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(* Title: HOL/Auth/Public_lemmas
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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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Theory of Public Keys (common to all symmetric-key protocols)
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Server keys; initial states of agents; new nonces and keys; function "sees"
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*)
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val inj_pubK = thm "inj_pubK";
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val priK_neq_pubK = thm "priK_neq_pubK";
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(*** Basic properties of pubK & priK ***)
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AddIffs [inj_pubK RS inj_eq];
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Goal "(priK A = priK B) = (A=B)";
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by Safe_tac;
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by (dres_inst_tac [("f","invKey")] arg_cong 1);
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by (Full_simp_tac 1);
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qed "priK_inj_eq";
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AddIffs [priK_inj_eq];
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AddIffs [priK_neq_pubK, priK_neq_pubK RS not_sym];
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Goalw [symKeys_def] "pubK A \\<notin> symKeys";
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by (Simp_tac 1);
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qed "not_symKeys_pubK";
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Goalw [symKeys_def] "priK A \\<notin> symKeys";
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by (Simp_tac 1);
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qed "not_symKeys_priK";
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AddIffs [not_symKeys_pubK, not_symKeys_priK];
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Goal "(K \\<in> symKeys) \\<noteq> (K' \\<in> symKeys) ==> K \\<noteq> K'";
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by (Blast_tac 1);
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qed "symKeys_neq_imp_neq";
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Goal "[| Crypt K X \\<in> analz H; K \\<in> symKeys; Key K \\<in> analz H |] \
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\ ==> X \\<in> analz H";
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by (auto_tac(claset(), simpset() addsimps [symKeys_def]));
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qed "analz_symKeys_Decrypt";
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(** "Image" equations that hold for injective functions **)
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Goal "(invKey x : invKey`A) = (x:A)";
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by Auto_tac;
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qed "invKey_image_eq";
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(*holds because invKey is injective*)
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Goal "(pubK x : pubK`A) = (x:A)";
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by Auto_tac;
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qed "pubK_image_eq";
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Goal "(priK x ~: pubK`A)";
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by Auto_tac;
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qed "priK_pubK_image_eq";
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Addsimps [invKey_image_eq, pubK_image_eq, priK_pubK_image_eq];
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(** Rewrites should not refer to initState(Friend i)
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-- not in normal form! **)
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Goalw [keysFor_def] "keysFor (parts (initState C)) = {}";
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by (induct_tac "C" 1);
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by (auto_tac (claset() addIs [range_eqI], simpset()));
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qed "keysFor_parts_initState";
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Addsimps [keysFor_parts_initState];
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(*** Function "spies" ***)
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(*Agents see their own private keys!*)
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Goal "Key (priK A) : initState A";
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by (induct_tac "A" 1);
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by Auto_tac;
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qed "priK_in_initState";
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AddIffs [priK_in_initState];
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(*All public keys are visible*)
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Goal "Key (pubK A) : spies evs";
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by (induct_tac "evs" 1);
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by (ALLGOALS (asm_simp_tac
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(simpset() addsimps [imageI, knows_Cons]
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addsplits [expand_event_case])));
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qed_spec_mp "spies_pubK";
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(*Spy sees private keys of bad agents!*)
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Goal "A: bad ==> Key (priK A) : spies evs";
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by (induct_tac "evs" 1);
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by (ALLGOALS (asm_simp_tac
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(simpset() addsimps [imageI, knows_Cons]
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addsplits [expand_event_case])));
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qed "Spy_spies_bad";
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AddIffs [spies_pubK, spies_pubK RS analz.Inj];
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AddSIs [Spy_spies_bad];
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(*** Fresh nonces ***)
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Goal "Nonce N ~: parts (initState B)";
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by (induct_tac "B" 1);
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by Auto_tac;
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qed "Nonce_notin_initState";
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AddIffs [Nonce_notin_initState];
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Goal "Nonce N ~: used []";
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by (simp_tac (simpset() addsimps [used_Nil]) 1);
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qed "Nonce_notin_used_empty";
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Addsimps [Nonce_notin_used_empty];
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(*** Supply fresh nonces for possibility theorems. ***)
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(*In any trace, there is an upper bound N on the greatest nonce in use.*)
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Goal "EX N. ALL n. N<=n --> Nonce n ~: used evs";
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by (induct_tac "evs" 1);
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by (res_inst_tac [("x","0")] exI 1);
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by (ALLGOALS (asm_simp_tac
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(simpset() addsimps [used_Cons]
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addsplits [expand_event_case])));
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by Safe_tac;
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by (ALLGOALS (rtac (msg_Nonce_supply RS exE)));
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by (ALLGOALS (blast_tac (claset() addSEs [add_leE])));
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val lemma = result();
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Goal "EX N. Nonce N ~: used evs";
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by (rtac (lemma RS exE) 1);
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by (Blast_tac 1);
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qed "Nonce_supply1";
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Goal "Nonce (@ N. Nonce N ~: used evs) ~: used evs";
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by (rtac (lemma RS exE) 1);
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by (rtac someI 1);
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by (Fast_tac 1);
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qed "Nonce_supply";
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(*Tactic for possibility theorems*)
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fun possibility_tac st = st |>
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REPEAT (*omit used_Says so that Nonces start from different traces!*)
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(ALLGOALS (simp_tac (simpset() delsimps [used_Says]))
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THEN
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REPEAT_FIRST (eq_assume_tac ORELSE'
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resolve_tac [refl, conjI, Nonce_supply]));
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(*** Specialized rewriting for the analz_image_... theorems ***)
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Goal "insert (Key K) H = Key ` {K} Un H";
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by (Blast_tac 1);
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qed "insert_Key_singleton";
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Goal "insert (Key K) (Key`KK Un C) = Key ` (insert K KK) Un C";
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by (Blast_tac 1);
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qed "insert_Key_image";
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(*Reverse the normal simplification of "image" to build up (not break down)
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the set of keys. Based on analz_image_freshK_ss, but simpler.*)
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val analz_image_keys_ss =
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simpset() delsimps [image_insert, image_Un]
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delsimps [thm "imp_disjL"] (*reduces blow-up*)
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addsimps [image_insert RS sym, image_Un RS sym,
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rangeI,
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insert_Key_singleton,
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insert_Key_image, thm "Un_assoc" RS sym];
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