src/HOL/Auth/Yahalom.ML
author paulson
Fri Dec 20 10:23:48 1996 +0100 (1996-12-20)
changeset 2454 92f43ed48935
parent 2451 ce85a2aafc7a
child 2516 4d68fbe6378b
permissions -rw-r--r--
Corrected comments
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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 "otway" 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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open Yahalom;
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proof_timing:=true;
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HOL_quantifiers := false;
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Pretty.setdepth 20;
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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 yahalom.YM4) 2);
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by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
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by (ALLGOALS Fast_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 {|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 {|Crypt (shrK A) {|B, K, NA, NB|}, 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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(*** Future keys can't be seen or used! ***)
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(*Nobody can have SEEN keys that will be generated in the future. *)
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goal thy "!!i. evs : yahalom lost ==>        \
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\              length evs <= i --> Key(newK i) ~: parts (sees lost Spy evs)";
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by (parts_induct_tac 1);
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by (REPEAT_FIRST (best_tac (!claset addEs [leD RS notE]
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				    addDs [impOfSubs analz_subset_parts,
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                                           impOfSubs parts_insert_subset_Un,
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                                           Suc_leD]
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                                    addss (!simpset))));
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qed_spec_mp "new_keys_not_seen";
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Addsimps [new_keys_not_seen];
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(*Variant: old messages must contain old keys!*)
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goal thy 
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 "!!evs. [| Says A B X : set_of_list evs;          \
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\           Key (newK i) : parts {X};    \
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\           evs : yahalom lost                     \
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\        |] ==> i < length evs";
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by (rtac ccontr 1);
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by (dtac leI 1);
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by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
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                      addIs  [impOfSubs parts_mono]) 1);
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qed "Says_imp_old_keys";
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(*Ready-made for the classical reasoner*)
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goal thy "!!evs. [| Says A B {|Crypt K {|b,Key(newK(length evs)),na,nb|}, X|}\
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\                   : set_of_list evs;  evs : yahalom lost |]              \
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\                ==> R";
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by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
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                      addss (!simpset addsimps [parts_insertI])) 1);
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qed "Says_too_new_key";
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AddSEs [Says_too_new_key];
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(*Nobody can have USED keys that will be generated in the future.
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  ...very like new_keys_not_seen*)
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goal thy "!!i. evs : yahalom lost ==>        \
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\             length evs <= i --> newK i ~: keysFor(parts(sees lost Spy evs))";
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by (parts_induct_tac 1);
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(*YM1, YM2 and YM3*)
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by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,3,2]));
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(*Fake and Oops: these messages send unknown (X) components*)
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by (EVERY
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    (map (fast_tac
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          (!claset addDs [impOfSubs analz_subset_parts,
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                          impOfSubs (analz_subset_parts RS keysFor_mono),
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                          impOfSubs (parts_insert_subset_Un RS keysFor_mono),
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                          Suc_leD]
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                   addss (!simpset))) [3,1]));
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(*YM4: if K was used then it had been seen, contradicting new_keys_not_seen*)
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by (fast_tac
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      (!claset addSEs partsEs
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               addSDs [Says_imp_sees_Spy RS parts.Inj]
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               addEs [new_keys_not_seen RSN(2,rev_notE)]
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               addDs [Suc_leD]) 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                                                    \
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\            {|Crypt (shrK A) {|Agent B, K, NA, NB|}, X|} : set_of_list evs; \
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\           evs : yahalom lost |]                                            \
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\        ==> EX i. K = Key(newK i)";
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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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    forw_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 Full_simp_tac 7 THEN
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    REPEAT ((etac exE 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 (newK i)) (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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(*NOT useful in this form, but it says that session keys are not used
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  to encrypt messages containing other keys, in the actual protocol.
