Says S A {|NB, Crypt (shrK A) {|B, K, NA|}, X|} \
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\                  : set_of_list evs ==> \
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\                K : parts (sees lost Spy evs)";
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by (fast_tac (!claset addSEs partsEs
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                      addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
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qed "YM4_Key_parts_sees_Spy";
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(*We instantiate the variable to "lost".  Leaving it as a Var makes proofs
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  harder: the simplifier does less.*)
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val parts_Fake_tac = 
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    forw_inst_tac [("lost","lost")] YM4_parts_sees_Spy 6 THEN
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    forw_inst_tac [("lost","lost")] YM4_Key_parts_sees_Spy 7;
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(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
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fun parts_induct_tac i = SELECT_GOAL
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    (DETERM (etac yahalom.induct 1 THEN parts_Fake_tac THEN
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             (*Fake message*)
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             TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
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                                           impOfSubs Fake_parts_insert]
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                                    addss (!simpset)) 2)) THEN
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     (*Base case*)
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     fast_tac (!claset addss (!simpset)) 1 THEN
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     ALLGOALS Asm_simp_tac) i;
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(** Theorems of the form X ~: parts (sees lost 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 lost at start)*)
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goal thy 
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 "!!evs. evs : yahalom lost \
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\        ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
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by (parts_induct_tac 1);
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by (Auto_tac());
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qed "Spy_see_shrK";
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Addsimps [Spy_see_shrK];
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goal thy 
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 "!!evs. evs : yahalom lost \
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\        ==> (Key (shrK A) : analz (sees lost Spy evs)) = (A : lost)";
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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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goal thy  "!!A. [| Key (shrK A) : parts (sees lost Spy evs);       \
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\                  evs : yahalom lost |] ==> A:lost";
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by (fast_tac (!claset addDs [Spy_see_shrK]) 1);
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qed "Spy_see_shrK_D";
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bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
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AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
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(*Nobody can have used non-existent keys!*)
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goal thy "!!evs. evs : yahalom lost ==>          \
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\         Key K ~: used evs --> K ~: keysFor (parts (sees lost Spy evs))";
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by (parts_induct_tac 1);
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(*YM4: Key K is not fresh!*)
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by (fast_tac (!claset addSEs sees_Spy_partsEs) 3);
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(*YM3*)
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by (fast_tac (!claset addss (!simpset)) 2);
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(*Fake*)
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by (best_tac
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      (!claset addIs [impOfSubs analz_subset_parts]
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               addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
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                      impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
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               addss (!simpset)) 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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(*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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goal thy 
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 "!!evs. [| Says Server A {|NB', Crypt (shrK A) {|Agent B, Key K, NA|}, X|} \
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\            : set_of_list evs;                                         \
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\           evs : yahalom lost |]                                       \
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\        ==> K ~: range shrK & A ~= B";
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by (etac rev_mp 1);
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by (etac yahalom.induct 1);
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by (ALLGOALS (fast_tac (!claset addss (!simpset))));
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qed "Says_Server_message_form";
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(*For proofs involving analz.  We again instantiate the variable to "lost".*)
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val analz_Fake_tac = 
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    dres_inst_tac [("lost","lost")] YM4_analz_sees_Spy 6 THEN
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    forw_inst_tac [("lost","lost")] Says_Server_message_form 7 THEN
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    assume_tac 7 THEN
