1934
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(* Title: HOL/Auth/NS_Shared
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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 "ns_shared" for Needham-Schroeder Shared-Key protocol.
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From page 247 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 NS_Shared;
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1943
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proof_timing:=true;
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1997
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HOL_quantifiers := false;
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2015
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(*Weak liveness: there are traces that reach the end*)
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1997
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goal thy
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"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \
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2032
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\ ==> EX N K. EX evs: ns_shared lost. \
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1997
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\ Says A B (Crypt {|Nonce N, Nonce N|} K) : set_of_list evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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2032
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by (rtac (ns_shared.Nil RS ns_shared.NS1 RS ns_shared.NS2 RS ns_shared.NS3 RS ns_shared.NS4 RS ns_shared.NS5) 2);
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1997
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by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
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by (REPEAT_FIRST (resolve_tac [refl, conjI]));
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by (ALLGOALS (fast_tac (!claset addss (!simpset setsolver safe_solver))));
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2015
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result();
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1943
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1934
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(**** Inductive proofs about ns_shared ****)
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2032
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goal thy "!!evs. lost' <= lost ==> ns_shared lost' <= ns_shared lost";
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by (rtac subsetI 1);
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by (etac ns_shared.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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:: ns_shared.intrs))));
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qed "ns_shared_mono";
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(*The Spy can see more than anybody else, except for their initial state*)
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1934
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goal thy
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2032
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"!!evs. evs : ns_shared lost ==> \
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\ sees lost A evs <= initState lost A Un sees lost Spy evs";
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by (etac ns_shared.induct 1);
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1934
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by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD]
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2032
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addss (!simpset))));
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qed "sees_agent_subset_sees_Spy";
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(*The Spy can see more than anybody else who's lost their key!*)
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goal thy
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"!!evs. evs : ns_shared lost ==> \
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\ A: lost --> A ~= Server --> sees lost A evs <= sees lost Spy evs";
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by (etac ns_shared.induct 1);
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by (agent.induct_tac "A" 1);
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by (auto_tac (!claset addDs [sees_Says_subset_insert RS subsetD], (!simpset)));
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qed_spec_mp "sees_lost_agent_subset_sees_Spy";
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1934
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(*Nobody sends themselves messages*)
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2032
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goal thy "!!evs. evs : ns_shared lost ==> ALL A X. Says A A X ~: set_of_list evs";
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by (etac ns_shared.induct 1);
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1934
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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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1943
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(*For reasoning about the encrypted portion of message NS3*)
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1934
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goal thy "!!evs. (Says S A (Crypt {|N, B, K, X|} KA)) : set_of_list evs ==> \
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2032
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\ X : parts (sees lost Spy evs)";
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1934
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by (fast_tac (!claset addSEs partsEs
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2032
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addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
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qed "NS3_msg_in_parts_sees_Spy";
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val parts_Fake_tac =
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dres_inst_tac [("lost","lost")] NS3_msg_in_parts_sees_Spy 5;
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1934
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2032
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(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
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2015
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sends messages containing X! **)
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1934
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2032
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(*Spy never sees lost another agent's shared key!*)
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1934
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goal thy
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2032
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"!!evs. [| evs : ns_shared lost; A ~: lost |] \
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\ ==> Key (shrK A) ~: parts (sees lost Spy evs)";
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by (etac ns_shared.induct 1);
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by parts_Fake_tac;
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1934
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by (Auto_tac());
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(*Deals with Fake message*)
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by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
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2032
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impOfSubs Fake_parts_insert]) 1);
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qed "Spy_not_see_shrK";
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1934
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2032
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bind_thm ("Spy_not_analz_shrK",
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[analz_subset_parts, Spy_not_see_shrK] MRS contra_subsetD);
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1934
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2032
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Addsimps [Spy_not_see_shrK, Spy_not_analz_shrK];
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1934
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1965
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(*We go to some trouble to preserve R in the 3rd and 4th subgoals
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1967
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As usual fast_tac cannot be used because it uses the equalities too soon*)
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1934
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val major::prems =
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2032