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  We require that agents should behave like this subsequently also.*)
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goal thy 
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 "!!evs. evs : yahalom lost ==> \
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\        (Crypt (newK i) X) : parts (sees lost Spy evs) & \
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\        Key K : parts {X} --> Key K : parts (sees lost Spy evs)";
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by (etac yahalom.induct 1);
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by parts_Fake_tac;
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by (ALLGOALS Asm_simp_tac);
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(*Deals with Faked messages*)
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by (best_tac (!claset addSEs partsEs
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                      addDs [impOfSubs parts_insert_subset_Un]
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                      addss (!simpset)) 2);
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(*Base case*)
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by (Auto_tac());
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result();
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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 E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
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\           (K : newK``E | 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, analz_image_newK_lemma]));
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by (ALLGOALS (*Takes 11 secs*)
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    (asm_simp_tac 
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     (!simpset addsimps [Un_assoc RS sym, 
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			 insert_Key_singleton, insert_Key_image, pushKey_newK]
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               setloop split_tac [expand_if])));
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(*YM4, Fake*) 
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by (EVERY (map spy_analz_tac [4, 2]));
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(*Oops, YM3, Base*)
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by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
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qed_spec_mp "analz_image_newK";
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goal thy
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 "!!evs. evs : yahalom lost ==>                               \
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\        Key K : analz (insert (Key(newK i)) (sees lost Spy evs)) = \
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\        (K = newK i | Key K : analz (sees lost Spy evs))";
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by (asm_simp_tac (HOL_ss addsimps [analz_image_newK, 
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                                   insert_Key_singleton]) 1);
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by (Fast_tac 1);
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qed "analz_insert_Key_newK";
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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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\           {|Crypt (shrK A) {|Agent B, Key K, NA, NB|}, 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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by (ex_strip_tac 2);
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by (Fast_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 very new message, and handle this case by contradiction*)
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by (Fast_tac 1);  (*uses Says_too_new_key*)
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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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\           {|Crypt (shrK A) {|Agent B, Key K, NA, NB|}, X|}        \
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\           : set_of_list evs;                                      \
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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_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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(*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, NB|}                  \
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\            : parts (sees lost Spy evs);                              \
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\           A ~: lost;  evs : yahalom lost |]                          \
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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_of_list evs";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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qed "A_trusts_YM3";
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(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
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goal thy 
paulson@2133
   316
 "!!evs. [| A ~: lost;  B ~: lost;  evs : yahalom lost |]         \
paulson@2051
   317
\        ==> Says Server A                                        \
paulson@2284
   318
\              {|Crypt (shrK A) {|Agent B, Key K, NA, NB|},       \
paulson@2284
   319
\                Crypt (shrK B) {|Agent A, Key K|}|}              \
paulson@2110
   320
\             : set_of_list evs -->                               \
paulson@2110
   321
\            Says A Spy {|NA, NB, Key K|} ~: set_of_list evs -->  \
