src/HOL/Auth/NS_Public_Bad.ML
author paulson
Tue Jan 07 10:18:20 1997 +0100 (1997-01-07)
changeset 2480 f9be937df511
parent 2451 ce85a2aafc7a
child 2497 47de509bdd55
permissions -rw-r--r--
Tidied up the unicity proofs
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(*  Title:      HOL/Auth/NS_Public_Bad
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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_public" for the Needham-Schroeder Public-Key protocol.
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Flawed version, vulnerable to Lowe's attack.
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From page 260 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_Public_Bad;
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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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AddIffs [Spy_in_lost];
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(*Replacing the variable by a constant improves search speed by 50%!*)
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val Says_imp_sees_Spy' = 
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    read_instantiate_sg (sign_of thy) [("lost","lost")] Says_imp_sees_Spy;
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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 ==> EX NB. EX evs: ns_public.               \
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\                     Says A B (Crypt (pubK B) (Nonce NB)) : set_of_list evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (ns_public.Nil RS ns_public.NS1 RS ns_public.NS2 RS ns_public.NS3) 2);
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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 (REPEAT_FIRST (fast_tac (!claset addss (!simpset setsolver safe_solver))));
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result();
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(**** Inductive proofs about ns_public ****)
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(*Nobody sends themselves messages*)
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goal thy "!!evs. evs : ns_public ==> ALL A X. Says A A X ~: set_of_list evs";
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by (etac ns_public.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 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 ns_public.induct 1 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 private key! (unless it's lost at start)*)
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goal thy 
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 "!!evs. evs : ns_public \
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\        ==> (Key (priK 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_priK";
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Addsimps [Spy_see_priK];
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goal thy 
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 "!!evs. evs : ns_public \
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\        ==> (Key (priK 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_priK";
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Addsimps [Spy_analz_priK];
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goal thy  "!!A. [| Key (priK A) : parts (sees lost Spy evs);       \
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\                  evs : ns_public |] ==> A:lost";
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by (fast_tac (!claset addDs [Spy_see_priK]) 1);
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qed "Spy_see_priK_D";
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bind_thm ("Spy_analz_priK_D", analz_subset_parts RS subsetD RS Spy_see_priK_D);
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AddSDs [Spy_see_priK_D, Spy_analz_priK_D];
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(*** Future nonces can't be seen or used! ***)
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goal thy "!!i. evs : ns_public ==>        \
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\             length evs <= i --> Nonce (newN i) ~: parts (sees lost Spy evs)";
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by (parts_induct_tac 1);
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by (REPEAT_FIRST (fast_tac (!claset 
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                              addSEs partsEs
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                              addSDs  [Says_imp_sees_Spy' RS parts.Inj]
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                              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_nonces_not_seen";
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Addsimps [new_nonces_not_seen];
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val nonce_not_seen_now = le_refl RSN (2, new_nonces_not_seen) RSN (2,rev_notE);
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fun analz_induct_tac i = 
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    etac ns_public.induct i	THEN
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    ALLGOALS (asm_simp_tac 
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	      (!simpset addsimps [not_parts_not_analz]
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                        setloop split_tac [expand_if]));
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(**** Authenticity properties obtained from NS2 ****)
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(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
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  is secret.  (Honest users generate fresh nonces.)*)