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goal thy "[| Key (shrK A) : parts (sees lost Spy evs); \
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\ evs : ns_shared lost; \
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\ A:lost ==> R \
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1934
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\ |] ==> R";
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2032
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by (rtac ccontr 1);
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by (rtac ([major, Spy_not_see_shrK] MRS rev_notE) 1);
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1934
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by (swap_res_tac prems 2);
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1967
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by (ALLGOALS (fast_tac (!claset addIs prems)));
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2032
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qed "Spy_see_shrK_E";
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1934
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2032
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bind_thm ("Spy_analz_shrK_E",
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analz_subset_parts RS subsetD RS Spy_see_shrK_E);
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1934
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2032
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AddSEs [Spy_see_shrK_E, Spy_analz_shrK_E];
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1934
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goal thy
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2032
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"!!evs. evs : ns_shared lost ==> \
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\ (Key (shrK A) : analz (sees lost Spy evs)) = (A : lost)";
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2027
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by (best_tac (!claset addIs [impOfSubs (subset_insertI RS analz_mono)]
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2032
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addss (!simpset)) 1);
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1943
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qed "shrK_mem_analz";
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1934
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1967
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Addsimps [shrK_mem_analz];
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1934
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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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This has to be proved anew for each protocol description,
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but should go by similar reasoning every time. Hardest case is the
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standard Fake rule.
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The length comparison, and Union over C, are essential for the
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induction! *)
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2032
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goal thy "!!evs. evs : ns_shared lost ==> \
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1934
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\ length evs <= length evs' --> \
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2032
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\ Key (newK evs') ~: (UN C. parts (sees lost C evs))";
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by (etac ns_shared.induct 1);
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by parts_Fake_tac;
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1934
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(*auto_tac does not work here, as it performs safe_tac first*)
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by (ALLGOALS Asm_simp_tac);
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2015
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by (ALLGOALS (fast_tac (!claset addDs [impOfSubs analz_subset_parts,
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2032
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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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1934
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val lemma = result();
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(*Variant needed for the main theorem below*)
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goal thy
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2032
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"!!evs. [| evs : ns_shared lost; length evs <= length evs' |] \
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\ ==> Key (newK evs') ~: parts (sees lost C evs)";
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1934
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by (fast_tac (!claset addDs [lemma]) 1);
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qed "new_keys_not_seen";
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Addsimps [new_keys_not_seen];
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(*Another 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 evt) : parts {X}; \
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2032
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\ evs : ns_shared lost \
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1934
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\ |] ==> length evt < length evs";
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2032
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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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2015
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addIs [impOfSubs parts_mono]) 1);
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1934
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qed "Says_imp_old_keys";
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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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2032
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goal thy "!!evs. evs : ns_shared lost ==> \
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1934
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\ length evs <= length evs' --> \
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2032
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\ newK evs' ~: keysFor (UN C. parts (sees lost C evs))";
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by (etac ns_shared.induct 1);
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by parts_Fake_tac;
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1934
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by (ALLGOALS Asm_simp_tac);
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(*NS1 and NS2*)
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map (by o fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [3,2];
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(*Fake and NS3*)
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map (by o best_tac
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1997
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(!claset addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
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2027
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impOfSubs (parts_insert_subset_Un RS keysFor_mono),
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2032
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Suc_leD]
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addEs [new_keys_not_seen RS not_parts_not_analz RSN (2,rev_notE)]
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addss (!simpset)))
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1934
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[2,1];
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(*NS4 and NS5: nonce exchange*)
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1997
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by (ALLGOALS (deepen_tac (!claset addSDs [Says_imp_old_keys]
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2032
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addIs [less_SucI, impOfSubs keysFor_mono]
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addss (!simpset addsimps [le_def])) 0));
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1934
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val lemma = result();
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goal thy
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2032
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"!!evs. [| evs : ns_shared lost; length evs <= length evs' |] \
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\ ==> newK evs' ~: keysFor (parts (sees lost C evs))";