paulson@2051
   322
\            Key K ~: analz (sees lost Spy evs)";
paulson@2032
   323
by (etac yahalom.induct 1);
paulson@2110
   324
by analz_Fake_tac;
paulson@2013
   325
by (ALLGOALS
paulson@2013
   326
    (asm_simp_tac 
paulson@2451
   327
     (!simpset addsimps [not_parts_not_analz, analz_insert_Key_newK]
paulson@2013
   328
               setloop split_tac [expand_if])));
paulson@2013
   329
(*YM3*)
paulson@2133
   330
by (Fast_tac 2);  (*uses Says_too_new_key*)
paulson@2133
   331
(*OR4, Fake*) 
paulson@2377
   332
by (REPEAT_FIRST spy_analz_tac);
paulson@2451
   333
(*Oops*)
paulson@2110
   334
by (fast_tac (!claset delrules [disjE] 
paulson@2170
   335
                      addDs [unique_session_keys]
paulson@2170
   336
                      addss (!simpset)) 1);
paulson@2110
   337
val lemma = result() RS mp RS mp RSN(2,rev_notE);
paulson@2013
   338
paulson@2013
   339
paulson@2013
   340
(*Final version: Server's message in the most abstract form*)
paulson@1985
   341
goal thy 
paulson@2110
   342
 "!!evs. [| Says Server A                                               \
paulson@2284
   343
\            {|Crypt (shrK A) {|Agent B, K, NA, NB|},                   \
paulson@2284
   344
\              Crypt (shrK B) {|Agent A, K|}|} : set_of_list evs;       \
paulson@2110
   345
\           Says A Spy {|NA, NB, K|} ~: set_of_list evs;                \
paulson@2110
   346
\           A ~: lost;  B ~: lost;  evs : yahalom lost |] ==>           \
paulson@2032
   347
\     K ~: analz (sees lost Spy evs)";
paulson@2013
   348
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
paulson@2013
   349
by (fast_tac (!claset addSEs [lemma]) 1);
paulson@2032
   350
qed "Spy_not_see_encrypted_key";
paulson@2001
   351
paulson@2001
   352
paulson@2045
   353
goal thy 
paulson@2045
   354
 "!!evs. [| C ~: {A,B,Server};                                          \
paulson@2045
   355
\           Says Server A                                               \
paulson@2284
   356
\            {|Crypt (shrK A) {|Agent B, K, NA, NB|},                   \
paulson@2284
   357
\              Crypt (shrK B) {|Agent A, K|}|} : set_of_list evs;       \
paulson@2110
   358
\           Says A Spy {|NA, NB, K|} ~: set_of_list evs;                \
paulson@2045
   359
\           A ~: lost;  B ~: lost;  evs : yahalom lost |] ==>           \
paulson@2045
   360
\     K ~: analz (sees lost C evs)";
paulson@2045
   361
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
paulson@2045
   362
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
paulson@2045
   363
by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
paulson@2045
   364
by (REPEAT_FIRST (fast_tac (!claset addIs [yahalom_mono RS subsetD])));
paulson@2045
   365
qed "Agent_not_see_encrypted_key";
paulson@2045
   366
paulson@2045
   367
paulson@2110
   368
(*** Security Guarantee for B upon receiving YM4 ***)
paulson@2013
   369
paulson@2110
   370
(*B knows, by the first part of A's message, that the Server distributed 
paulson@2110
   371
  the key for A and B.  But this part says nothing about nonces.*)
paulson@2001
   372
goal thy 
paulson@2284
   373
 "!!evs. [| Crypt (shrK B) {|Agent A, Key K|} : parts (sees lost Spy evs); \
paulson@2051
   374
\           B ~: lost;  evs : yahalom lost |]                           \
paulson@2001
   375
\        ==> EX NA NB. Says Server A                                    \
paulson@2451
   376
\                        {|Crypt (shrK A) {|Agent B, Key K,             \
paulson@2451
   377
\                                  Nonce NA, Nonce NB|},                \
paulson@2284
   378
\                          Crypt (shrK B) {|Agent A, Key K|}|}          \
paulson@2013
   379
\                       : set_of_list evs";
paulson@2032
   380
by (etac rev_mp 1);
paulson@2110
   381
by (parts_induct_tac 1);
paulson@2110
   382
(*YM3*)
paulson@2110
   383
by (Fast_tac 1);
paulson@2110
   384
qed "B_trusts_YM4_shrK";
paulson@2110
   385
paulson@2133
   386
paulson@2133
   387
(*** General properties of nonces ***)
paulson@2133
   388
paulson@2451
   389
goal thy "!!evs. evs : yahalom lost ==>       \
paulson@2451
   390
\                length evs <= i --> \
paulson@2451
   391
\                Nonce (newN i) ~: parts (sees lost Spy evs)";
paulson@2160
   392
by (parts_induct_tac 1);
paulson@2160
   393
by (REPEAT_FIRST (fast_tac (!claset 
paulson@2160
   394
                              addSEs partsEs
paulson@2160
   395
                              addSDs  [Says_imp_sees_Spy RS parts.Inj]
paulson@2269
   396
                              addEs  [leD RS notE]
paulson@2269
   397
			      addDs  [impOfSubs analz_subset_parts,
paulson@2160
   398
                                      impOfSubs parts_insert_subset_Un,
paulson@2160
   399
                                      Suc_leD]
paulson@2160
   400
                              addss (!simpset))));
paulson@2160
   401
qed_spec_mp "new_nonces_not_seen";
paulson@2133
   402
Addsimps [new_nonces_not_seen];
paulson@2133
   403
paulson@2160
   404
(*Variant: old messages must contain old nonces!*)
paulson@2133
   405
goal thy 
paulson@2451
   406
 "!!evs. [| Says A B X : set_of_list evs;              \
paulson@2451
   407
\           Nonce (newN i) : parts {X};      \
paulson@2451
   408
\           evs : yahalom lost                         \
paulson@2451
   409
\        |] ==> i < length evs";
paulson@2133
   410
by (rtac ccontr 1);
paulson@2133
   411
by (dtac leI 1);
paulson@2133
   412
by (fast_tac (!claset addSDs [new_nonces_not_seen, Says_imp_sees_Spy]
paulson@2133
   413
                      addIs  [impOfSubs parts_mono]) 1);
paulson@2133
   414
qed "Says_imp_old_nonces";
paulson@2133
   415
paulson@2133
   416
paulson@2133