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goal thy 
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 "!!evs. evs : ns_public                       \
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\ ==> Nonce NA ~: analz (sees lost Spy evs) -->           \
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\     Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
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\     Crypt (pubK C) {|NA', Nonce NA|} ~: parts (sees lost Spy evs)";
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by (analz_induct_tac 1);
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(*NS3*)
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by (fast_tac (!claset addSEs partsEs
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                      addEs  [nonce_not_seen_now]) 4);
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(*NS2*)
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by (fast_tac (!claset addSEs partsEs
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                      addEs  [nonce_not_seen_now]) 3);
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(*Fake*)
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by (best_tac (!claset addIs [analz_insertI]
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		      addDs [impOfSubs analz_subset_parts,
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			     impOfSubs Fake_parts_insert]
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	              addss (!simpset)) 2);
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(*Base*)
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by (fast_tac (!claset addss (!simpset)) 1);
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bind_thm ("no_nonce_NS1_NS2", result() RSN (2, rev_mp) RSN (2, rev_mp));
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(*Unicity for NS1: nonce NA identifies agents A and B*)
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goal thy 
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 "!!evs. evs : ns_public                                                    \
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\ ==> Nonce NA ~: analz (sees lost Spy evs) -->                             \
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\     (EX A' B'. ALL A B.                                                   \
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\      Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
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\      A=A' & B=B')";
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by (analz_induct_tac 1);
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(*NS1*)
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by (expand_case_tac "NA = ?y" 3 THEN
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    REPEAT (fast_tac (!claset addSEs (nonce_not_seen_now::partsEs)) 3));
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(*Base*)
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by (fast_tac (!claset addss (!simpset)) 1);
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(*Fake*)
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by (simp_tac (!simpset addsimps [all_conj_distrib,parts_insert_sees]) 1);
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by (step_tac (!claset addSIs [analz_insertI]) 1);
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by (ex_strip_tac 1);
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by (best_tac (!claset delrules [conjI]
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		      addSDs [impOfSubs Fake_parts_insert]
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	              addDs  [impOfSubs analz_subset_parts]
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                      addss (!simpset)) 1);
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val lemma = result();
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goal thy 
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 "!!evs. [| Crypt(pubK B)  {|Nonce NA, Agent A|}  : parts(sees lost Spy evs); \
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\           Crypt(pubK B') {|Nonce NA, Agent A'|} : parts(sees lost Spy evs); \
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\           Nonce NA ~: analz (sees lost Spy evs);                            \
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\           evs : ns_public |]                                                \
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\        ==> A=A' & B=B'";
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by (prove_unique_tac lemma 1);
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qed "unique_NA";
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(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure*)
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goal thy 
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 "!!evs. [| A ~: lost;  B ~: lost;  evs : ns_public |]   \
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\ ==> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs \
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\     --> Nonce NA ~: analz (sees lost Spy evs)";
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by (analz_induct_tac 1);
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(*NS3*)
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by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
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                      addEs  [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 4);
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(*NS1*)