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1934
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by (fast_tac (!claset addSDs [lemma] addss (!simpset)) 1);
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qed "new_keys_not_used";
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bind_thm ("new_keys_not_analzd",
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2032
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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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1934
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Addsimps [new_keys_not_used, new_keys_not_analzd];
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(** Lemmas concerning the form of items passed in messages **)
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2015
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211 |
(*Describes the form of K, X and K' when the Server sends this message.*)
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1934
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goal thy
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"!!evs. [| Says Server A (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
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2032
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\ evs : ns_shared lost |] \
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\ ==> (EX evt: ns_shared lost. \
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2015
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216 |
\ K = Key(newK evt) & \
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\ X = (Crypt {|K, Agent A|} (shrK B)) & \
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\ K' = shrK A)";
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2032
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219 |
by (etac rev_mp 1);
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by (etac ns_shared.induct 1);
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2015
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221 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
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1934
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qed "Says_Server_message_form";
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1943
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225 |
(*Describes the form of X when the following message is sent. The use of
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1965
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226 |
"parts" strengthens the induction hyp for proving the Fake case. The
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assumptions on A are needed to prevent its being a Faked message.*)
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1934
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goal thy
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2032
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229 |
"!!evs. evs : ns_shared lost ==> \
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1967
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\ Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A) \
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2032
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\ : parts (sees lost Spy evs) & \
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\ A ~: lost --> \
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\ (EX evt: ns_shared lost. K = newK evt & \
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1943
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234 |
\ X = (Crypt {|Key K, Agent A|} (shrK B)))";
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2032
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by (etac ns_shared.induct 1);
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by parts_Fake_tac;
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2015
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237 |
(*Fake case*)
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1967
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238 |
by (best_tac (!claset addSDs [impOfSubs Fake_parts_insert]
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2032
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addDs [impOfSubs analz_subset_parts]
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addss (!simpset)) 2);
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2015
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241 |
by (Auto_tac());
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val lemma = result() RS mp;
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1934
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243 |
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244 |
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2015
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245 |
(*The following theorem is proved by cases. If the message was sent with a
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2032
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246 |
bad key then the Spy reads it -- even if he didn't send it in the first
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2015
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247 |
place.*)
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1934
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248 |
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1965
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249 |
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(*EITHER describes the form of X when the following message is sent,
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OR reduces it to the Fake case.
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Use Says_Server_message_form if applicable.*)
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1934
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253 |
goal thy
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1943
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254 |
"!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A)) \
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2032
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255 |
\ : set_of_list evs; evs : ns_shared lost |] \
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\ ==> (EX evt: ns_shared lost. K = newK evt & length evt < length evs & \
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1965
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257 |
\ X = (Crypt {|Key K, Agent A|} (shrK B))) | \
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2032
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258 |
\ X : analz (sees lost Spy evs)";
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259 |
by (excluded_middle_tac "A : lost" 1);
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260 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
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addss (!simpset)) 2);
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2015
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262 |
by (forward_tac [lemma] 1);
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by (fast_tac (!claset addEs partsEs
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2032
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264 |
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
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2015
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265 |
by (fast_tac (!claset addIs [Says_imp_old_keys] addss (!simpset)) 1);
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1934
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266 |
qed "Says_S_message_form";
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267 |
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268 |
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269 |
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(****
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The following is to prove theorems of the form
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272 |
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2032
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273 |
Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==>
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Key K : analz (sees lost Spy evs)
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1934
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275 |
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A more general formula must be proved inductively.
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277 |
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****)
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279 |
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280 |
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281 |
(*NOT useful in this form, but it says that session keys are not used
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282 |
to encrypt messages containing other keys, in the actual protocol.