   417
(** The Nonce NB uniquely identifies B's message. **)
paulson@2133
   418
paulson@2133
   419
val nonce_not_seen_now = le_refl RSN (2, new_nonces_not_seen) RSN (2,rev_notE);
paulson@2133
   420
paulson@2133
   421
goal thy 
paulson@2451
   422
 "!!evs. evs : yahalom lost ==> \
paulson@2451
   423
\   EX NA' A' B'. ALL NA A B. \
paulson@2284
   424
\      Crypt (shrK B) {|Agent A, Nonce NA, NB|} : parts(sees lost Spy evs) \
paulson@2133
   425
\      --> B ~: lost --> NA = NA' & A = A' & B = B'";
paulson@2451
   426
by (etac yahalom.induct 1 THEN parts_Fake_tac);
paulson@2451
   427
(*Fake: the tactic in parts_induct_tac works, but takes 4 times longer*)
paulson@2451
   428
by (REPEAT (etac exE 2) THEN 
paulson@2451
   429
    best_tac (!claset addSIs [exI]
paulson@2451
   430
		      addDs [impOfSubs Fake_parts_insert]
paulson@2451
   431
      	              addss (!simpset)) 2);
paulson@2451
   432
(*Base case*)
paulson@2451
   433
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2451
   434
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib]))); 
paulson@2133
   435
(*YM2: creation of new Nonce.  Move assertion into global context*)
paulson@2133
   436
by (expand_case_tac "NB = ?y" 1);
paulson@2133
   437
by (fast_tac (!claset addSEs (nonce_not_seen_now::partsEs)) 1);
paulson@2133
   438
val lemma = result();
paulson@2133
   439
paulson@2110
   440
goal thy 
paulson@2284
   441
 "!!evs.[| Crypt (shrK B) {|Agent A, Nonce NA, NB|} \
paulson@2133
   442
\                  : parts (sees lost Spy evs);         \
paulson@2284
   443
\          Crypt (shrK B') {|Agent A', Nonce NA', NB|} \
paulson@2133
   444
\                  : parts (sees lost Spy evs);         \
paulson@2133
   445
\          evs : yahalom lost;  B ~: lost;  B' ~: lost |]  \
paulson@2133
   446
\        ==> NA' = NA & A' = A & B' = B";
paulson@2451
   447
by (prove_unique_tac lemma 1);
paulson@2133
   448
qed "unique_NB";
paulson@2133
   449
paulson@2377
   450
(*OLD VERSION
paulson@2133
   451
fun lost_tac s =
paulson@2133
   452
    case_tac ("(" ^ s ^ ") : lost") THEN'
paulson@2133
   453
    SELECT_GOAL 
paulson@2133
   454
      (REPEAT_DETERM (dtac (Says_imp_sees_Spy RS analz.Inj) 1) THEN
paulson@2133
   455
       REPEAT_DETERM (etac MPair_analz 1) THEN
paulson@2133
   456
       dres_inst_tac [("A", s)] Crypt_Spy_analz_lost 1 THEN
paulson@2133
   457
       assume_tac 1 THEN Fast_tac 1);
paulson@2377
   458
*)
paulson@2133
   459
paulson@2133
   460
fun lost_tac s =
paulson@2133
   461
    case_tac ("(" ^ s ^ ") : lost") THEN'
paulson@2133
   462
    SELECT_GOAL 
paulson@2133
   463
      (REPEAT_DETERM (dtac (Says_imp_sees_Spy RS analz.Inj) 1) THEN
paulson@2133
   464
       REPEAT_DETERM (etac MPair_analz 1) THEN
paulson@2133
   465
       THEN_BEST_FIRST 
paulson@2133
   466
         (dres_inst_tac [("A", s)] Crypt_Spy_analz_lost 1 THEN assume_tac 1)
paulson@2170
   467
         (has_fewer_prems 1, size_of_thm)
paulson@2170
   468
         (Step_tac 1));
paulson@2133
   469
paulson@2133
   470
paulson@2133
   471
(*Variant useful for proving secrecy of NB*)
paulson@2133
   472
goal thy 
paulson@2284
   473
 "!!evs.[| Says C D   {|X,  Crypt (shrK B) {|Agent A, Nonce NA, NB|}|} \
paulson@2133
   474
\          : set_of_list evs;  B ~: lost;         \
paulson@2284
   475
\          Says C' D' {|X', Crypt (shrK B') {|Agent A', Nonce NA', NB|}|} \
paulson@2133
   476
\          : set_of_list evs;                           \
paulson@2133
   477
\          NB ~: analz (sees lost Spy evs);             \
paulson@2133
   478
\          evs : yahalom lost |]  \
paulson@2133
   479
\        ==> NA' = NA & A' = A & B' = B";
paulson@2133
   480
by (lost_tac "B'" 1);
paulson@2133
   481
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]
paulson@2133
   482
                      addSEs [MPair_parts]
paulson@2133
   483
                      addDs  [unique_NB]) 1);
paulson@2133
   484
qed "Says_unique_NB";
paulson@2133
   485
paulson@2133
   486
goal thy 
paulson@2133
   487
 "!!evs. [| B ~: lost;  evs : yahalom lost  |]               \
paulson@2133
   488
\ ==>  Nonce NB ~: analz (sees lost Spy evs) -->  \
paulson@2284
   489
\      Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|} : parts (sees lost Spy evs) \
paulson@2284
   490
\ --> Crypt (shrK B') {|Agent A', Nonce NB, NB'|} ~: parts (sees lost Spy evs)";
paulson@2133
   491
by (etac yahalom.induct 1);
paulson@2133
   492
by parts_Fake_tac;
paulson@2133
   493
by (REPEAT_FIRST 
paulson@2133
   494
    (rtac impI THEN' 
paulson@2133
   495
     dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD) THEN'
paulson@2133
   496
     mp_tac));
paulson@2133
   497
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
paulson@2170
   498
                             impOfSubs Fake_parts_insert]
paulson@2170
   499
                      addss (!simpset)) 2);
paulson@2133
   500
by (ALLGOALS Asm_simp_tac);
paulson@2133
   501
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2133
   502
by (fast_tac (!claset addDs [Says_imp_sees_Spy RS analz.Inj]
paulson@2170
   503
                      addSIs [parts_insertI]
paulson@2133
   504
                      addSEs partsEs
paulson@2133
   505
                      addEs [Says_imp_old_nonces RS less_irrefl]
paulson@2133
   506
                      addss (!simpset)) 1);
paulson@2133
   507
val no_nonce_YM1_YM2 = standard (result() RS mp RSN (2, rev_mp) RS notE);
paulson@2133
   508
paulson@2133
   509
paulson@2133
   510
paulson@2133
   511
(**** Towards proving secrecy of Nonce NB ****)
paulson@2133
   512
paulson@2133
   513
(*B knows, by the second part of A's message, that the Server distributed 
paulson@2133
   514
  the key quoting nonce NB.  This part says nothing about agent names. 