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by (fast_tac (!claset addSEs partsEs
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                      addSDs [new_nonces_not_seen, 
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			      Says_imp_sees_Spy' RS parts.Inj]) 2);
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(*Fake*)
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by (REPEAT_FIRST spy_analz_tac);
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(*NS2*)
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by (Step_tac 1);
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by (fast_tac (!claset addSEs partsEs
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                      addSDs [new_nonces_not_seen, 
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			      Says_imp_sees_Spy' RS parts.Inj]) 2);
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by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
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                      addDs  [unique_NA]) 1);
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bind_thm ("Spy_not_see_NA", result() RSN (2, rev_mp));
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(*Authentication for A: if she receives message 2 and has used NA
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  to start a run, then B has sent message 2.*)
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goal thy 
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 "!!evs. [| A ~: lost;  B ~: lost;  evs : ns_public |]       \
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\ ==> Crypt (pubK A) {|Nonce NA, Nonce NB|} : parts (sees lost Spy evs) \
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\     --> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs \
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\     --> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs";
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by (analz_induct_tac 1);
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by (ALLGOALS Asm_simp_tac);
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(*NS1*)
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by (fast_tac (!claset addSEs partsEs
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                      addSDs [new_nonces_not_seen, 
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			      Says_imp_sees_Spy' RS parts.Inj]) 2);
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(*Fake*)
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by (REPEAT_FIRST (resolve_tac [impI, conjI]));
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by (fast_tac (!claset addss (!simpset)) 1);
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by (forward_tac [Spy_not_see_NA] 1 THEN REPEAT (assume_tac 1));
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by (best_tac (!claset addSIs [disjI2]
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		      addSDs [impOfSubs Fake_parts_insert]
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		      addDs  [impOfSubs analz_subset_parts]
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	              addss (!simpset)) 1);
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(*NS2*)
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by (Step_tac 1);
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by (forward_tac [Spy_not_see_NA] 1 THEN REPEAT (assume_tac 1));
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by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
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                      addDs  [unique_NA]) 1);
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qed_spec_mp "NA_decrypt_imp_B_msg";
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(*Corollary: if A receives B's message NS2 and the nonce NA agrees
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  then that message really originated with B.*)
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goal thy 
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 "!!evs. [| Says B' A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set_of_list evs;\
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\           Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs;\
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\           A ~: lost;  B ~: lost;  evs : ns_public |]  \
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\        ==> Says B A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set_of_list evs";
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by (fast_tac (!claset addSIs [NA_decrypt_imp_B_msg]
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                      addEs  partsEs
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                      addDs  [Says_imp_sees_Spy' RS parts.Inj]) 1);
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qed "A_trusts_NS2";
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(*If the encrypted message appears then it originated with Alice in NS1*)
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goal thy 
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 "!!evs. evs : ns_public                   \
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\    ==> Nonce NA ~: analz (sees lost Spy evs) --> \
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\        Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
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\        Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs";
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by (analz_induct_tac 1);