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    REPEAT ((etac conjE ORELSE' hyp_subst_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) (sees lost Spy evs)) ==>
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          Key K : analz (sees lost 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 thy  
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 "!!evs. evs : yahalom lost ==> \
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\  ALL K KK. KK <= Compl (range shrK) -->                      \
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\            (Key K : analz (Key``KK Un (sees lost Spy evs))) = \
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\            (K : KK | Key K : analz (sees lost Spy evs))";
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by (etac yahalom.induct 1);
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by analz_Fake_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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(*Base*)
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by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
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(*YM4, Fake*) 
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by (REPEAT (spy_analz_tac 1));
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qed_spec_mp "analz_image_freshK";
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goal thy
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 "!!evs. [| evs : yahalom lost;  KAB ~: range shrK |] ==>             \
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\        Key K : analz (insert (Key KAB) (sees lost Spy evs)) = \
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\        (K = KAB | Key K : analz (sees lost 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 thy 
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 "!!evs. evs : yahalom lost ==>                                     \
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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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\           {|NB, Crypt (shrK A) {|Agent B, Key K, NA|}, X|}        \
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\          : set_of_list 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 (Step_tac 1);
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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 (fast_tac (!claset addSEs sees_Spy_partsEs
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                      delrules [conjI]    (*prevent split-up into 4 subgoals*)
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                      addss (!simpset addsimps [parts_insertI])) 1);
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val lemma = result();
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goal thy 
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"!!evs. [| Says Server A                                            \
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\           {|NB, Crypt (shrK A) {|Agent B, Key K, NA|}, X|}        \
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\           : set_of_list evs;                                      \
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\          Says Server A'                                           \
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\           {|NB', Crypt (shrK A') {|Agent B', Key K, NA'|}, X'|}   \
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\           : set_of_list evs;                                      \
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\          evs : yahalom lost |]                                    \
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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 thy 
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 "!!evs. [| A ~: lost;  B ~: lost;  A ~= B;                          \
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\           evs : yahalom lost |]            \
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\        ==> Says Server A                                           \
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\              {|NB, Crypt (shrK A) {|Agent B, Key K, NA|},           \
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\                    Crypt (shrK B) {|NB, Key K, Agent A|}|}          \
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\             : set_of_list evs -->                               \
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\            Says A Spy {|NA, NB, Key K|} ~: set_of_list evs -->  \
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\            Key K ~: analz (sees lost Spy evs)";
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by (etac yahalom.induct 1);
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by analz_Fake_tac;
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by (ALLGOALS
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    (asm_simp_tac 
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     (!simpset addsimps [not_parts_not_analz, analz_insert_freshK]
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               setloop split_tac [expand_if])));
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(*YM3*)
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by (fast_tac (!claset delrules [impCE]
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                      addSEs sees_Spy_partsEs
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                      addIs [impOfSubs analz_subset_parts]
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                      addss (!simpset addsimps [parts_insert2])) 2);
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(*OR4, Fake*) 
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by (REPEAT_FIRST spy_analz_tac);
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(*Oops*)
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by (fast_tac (!claset delrules [disjE] 
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                      addDs [unique_session_keys]
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                      addss (!simpset)) 1);
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val lemma = result() RS mp RS mp RSN(2,rev_notE);
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(*Final version: Server's message in the most abstract form*)
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goal thy 
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 "!!evs. [| Says Server A                                         \