paulson@2133
   515
  Secrecy of NB is crucial.*)
paulson@2133
   516
goal thy 
paulson@2133
   517
 "!!evs. evs : yahalom lost                                             \
paulson@2133
   518
\        ==> Nonce NB ~: analz (sees lost Spy evs) -->                  \
paulson@2284
   519
\            Crypt K (Nonce NB) : parts (sees lost Spy evs) -->         \
paulson@2133
   520
\            (EX A B NA. Says Server A                                  \
paulson@2284
   521
\                        {|Crypt (shrK A) {|Agent B, Key K,                      \
paulson@2284
   522
\                                  Nonce NA, Nonce NB|},       \
paulson@2284
   523
\                          Crypt (shrK B) {|Agent A, Key K|}|}          \
paulson@2133
   524
\                       : set_of_list evs)";
paulson@2133
   525
by (etac yahalom.induct 1);
paulson@2133
   526
by parts_Fake_tac;
paulson@2133
   527
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2133
   528
by (REPEAT_FIRST
paulson@2133
   529
    (rtac impI THEN'
paulson@2133
   530
     dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD)));
paulson@2133
   531
by (ALLGOALS Asm_simp_tac);
paulson@2133
   532
(*Fake, YM3, YM4*)
paulson@2133
   533
by (Fast_tac 2);
paulson@2133
   534
by (fast_tac (!claset addSDs [impOfSubs Fake_parts_insert]
paulson@2133
   535
                      addDs [impOfSubs analz_subset_parts]
paulson@2133
   536
                      addss (!simpset)) 1);
paulson@2133
   537
(*YM4*)
paulson@2133
   538
by (Step_tac 1);
paulson@2133
   539
by (lost_tac "A" 1);
paulson@2133
   540
by (fast_tac (!claset addDs [Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS
paulson@2322
   541
                             A_trusts_YM3]) 1);
paulson@2133
   542
val B_trusts_YM4_newK = result() RS mp RSN (2, rev_mp);
paulson@2133
   543
paulson@2133
   544
paulson@2133
   545
(*This is the original version of the result above.  But it is of little
paulson@2133
   546
  value because it assumes secrecy of K, which we cannot be assured of
paulson@2133
   547
  until we know that K is fresh -- which we do not know at the point this
paulson@2133
   548
  result is applied.*)
paulson@2133
   549
goal thy 
paulson@2133
   550
 "!!evs. evs : yahalom lost                                             \
paulson@2133
   551
\        ==> Key K ~: analz (sees lost Spy evs) -->                     \
paulson@2284
   552
\            Crypt K (Nonce NB) : parts (sees lost Spy evs) -->         \
paulson@2110
   553
\            (EX A B NA. Says Server A                                  \
paulson@2284
   554
\                        {|Crypt (shrK A) {|Agent B, Key K,                      \
paulson@2284
   555
\                                  Nonce NA, Nonce NB|},       \
paulson@2284
   556
\                          Crypt (shrK B) {|Agent A, Key K|}|}          \
paulson@2110
   557
\                       : set_of_list evs)";
paulson@2032
   558
by (etac yahalom.induct 1);
paulson@2110
   559
by parts_Fake_tac;
paulson@2001
   560
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2110
   561
by (TRYALL (rtac impI));
paulson@2110
   562
by (REPEAT_FIRST
paulson@2110
   563
    (dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD)));
paulson@2110
   564
by (ALLGOALS Asm_simp_tac);
paulson@2110
   565
(*Fake, YM3, YM4*)
paulson@2110
   566
by (fast_tac (!claset addSDs [Crypt_Fake_parts_insert]
paulson@2133
   567
                      addDs  [impOfSubs analz_subset_parts]) 1);
paulson@2110
   568
by (Fast_tac 1);
paulson@2110
   569
(*YM4*)
paulson@2110
   570
by (Step_tac 1);
paulson@2133
   571
by (lost_tac "A" 1);
paulson@2110
   572
by (fast_tac (!claset addDs [Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS
paulson@2322
   573
                             A_trusts_YM3]) 1);
paulson@2133
   574
result() RS mp RSN (2, rev_mp);
paulson@2133
   575
paulson@2133
   576
paulson@2133
   577
(*YM3 can only be triggered by YM2*)
paulson@2133
   578
goal thy 
paulson@2133
   579
 "!!evs. [| Says Server A                                           \
paulson@2284
   580
\            {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} : set_of_list evs; \