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(*Fake*)
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by (best_tac (!claset addSIs [disjI2]
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		      addSDs [impOfSubs Fake_parts_insert]
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		      addIs  [analz_insertI]
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		      addDs  [impOfSubs analz_subset_parts]
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	              addss (!simpset)) 2);
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(*Base*)
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by (fast_tac (!claset addss (!simpset)) 1);
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qed_spec_mp "B_trusts_NS1";
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(**** Authenticity properties obtained from NS2 ****)
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(*Unicity for NS2: nonce NB identifies agent A and nonce NA
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  [proof closely follows that for unique_NA] *)
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goal thy 
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 "!!evs. evs : ns_public                                                    \
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\ ==> Nonce NB ~: analz (sees lost Spy evs) -->                             \
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\     (EX A' NA'. ALL A NA.                                                 \
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\      Crypt (pubK A) {|Nonce NA, Nonce NB|} : parts (sees lost Spy evs) --> \
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\      A=A' & NA=NA')";
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by (analz_induct_tac 1);
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(*NS2*)
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by (expand_case_tac "NB = ?y" 3 THEN
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    REPEAT (fast_tac (!claset addSEs (nonce_not_seen_now::partsEs)) 3));
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(*Base*)
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by (fast_tac (!claset addss (!simpset)) 1);
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(*Fake*)
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by (simp_tac (!simpset addsimps [all_conj_distrib,parts_insert_sees]) 1);
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by (step_tac (!claset addSIs [analz_insertI]) 1);
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by (ex_strip_tac 1);
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by (best_tac (!claset delrules [conjI]
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	              addSDs [impOfSubs Fake_parts_insert]
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	              addDs  [impOfSubs analz_subset_parts] 
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	              addss (!simpset)) 1);
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val lemma = result();
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goal thy 
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 "!!evs. [| Crypt(pubK A) {|Nonce NA, Nonce NB|}  : parts(sees lost Spy evs); \
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\           Crypt(pubK A'){|Nonce NA', Nonce NB|} : parts(sees lost Spy evs); \
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\           Nonce NB ~: analz (sees lost Spy evs);                            \
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\           evs : ns_public |]                                                \
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\        ==> A=A' & NA=NA'";
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by (prove_unique_tac lemma 1);
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qed "unique_NB";
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(*NB remains secret PROVIDED Alice never responds with round 3*)
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goal thy 
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 "!!evs. [| A ~: lost;  B ~: lost;  evs : ns_public |]   \
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\ ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs --> \
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\     (ALL C. Says A C (Crypt (pubK C) (Nonce NB)) ~: set_of_list evs) --> \
paulson@2318
   298
\     Nonce NB ~: analz (sees lost Spy evs)";
paulson@2418
   299
by (analz_induct_tac 1);
paulson@2318
   300
(*NS1*)
paulson@2318
   301
by (fast_tac (!claset addSEs partsEs
paulson@2318
   302
                      addSDs [new_nonces_not_seen, 
paulson@2318
   303
			      Says_imp_sees_Spy' RS parts.Inj]) 2);
paulson@2318
   304
(*Fake*)
paulson@2374
   305
by (REPEAT_FIRST spy_analz_tac);
paulson@2318
   306
(*NS2 and NS3*)
paulson@2318
   307
by (Step_tac 1);
paulson@2318
   308
(*NS2*)
paulson@2318
   309
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
paulson@2318
   310
                      addEs  [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 3);
paulson@2318
   311
by (fast_tac (!claset addSEs partsEs
paulson@2318
   312
                      addSDs [new_nonces_not_seen, 
paulson@2318
   313
                              Says_imp_sees_Spy' RS parts.Inj]) 2);
paulson@2318
   314
by (Fast_tac 1);
paulson@2318
   315
(*NS3*)
paulson@2318
   316
by (Fast_tac 2);
paulson@2318
   317
by (fast_tac (!claset addSEs partsEs
paulson@2318
   318
                      addSDs [Says_imp_sees_Spy' RS parts.Inj,
paulson@2318
   319
			      new_nonces_not_seen, 
paulson@2318
   320
                              impOfSubs analz_subset_parts]) 1);