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\              {|NB, Crypt (shrK A) {|Agent B, Key K, NA|},       \
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\                    Crypt (shrK B) {|NB, Key K, Agent A|}|}      \
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\           : set_of_list evs;                                    \
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\           Says A Spy {|NA, NB, Key K|} ~: set_of_list evs;      \
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\           A ~: lost;  B ~: lost;  evs : yahalom lost |]         \
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\        ==> Key K ~: analz (sees lost Spy evs)";
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by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
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by (fast_tac (!claset addSEs [lemma]) 1);
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qed "Spy_not_see_encrypted_key";
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goal thy 
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 "!!evs. [| C ~: {A,B,Server};                                    \
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\           Says Server A                                         \
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\              {|NB, Crypt (shrK A) {|Agent B, Key K, NA|},       \
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\                    Crypt (shrK B) {|NB, Key K, Agent A|}|}      \
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\           : set_of_list evs;                                    \
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\           Says A Spy {|NA, NB, Key K|} ~: set_of_list evs;      \
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\           A ~: lost;  B ~: lost;  evs : yahalom lost |]         \
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\        ==> Key K ~: analz (sees lost C evs)";
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by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
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by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
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by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
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by (REPEAT_FIRST (fast_tac (!claset addIs [yahalom_mono RS subsetD])));
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qed "Agent_not_see_encrypted_key";
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(*** Security Guarantees for A and B ***)
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(*If the encrypted message appears then it originated with the Server.*)
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goal thy
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 "!!evs. [| Crypt (shrK A) {|Agent B, Key K, NA|}                \
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\            : parts (sees lost Spy evs);                              \
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\           A ~: lost;  evs : yahalom lost |]                          \
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\         ==> EX NB. Says Server A                                     \
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\                      {|NB, Crypt (shrK A) {|Agent B, Key K, NA|},    \
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\                            Crypt (shrK B) {|NB, Key K, Agent A|}|}   \
paulson@2155
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\                    : set_of_list evs";
paulson@2155
   304
by (etac rev_mp 1);
paulson@2155
   305
by (parts_induct_tac 1);
paulson@2155
   306
(*The nested conjunctions are entirely useless*)
paulson@2155
   307
by (REPEAT_FIRST (rtac conjI ORELSE' fast_tac (!claset delrules [conjI])));
paulson@2323
   308
qed "A_trusts_YM3";
paulson@2155
   309
paulson@2111
   310
paulson@2111
   311
(*B knows, by the first part of A's message, that the Server distributed 
paulson@2155
   312
  the key for A and B. *)
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goal thy 
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   314
 "!!evs. [| Crypt (shrK B) {|Nonce NB, Key K, Agent A|}              \
paulson@2155
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\            : parts (sees lost Spy evs);                            \
paulson@2155
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\           B ~: lost;  evs : yahalom lost |]                        \
paulson@2111
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\        ==> EX NA. Says Server A                                    \
paulson@2155
   318
\                    {|Nonce NB,                                     \
paulson@2284
   319
\                      Crypt (shrK A) {|Agent B, Key K, Nonce NA|},  \
paulson@2284
   320
\                      Crypt (shrK B) {|Nonce NB, Key K, Agent A|}|} \
paulson@2111
   321
\                       : set_of_list evs";
paulson@2111
   322
by (etac rev_mp 1);
paulson@2111
   323
by (parts_induct_tac 1);
paulson@2111
   324
(*YM3*)
paulson@2111
   325
by (Fast_tac 1);
paulson@2111
   326
qed "B_trusts_YM4_shrK";
paulson@2111
   327
paulson@2111
   328
(*With this variant we don't bother to use the 2nd part of YM4 at all, since
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   329
  Nonce NB is available in the first part.  However the 2nd part does assure B
paulson@2111
   330
  of A's existence.*)
paulson@2111
   331
paulson@2155
   332
(*What can B deduce from receipt of YM4?  Stronger and simpler than Yahalom
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   333
  because we do not have to show that NB is secret. *)
paulson@2111
   334
goal thy 
paulson@2284
   335
 "!!evs. [| Says A' B {|Crypt (shrK B) {|Nonce NB, Key K, Agent A|},    \
paulson@2284
   336
\                       Crypt K (Nonce NB)|} : set_of_list evs;         \
paulson@2111
   337
\           A ~: lost;  B ~: lost;  evs : yahalom lost |]               \
paulson@2111
   338
\        ==> EX NA. Says Server A                                       \
paulson@2155
   339
\                    {|Nonce NB,                                        \
paulson@2284
   340
\                      Crypt (shrK A) {|Agent B, Key K, Nonce NA|},     \
paulson@2284
   341
\                      Crypt (shrK B) {|Nonce NB, Key K, Agent A|}|}    \
paulson@2111
   342
\                   : set_of_list evs";
paulson@2170
   343
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1);
paulson@2111
   344
by (fast_tac (!claset addSDs [B_trusts_YM4_shrK]) 1);
paulson@2323
   345
qed "B_trusts_YM4";