paulson@2133
   581
\           evs : yahalom lost |]                                        \
paulson@2133
   582
\        ==> EX B'. Says B' Server                                       \
paulson@2284
   583
\                      {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |} \
paulson@2133
   584
\                   : set_of_list evs";
paulson@2133
   585
by (etac rev_mp 1);
paulson@2133
   586
by (etac yahalom.induct 1);
paulson@2133
   587
by (ALLGOALS Asm_simp_tac);
paulson@2133
   588
by (ALLGOALS Fast_tac);
paulson@2133
   589
qed "Says_Server_imp_YM2";
paulson@2133
   590
paulson@2133
   591
paulson@2133
   592
(** Dedicated tactics for the nonce secrecy proofs **)
paulson@2133
   593
paulson@2133
   594
val no_nonce_tac = SELECT_GOAL
paulson@2133
   595
   (REPEAT (resolve_tac [impI, notI] 1) THEN
paulson@2133
   596
    REPEAT (hyp_subst_tac 1) THEN
paulson@2133
   597
    etac (Says_imp_sees_Spy RS parts.Inj RS parts.Snd RS no_nonce_YM1_YM2) 1
paulson@2133
   598
    THEN
paulson@2133
   599
    etac (Says_imp_sees_Spy RS parts.Inj RS parts.Snd) 4
paulson@2133
   600
    THEN 
paulson@2133
   601
    REPEAT_FIRST assume_tac);
paulson@2133
   602
paulson@2133
   603
val not_analz_insert = subset_insertI RS analz_mono RS contra_subsetD;
paulson@2133
   604
paulson@2133
   605
fun grind_tac i = 
paulson@2133
   606
 SELECT_GOAL
paulson@2133
   607
  (REPEAT_FIRST 
paulson@2133
   608
   (Safe_step_tac ORELSE' (dtac spec THEN' mp_tac) ORELSE'
paulson@2133
   609
    assume_tac ORELSE'
paulson@2133
   610
    depth_tac (!claset delrules [conjI]
paulson@2377
   611
                       addSIs [exI, analz_insertI,
paulson@2170
   612
                               impOfSubs (Un_upper2 RS analz_mono)]) 2)) i;
paulson@2133
   613
paulson@2133
   614
(*The only nonces that can be found with the help of session keys are
paulson@2133
   615
  those distributed as nonce NB by the Server.  The form of the theorem
paulson@2133
   616
  recalls analz_image_newK, but it is much more complicated.*)
paulson@2133
   617
goal thy 
paulson@2133
   618
 "!!evs. evs : yahalom lost ==>                                           \
paulson@2133
   619
\     ALL E. Nonce NB : analz (Key``(newK``E) Un (sees lost Spy evs)) --> \
paulson@2133
   620
\     (EX K: newK``E. EX A B na X.                                        \
paulson@2133
   621
\            Says Server A                                                \
paulson@2284
   622
\                {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}   \
paulson@2133
   623
\            : set_of_list evs)  |  Nonce NB : analz (sees lost Spy evs)";
paulson@2133
   624
by (etac yahalom.induct 1);
paulson@2133
   625
by analz_Fake_tac;
paulson@2451
   626
by (ALLGOALS  (*22 seconds*)
paulson@2133
   627
    (asm_simp_tac 
paulson@2454
   628
     (!simpset addsimps ([not_parts_not_analz, analz_image_newK,
paulson@2170
   629
                          insert_Key_singleton, insert_Key_image]
paulson@2170
   630
                         @ pushes)
paulson@2133
   631
               setloop split_tac [expand_if])));
paulson@2133
   632
(*Base*)
paulson@2133
   633
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2133
   634
(*Fake*) (** LEVEL 4 **)
paulson@2133
   635
by (spy_analz_tac 1);
paulson@2451
   636
(*YM1-YM3*) (*24 seconds*)
paulson@2133
   637
by (EVERY (map grind_tac [3,2,1]));
paulson@2133
   638
(*Oops*)
paulson@2133
   639
by (Full_simp_tac 2);
paulson@2133
   640
by (simp_tac (!simpset addsimps [insert_Key_image]) 2);
paulson@2133
   641
by (grind_tac 2);
paulson@2133
   642
by (fast_tac (!claset delrules [bexI] 
paulson@2170
   643
                      addDs [unique_session_keys]
paulson@2170
   644
                      addss (!simpset)) 2);
paulson@2133
   645
(*YM4*)
paulson@2454
   646
(** LEVEL 10 **)
paulson@2170
   647
by (rtac (impI RS allI) 1);
paulson@2377
   648
by (dtac (impOfSubs Fake_analz_insert) 1 THEN etac synth.Inj 1);
paulson@2133
   649
by (eres_inst_tac [("P","Nonce NB : ?HH")] rev_mp 1);
paulson@2133
   650
by (asm_simp_tac (!simpset addsimps [analz_image_newK]