paulson@2318
   321
paulson@2318
   322
by (forw_inst_tac [("A'","A")] (Says_imp_sees_Spy' RS parts.Inj RS unique_NB) 1
paulson@2318
   323
    THEN REPEAT (eresolve_tac [asm_rl, Says_imp_sees_Spy' RS parts.Inj] 1));
paulson@2318
   324
by (Fast_tac 1);
paulson@2318
   325
bind_thm ("Spy_not_see_NB", result() RSN (2, rev_mp) RS mp);
paulson@2318
   326
paulson@2318
   327
paulson@2318
   328
paulson@2318
   329
(*Authentication for B: if he receives message 3 and has used NB
paulson@2318
   330
  in message 2, then A has sent message 3.*)
paulson@2318
   331
goal thy 
paulson@2318
   332
 "!!evs. [| A ~: lost;  B ~: lost;  evs : ns_public |]       \
paulson@2318
   333
\ ==> Crypt (pubK B) (Nonce NB) : parts (sees lost Spy evs) \
paulson@2318
   334
\     --> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs \
paulson@2318
   335
\     --> (EX C. Says A C (Crypt (pubK C) (Nonce NB)) : set_of_list evs)";
paulson@2418
   336
by (analz_induct_tac 1);
paulson@2318
   337
by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
paulson@2318
   338
by (ALLGOALS Asm_simp_tac);
paulson@2318
   339
(*NS1*)
paulson@2318
   340
by (fast_tac (!claset addSEs partsEs
paulson@2318
   341
                      addSDs [new_nonces_not_seen, 
paulson@2318
   342
			      Says_imp_sees_Spy' RS parts.Inj]) 2);
paulson@2318
   343
(*Fake*)
paulson@2318
   344
by (REPEAT_FIRST (resolve_tac [impI, conjI]));
paulson@2318
   345
by (fast_tac (!claset addss (!simpset)) 1);
paulson@2318
   346
br (ccontr RS disjI2) 1;
paulson@2318
   347
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1));
paulson@2318
   348
by (Fast_tac 1);
paulson@2318
   349
by (best_tac (!claset addSDs [impOfSubs Fake_parts_insert]
paulson@2318
   350
	              addDs  [impOfSubs analz_subset_parts] 
paulson@2318
   351
	              addss (!simpset)) 1);
paulson@2318
   352
(*NS3*)
paulson@2318
   353
by (Step_tac 1);
paulson@2318
   354
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT1 (assume_tac 1));
paulson@2318
   355
by (Fast_tac 1);
paulson@2318
   356
by (best_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
paulson@2318
   357
	              addDs  [unique_NB]) 1);
paulson@2318
   358
qed_spec_mp "NB_decrypt_imp_A_msg";
paulson@2318
   359
paulson@2318
   360
paulson@2318
   361
(*Corollary: if B receives message NS3 and the nonce NB agrees
paulson@2318
   362
  then A sent NB to somebody....*)
paulson@2318
   363
goal thy 
paulson@2318
   364
 "!!evs. [| Says A' B (Crypt (pubK B) (Nonce NB)): set_of_list evs;    \
paulson@2318
   365
\           Says B A  (Crypt (pubK A) {|Nonce NA, Nonce NB|})          \
paulson@2318
   366
\             : set_of_list evs;                                       \
paulson@2318
   367
\           A ~: lost;  B ~: lost;  evs : ns_public |]                 \
paulson@2318
   368
\        ==> EX C. Says A C (Crypt (pubK C) (Nonce NB)) : set_of_list evs";
paulson@2318
   369
by (fast_tac (!claset addSIs [NB_decrypt_imp_A_msg]
paulson@2318
   370
                      addEs  partsEs
paulson@2318
   371
                      addDs  [Says_imp_sees_Spy' RS parts.Inj]) 1);
paulson@2318
   372
qed "B_trusts_NS3";
paulson@2318
   373
paulson@2318
   374
paulson@2318
   375
(*Can we strengthen the secrecy theorem?  NO*)
paulson@2318
   376
goal thy 
paulson@2318
   377
 "!!evs. [| A ~: lost;  B ~: lost;  evs : ns_public |]   \
paulson@2318
   378
\ ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs \
paulson@2318
   379
\     --> Nonce NB ~: analz (sees lost Spy evs)";
paulson@2418
   380
by (analz_induct_tac 1);
paulson@2318
   381
(*NS1*)
paulson@2318
   382
by (fast_tac (!claset addSEs partsEs
paulson@2318
   383
                      addSDs [new_nonces_not_seen, 
paulson@2318
   384
			      Says_imp_sees_Spy' RS parts.Inj]) 2);
paulson@2318
   385
(*Fake*)
paulson@2374
   386
by (REPEAT_FIRST spy_analz_tac);
paulson@2318
   387
(*NS2 and NS3*)
paulson@2318
   388
by (Step_tac 1);
paulson@2318
   389
(*NS2*)
paulson@2318
   390
by (fast_tac (!claset addSEs partsEs
paulson@2318
   391
                      addSDs [new_nonces_not_seen, 
paulson@2318
   392
                              Says_imp_sees_Spy' RS parts.Inj]) 2);
paulson@2318
   393
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
paulson@2318
   394
                      addEs  [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 1);
paulson@2318
   395
(*NS3*)
paulson@2318
   396
by (forw_inst_tac [("A'","A")] (Says_imp_sees_Spy' RS parts.Inj RS unique_NB) 1
paulson@2318
   397
    THEN REPEAT (eresolve_tac [asm_rl, Says_imp_sees_Spy' RS parts.Inj] 1));
paulson@2318
   398
by (Step_tac 1);
paulson@2318
   399
paulson@2318
   400
(*
paulson@2318
   401
THIS IS THE ATTACK!
paulson@2318
   402
Level 9
paulson@2318
   403
!!evs. [| A ~: lost; B ~: lost; evs : ns_public |]
paulson@2318
   404
       ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|})
paulson@2318
   405
           : set_of_list evs -->
paulson@2318
   406
           Nonce NB ~: analz (sees lost Spy evs)
paulson@2318
   407
 1. !!evs Aa Ba B' NAa NBa evsa.
paulson@2318
   408
       [| A ~: lost; B ~: lost; evsa : ns_public; A ~= Ba;
paulson@2318
   409
          Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evsa;
paulson@2318
   410
          Says A Ba (Crypt (pubK Ba) {|Nonce NA, Agent A|}) : set_of_list evsa;
paulson@2318
   411
          Ba : lost;
paulson@2318
   412
          Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evsa;
paulson@2318
   413
          Nonce NB ~: analz (sees lost Spy evsa) |]
paulson@2318
   414
       ==> False
paulson@2318
   415
*)
paulson@2318
   416
paulson@2318
   417
paulson@2318
   418