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283 |
We require that agents should behave like this subsequently also.*)
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284 |
goal thy
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2032
|
285 |
"!!evs. evs : ns_shared lost ==> \
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286 |
\ (Crypt X (newK evt)) : parts (sees lost Spy evs) & \
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287 |
\ Key K : parts {X} --> Key K : parts (sees lost Spy evs)";
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|
288 |
by (etac ns_shared.induct 1);
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|
289 |
by parts_Fake_tac;
|
1934
|
290 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps pushes)));
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291 |
(*Deals with Faked messages*)
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|
292 |
by (best_tac (!claset addSEs partsEs
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2032
|
293 |
addDs [impOfSubs parts_insert_subset_Un]
|
1965
|
294 |
addss (!simpset)) 2);
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|
295 |
(*Base, NS4 and NS5*)
|
1934
|
296 |
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
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297 |
result();
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298 |
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299 |
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|
300 |
(** Specialized rewriting for this proof **)
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301 |
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|
302 |
Delsimps [image_insert];
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303 |
Addsimps [image_insert RS sym];
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304 |
|
1965
|
305 |
Delsimps [image_Un];
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306 |
Addsimps [image_Un RS sym];
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307 |
|
2032
|
308 |
goal thy "insert (Key (newK x)) (sees lost A evs) = \
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|
309 |
\ Key `` (newK``{x}) Un (sees lost A evs)";
|
1934
|
310 |
by (Fast_tac 1);
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|
311 |
val insert_Key_singleton = result();
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|
312 |
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|
313 |
goal thy "insert (Key (f x)) (Key``(f``E) Un C) = \
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314 |
\ Key `` (f `` (insert x E)) Un C";
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|
315 |
by (Fast_tac 1);
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|
316 |
val insert_Key_image = result();
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317 |
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318 |
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|
319 |
(** Session keys are not used to encrypt other session keys **)
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|
320 |
|
1943
|
321 |
(*Lemma for the trivial direction of the if-and-only-if*)
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|
322 |
goal thy
|
1965
|
323 |
"!!evs. (Key K : analz (Key``nE Un sEe)) --> \
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|
324 |
\ (K : nE | Key K : analz sEe) ==> \
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|
325 |
\ (Key K : analz (Key``nE Un sEe)) = (K : nE | Key K : analz sEe)";
|
1943
|
326 |
by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 1);