paulson@2170
   651
                           setloop split_tac [expand_if]) 1);
paulson@2454
   652
(** LEVEL 14 **)
paulson@2133
   653
by (grind_tac 1);
paulson@2133
   654
by (REPEAT (dtac not_analz_insert 1));
paulson@2133
   655
by (lost_tac "A" 1);
paulson@2322
   656
by (dtac (Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1
paulson@2133
   657
    THEN REPEAT (assume_tac 1));
paulson@2133
   658
by (fast_tac (!claset delrules [allE, conjI] addSIs [bexI, exI]) 1);
paulson@2133
   659
by (fast_tac (!claset delrules [conjI]
paulson@2377
   660
                      addIs [analz_insertI]
paulson@2170
   661
                      addss (!simpset)) 1);
paulson@2133
   662
val Nonce_secrecy = result() RS spec RSN (2, rev_mp) |> standard;
paulson@2133
   663
paulson@2133
   664
paulson@2133
   665
(*Version required below: if NB can be decrypted using a session key then it
paulson@2133
   666
  was distributed with that key.  The more general form above is required
paulson@2133
   667
  for the induction to carry through.*)
paulson@2133
   668
goal thy 
paulson@2451
   669
 "!!evs. [| Says Server A                                                   \
paulson@2451
   670
\            {|Crypt (shrK A) {|Agent B, Key (newK i), na, Nonce NB'|}, X|} \
paulson@2451
   671
\           : set_of_list evs;                                              \
paulson@2451
   672
\           Nonce NB : analz (insert (Key (newK i)) (sees lost Spy evs));   \
paulson@2451
   673
\           evs : yahalom lost |]                                           \
paulson@2133
   674
\ ==> Nonce NB : analz (sees lost Spy evs) | NB = NB'";
paulson@2133
   675
by (asm_full_simp_tac (HOL_ss addsimps [insert_Key_singleton]) 1);
paulson@2133
   676
by (dtac Nonce_secrecy 1 THEN assume_tac 1);
paulson@2133
   677
by (fast_tac (!claset addDs [unique_session_keys]
paulson@2170
   678
                      addss (!simpset)) 1);
paulson@2133
   679
val single_Nonce_secrecy = result();
paulson@2133
   680
paulson@2133
   681
paulson@2133
   682
goal thy 
paulson@2133
   683
 "!!evs. [| A ~: lost;  B ~: lost;  Spy: lost;  evs : yahalom lost |]  \
paulson@2133
   684
\ ==> Says B Server                                                    \
paulson@2284
   685
\          {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
paulson@2133
   686
\     : set_of_list evs -->                               \
paulson@2133
   687
\     (ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set_of_list evs) -->  \
paulson@2133
   688
\     Nonce NB ~: analz (sees lost Spy evs)";
paulson@2133
   689
by (etac yahalom.induct 1);
paulson@2133
   690
by analz_Fake_tac;
paulson@2133
   691
by (ALLGOALS
paulson@2133
   692
    (asm_simp_tac 
paulson@2377
   693
     (!simpset addsimps ([not_parts_not_analz,
paulson@2133
   694
                          analz_insert_Key_newK] @ pushes)
paulson@2133
   695
               setloop split_tac [expand_if])));
paulson@2133
   696
by (fast_tac (!claset addSIs [parts_insertI]
paulson@2133
   697
                      addSEs partsEs
paulson@2133
   698
                      addEs [Says_imp_old_nonces RS less_irrefl]
paulson@2133
   699
                      addss (!simpset)) 2);
paulson@2377
   700
(*Proof of YM2*) (** LEVEL 4 **)
paulson@2133
   701
by (REPEAT (Safe_step_tac 2 ORELSE Fast_tac 2)); 
paulson@2133
   702
by (fast_tac (!claset addIs [parts_insertI]
paulson@2133
   703
                      addSEs partsEs
paulson@2133
   704
                      addEs [Says_imp_old_nonces RS less_irrefl]
paulson@2133
   705
                      addss (!simpset)) 3);
paulson@2133
   706
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 2);
paulson@2133
   707
(*Prove YM3 by showing that no NB can also be an NA*)
paulson@2133
   708
by (REPEAT (Safe_step_tac 2 ORELSE no_nonce_tac 2));
paulson@2133
   709
by (deepen_tac (!claset addDs [Says_unique_NB]) 1 2);
paulson@2377
   710
(*Fake*)
paulson@2377
   711
by (spy_analz_tac 1);
paulson@2377
   712
(*YM4*) (** LEVEL 10 **)
paulson@2133
   713
by (res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1);
paulson@2133