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|
327 |
val lemma = result();
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|
328 |
|
2015
|
329 |
(*The equality makes the induction hypothesis easier to apply*)
|
1934
|
330 |
goal thy
|
2032
|
331 |
"!!evs. evs : ns_shared lost ==> \
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|
332 |
\ ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
|
|
333 |
\ (K : newK``E | Key K : analz (sees lost Spy evs))";
|
|
334 |
by (etac ns_shared.induct 1);
|
|
335 |
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
|
1965
|
336 |
by (REPEAT ((eresolve_tac [bexE, conjE, disjE] ORELSE' hyp_subst_tac) 5));
|
1943
|
337 |
by (REPEAT_FIRST (resolve_tac [allI, lemma]));
|
1934
|
338 |
by (ALLGOALS
|
|
339 |
(asm_simp_tac
|
|
340 |
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
|
2032
|
341 |
@ pushes)
|
1934
|
342 |
setloop split_tac [expand_if])));
|
1943
|
343 |
(** LEVEL 5 **)
|
1965
|
344 |
(*NS3, Fake subcase*)
|
2032
|
345 |
by (spy_analz_tac 5);
|
1934
|
346 |
(*Cases NS2 and NS3!! Simple, thanks to auto case splits*)
|
|
347 |
by (REPEAT (Fast_tac 3));
|
1965
|
348 |
(*Fake case*) (** LEVEL 7 **)
|
2032
|
349 |
by (spy_analz_tac 2);
|
1934
|
350 |
(*Base case*)
|
|
351 |
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
|
|
352 |
qed_spec_mp "analz_image_newK";
|
|
353 |
|
|
354 |
|
|
355 |
goal thy
|
2032
|
356 |
"!!evs. evs : ns_shared lost ==> \
|
|
357 |
\ Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \
|
|
358 |
\ (K = newK evt | Key K : analz (sees lost Spy evs))";
|
1934
|
359 |
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK,
|
2032
|
360 |
insert_Key_singleton]) 1);
|
1934
|
361 |
by (Fast_tac 1);
|
|
362 |
qed "analz_insert_Key_newK";
|
|
363 |
|
|
364 |
|
2015
|
365 |
(** The session key K uniquely identifies the message, if encrypted
|
|
366 |
with a secure key **)
|
1965
|
367 |
|
|
368 |
fun ex_strip_tac i = REPEAT (ares_tac [exI, conjI] i) THEN assume_tac (i+1);
|
1934
|
369 |
|
|
370 |
goal thy
|
2032
|
371 |
"!!evs. evs : ns_shared lost ==> \
|
2015
|
372 |
\ EX X'. ALL A X N B. \
|
2032
|
373 |
\ A ~: lost --> \
|
|
374 |
\ Crypt {|N, Agent B, Key K, X|} (shrK A) : parts (sees lost Spy evs) --> \
|
2015
|
375 |
\ X=X'";
|
2032
|
376 |
by (Simp_tac 1); (*Miniscoping*)
|
|
377 |
by (etac ns_shared.induct 1);
|
|
378 |
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
|
1934
|
379 |
by (ALLGOALS
|
2015
|
380 |
(asm_simp_tac (!simpset addsimps [all_conj_distrib, ex_disj_distrib,
|
2032
|
381 |
imp_conj_distrib, parts_insert_sees])));
|
1965
|
382 |
by (REPEAT_FIRST (eresolve_tac [exE,disjE]));
|
2027
|
383 |
(*NS2: Cextraction of K = newK evsa to global context...*)
|
|
384 |
(** LEVEL 5 **)
|
|
385 |
by (excluded_middle_tac "K = newK evsa" 3);
|
|
386 |
by (Asm_simp_tac 3);
|
2032
|
387 |
by (etac exI 3);
|
1934
|
388 |
(*...we assume X is a very new message, and handle this case by contradiction*)
|
2015
|
389 |
by (fast_tac (!claset addSEs partsEs
|
2032
|
390 |
addEs [Says_imp_old_keys RS less_irrefl]
|
|
391 |
addss (!simpset)) 3);
|
2027
|
392 |
(*Base, Fake, NS3*) (** LEVEL 9 **)
|
|
393 |
by (REPEAT_FIRST ex_strip_tac);
|
2032
|
394 |
by (dtac synth.Inj 4);
|
2027
|
395 |
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs Fake_parts_insert]
|
2032
|
396 |
addss (!simpset))));
|
1934
|
397 |
val lemma = result();
|
|
398 |
|
|
399 |
(*In messages of this form, the session key uniquely identifies the rest*)
|
|
400 |
goal thy
|
|
401 |
"!!evs. [| Says S A \
|
1943
|
402 |
\ (Crypt {|N, Agent B, Key K, X|} (shrK C)) \
|
1965
|
403 |
\ : set_of_list evs; \
|
|
404 |
\ Says S' A' \
|
1943
|
405 |
\ (Crypt {|N', Agent B', Key K, X'|} (shrK C')) \
|
1934
|
406 |
\ : set_of_list evs; \
|
2032
|
407 |
\ evs : ns_shared lost; C ~: lost; C' ~: lost |] ==> X = X'";
|
|
408 |
by (dtac lemma 1);
|
|
409 |
by (etac exE 1);
|
1965
|
410 |
(*Duplicate the assumption*)
|
1934
|
411 |
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
|
2032
|
412 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
|
1934
|
413 |
qed "unique_session_keys";
|
|
414 |
|
|
415 |
|
|
416 |
|
2032
|
417 |
(** Crucial secrecy property: Spy does not see the keys sent in msg NS2 **)
|
2015
|
418 |
|
1934
|
419 |
goal thy
|
2032
|
420 |
"!!evs. [| A ~: lost; B ~: lost; \
|
|
421 |
\ evs : ns_shared lost; evt: ns_shared lost |] \
|
2015
|
422 |
\ ==> Says Server A \
|
|
423 |
\ (Crypt {|N, Agent B, Key(newK evt), \
|
|
424 |
\ Crypt {|Key(newK evt), Agent A|} (shrK B)|} (shrK A)) \
|
|
425 |
\ : set_of_list evs --> \
|
2032
|
426 |
\ Key(newK evt) ~: analz (sees lost Spy evs)";
|
|
427 |
by (etac ns_shared.induct 1);
|
1934
|
428 |
by (ALLGOALS
|
2015
|
429 |
(asm_simp_tac
|
1934
|
430 |
(!simpset addsimps ([analz_subset_parts RS contra_subsetD,
|
2032
|
431 |
analz_insert_Key_newK] @ pushes)
|
1934
|
432 |
setloop split_tac [expand_if])));
|
|
433 |
(*NS2*)
|
2015
|
434 |
by (fast_tac (!claset addIs [parts_insertI]
|
2032
|
435 |
addEs [Says_imp_old_keys RS less_irrefl]
|
|
436 |
addss (!simpset)) 2);
|
2015
|
437 |
(*Fake case*)
|
2032
|
438 |
by (spy_analz_tac 1);
|
1934
|
439 |
(*NS3: that message from the Server was sent earlier*)
|
|
440 |
by (forward_tac [Says_S_message_form] 1 THEN assume_tac 1);
|
1967
|
441 |
by (Step_tac 1);
|
2032
|
442 |
by (REPEAT_FIRST assume_tac);
|
|
443 |
by (spy_analz_tac 2); (*Prove the Fake subcase*)
|
1934
|
444 |
by (asm_full_simp_tac
|
|
445 |
(!simpset addsimps (mem_if::analz_insert_Key_newK::pushes)) 1);
|
1967
|
446 |
by (Step_tac 1);
|
2032
|
447 |
(**LEVEL 10 **)
|
|
448 |
by (excluded_middle_tac "Aa : lost" 1);
|
|
449 |
(*But this contradicts Key(newK evt) ~: analz (sees lost Spy evsa) *)
|
|
450 |
by (dtac (Says_imp_sees_Spy RS analz.Inj) 2);
|
2015
|
451 |
by (fast_tac (!claset addSDs [analz.Decrypt]
|
2032
|
452 |
addss (!simpset)) 2);
|
|
453 |
(*So now we have Aa ~: lost *)
|
|
454 |
by (dtac unique_session_keys 1);
|
1967
|
455 |
by (Auto_tac ());
|
2015
|
456 |
val lemma = result() RS mp RSN(2,rev_notE);
|
|
457 |
|
|
458 |
|
|
459 |
(*Final version: Server's message in the most abstract form*)
|
|
460 |
goal thy
|
|
461 |
"!!evs. [| Says Server A \
|
|
462 |
\ (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
|
2032
|
463 |
\ A ~: lost; B ~: lost; evs : ns_shared lost \
|
2015
|
464 |
\ |] ==> \
|
2032
|
465 |
\ K ~: analz (sees lost Spy evs)";
|
2015
|
466 |
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
|
|
467 |
by (fast_tac (!claset addSEs [lemma]) 1);
|
2032
|
468 |
qed "Spy_not_see_encrypted_key";
|
|
469 |
|
|
470 |
|
|
471 |
goal thy
|
|
472 |
"!!evs. [| C ~: {A,B,Server}; \
|
|
473 |
\ Says Server A \
|
|
474 |
\ (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
|
|
475 |
\ A ~: lost; B ~: lost; evs : ns_shared lost |] \
|
|
476 |
\ ==> K ~: analz (sees lost C evs)";
|
|
477 |
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
|
|
478 |
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
|
|
479 |
by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
|
|
480 |
by (REPEAT_FIRST (fast_tac (!claset addIs [ns_shared_mono RS subsetD])));
|
|
481 |
qed "Agent_not_see_encrypted_key";
|