   714
by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
paulson@2377
   715
by (SELECT_GOAL (REPEAT_FIRST (spy_analz_tac ORELSE' Safe_step_tac)) 1);
paulson@2377
   716
(** LEVEL 13 **)
paulson@2133
   717
by (lost_tac "Aa" 1);
paulson@2322
   718
by (dtac (Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1);
paulson@2133
   719
by (forward_tac [Says_Server_message_form] 3);
paulson@2133
   720
by (forward_tac [Says_Server_imp_YM2] 4);
paulson@2133
   721
by (REPEAT_FIRST ((eresolve_tac [asm_rl, bexE, exE, disjE])));
paulson@2133
   722
by (Full_simp_tac 1);
paulson@2377
   723
by (REPEAT_FIRST hyp_subst_tac);
paulson@2133
   724
(** LEVEL 20 **)
paulson@2133
   725
by (lost_tac "Ba" 1);
paulson@2170
   726
by (dtac (Says_imp_sees_Spy RS parts.Inj RS parts.Snd RS unique_NB) 1);
paulson@2133
   727
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]
paulson@2133
   728
                      addSEs [MPair_parts]) 1);
paulson@2133
   729
by (REPEAT (assume_tac 1 ORELSE Safe_step_tac 1)); 
paulson@2170
   730
by (dtac Spy_not_see_encrypted_key 1);
paulson@2133
   731
by (REPEAT (assume_tac 1 ORELSE Fast_tac 1)); 
paulson@2133
   732
by (spy_analz_tac 1);
paulson@2377
   733
(*Oops case*) (** LEVEL 27 **)
paulson@2133
   734
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 1);
paulson@2133
   735
by (step_tac (!claset delrules [disjE, conjI]) 1);
paulson@2133
   736
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1 THEN etac exE 1);
paulson@2133
   737
by (expand_case_tac "NB = NBa" 1);
paulson@2133
   738
by (deepen_tac (!claset addDs [Says_unique_NB]) 1 1);
paulson@2170
   739
by (rtac conjI 1);
paulson@2133
   740
by (no_nonce_tac 1);
paulson@2377
   741
(** LEVEL 34 **)
paulson@2133
   742
by (thin_tac "?PP|?QQ" 1);  (*subsumption!*)
paulson@2133
   743
by (fast_tac (!claset addSDs [single_Nonce_secrecy]) 1);
paulson@2133
   744
val Spy_not_see_NB = result() RSN(2,rev_mp) RSN(2,rev_mp) |> standard;
paulson@2133
   745
paulson@2001
   746
paulson@2110
   747
(*What can B deduce from receipt of YM4?  Note how the two components of
paulson@2110
   748
  the message contribute to a single conclusion about the Server's message.
paulson@2110
   749
  It's annoying that the "Says A Spy" assumption must quantify over 
paulson@2133
   750
  ALL POSSIBLE keys instead of our particular K (though at least the
paulson@2133
   751
  nonces are forced to agree with NA and NB). *)
paulson@2001
   752
goal thy 
paulson@2133
   753
 "!!evs. [| Says B Server                                        \
paulson@2284
   754
\            {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}  \
paulson@2133
   755
\           : set_of_list evs;       \
paulson@2284
   756
\           Says A' B {|Crypt (shrK B) {|Agent A, Key K|},              \
paulson@2284
   757
\                       Crypt K (Nonce NB)|} : set_of_list evs;         \
paulson@2133
   758
\           ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set_of_list evs; \
paulson@2133
   759
\           A ~: lost;  B ~: lost;  Spy: lost;  evs : yahalom lost |]   \
paulson@2133
   760
\         ==> Says Server A                                       \
paulson@2284
   761
\                     {|Crypt (shrK A) {|Agent B, Key K,                         \
paulson@2284
   762
\                               Nonce NA, Nonce NB|},          \
paulson@2284
   763
\                       Crypt (shrK B) {|Agent A, Key K|}|}             \
paulson@2001
   764
\                   : set_of_list evs";
paulson@2133
   765
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1));
paulson@2133
   766
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1 THEN
paulson@2133
   767
    dtac B_trusts_YM4_shrK 1);
paulson@2170
   768
by (dtac B_trusts_YM4_newK 3);
paulson@2110
   769
by (REPEAT_FIRST (eresolve_tac [asm_rl, exE]));
paulson@2133
   770
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1);
paulson@2170
   771
by (dtac unique_session_keys 1 THEN REPEAT (assume_tac 1));
paulson@2133
   772
by (deepen_tac (!claset addDs [Says_unique_NB] addss (!simpset)) 0 1);
paulson@2322
   773
qed "B_trusts_YM4";
paulson@2133
   774