src/HOL/Auth/NS_Public_Bad.ML
author paulson
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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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AddIffs [Spy_in_bad];
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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 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 possibility_tac;
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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 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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(*Induction for regularity theorems.  If induction formula has the form
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   X ~: analz (spies evs) --> ... then it shortens the proof by discarding
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   needless information about analz (insert X (spies evs))  *)
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fun parts_induct_tac i = 
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    etac ns_public.induct i
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    THEN 
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    REPEAT (FIRSTGOAL analz_mono_contra_tac)
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    THEN 
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    prove_simple_subgoals_tac i;
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(** Theorems of the form X ~: parts (spies 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 bad at start)*)
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goal thy 
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 "!!A. evs: ns_public ==> (Key (priK A) : parts (spies evs)) = (A : bad)";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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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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 "!!A. evs: ns_public ==> (Key (priK A) : analz (spies evs)) = (A : bad)";
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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 (spies evs);       \
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\                  evs : ns_public |] ==> A:bad";
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by (blast_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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(**** 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. [| Crypt (pubK B) {|Nonce NA, Agent A|} : parts (spies evs); \
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\           Nonce NA ~: analz (spies evs);   evs : ns_public |]       \
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\ ==> Crypt (pubK C) {|NA', Nonce NA|} ~: parts (spies evs)";
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by (etac rev_mp 1);
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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(*NS3*)
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by (blast_tac (!claset addSEs partsEs) 3);
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(*NS2*)
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by (blast_tac (!claset addSEs partsEs) 2);
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by (Fake_parts_insert_tac 1);
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qed "no_nonce_NS1_NS2";
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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. [| Nonce NA ~: analz (spies evs);  evs : ns_public |]      \
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\ ==> EX A' B'. ALL A B.                                               \
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\      Crypt (pubK B) {|Nonce NA, Agent A|} : parts (spies evs) --> \
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\      A=A' & B=B'";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (ALLGOALS
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    (asm_simp_tac (!simpset addsimps [all_conj_distrib, parts_insert_spies])));
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(*NS1*)
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by (expand_case_tac "NA = ?y" 2 THEN blast_tac (!claset addSEs partsEs) 2);
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(*Fake*)
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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 (Fake_parts_insert_tac 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(spies evs); \
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\           Crypt(pubK B') {|Nonce NA, Agent A'|} : parts(spies evs); \
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\           Nonce NA ~: analz (spies 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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(*Tactic for proving secrecy theorems*)
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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 setloop split_tac [expand_if]));
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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. [| Says A B (Crypt(pubK B) {|Nonce NA, Agent A|}) : set evs;     \
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\           A ~: bad;  B ~: bad;  evs : ns_public |]                    \
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\        ==>  Nonce NA ~: analz (spies evs)";
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by (etac rev_mp 1);
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by (analz_induct_tac 1);
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(*NS3*)
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by (blast_tac (!claset addDs  [Says_imp_spies RS parts.Inj]
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                       addEs  [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 4);
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(*NS2*)
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by (blast_tac (!claset addSEs [MPair_parts]
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		       addDs  [Says_imp_spies RS parts.Inj,
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			       parts.Body, unique_NA]) 3);
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(*NS1*)
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by (blast_tac (!claset addSEs spies_partsEs
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                       addIs  [impOfSubs analz_subset_parts]) 2);
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(*Fake*)
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by (spy_analz_tac 1);
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qed "Spy_not_see_NA";
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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. [| Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set evs;  \
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\           Says B' A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set evs;  \
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\           A ~: bad;  B ~: bad;  evs : ns_public |]                  \
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\        ==> Says B A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set evs";
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by (etac rev_mp 1);
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(*prepare induction over Crypt (pubK A) {|NA,NB|} : parts H*)
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by (etac (Says_imp_spies RS parts.Inj RS rev_mp) 1);
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by (etac ns_public.induct 1);
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by (ALLGOALS Asm_simp_tac);
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(*NS1*)
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by (blast_tac (!claset addSEs spies_partsEs) 2);
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(*Fake*)
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by (blast_tac (!claset addSDs [impOfSubs Fake_parts_insert]
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                       addDs  [Spy_not_see_NA, 
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			       impOfSubs analz_subset_parts]) 1);
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(*NS2; not clear why blast_tac needs to be preceeded by Step_tac*)
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by (Step_tac 1);
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by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj,
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			      Spy_not_see_NA, unique_NA]) 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. [| Crypt (pubK B) {|Nonce NA, Agent A|} : parts (spies evs); \
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\           Nonce NA ~: analz (spies evs);                            \
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\           evs : ns_public |]                                           \
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\   ==> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set evs";
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by (etac rev_mp 1);
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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qed "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. [| Nonce NB ~: analz (spies evs);  evs : ns_public |]  \
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\ ==> EX A' NA'. ALL A NA.                                         \
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\      Crypt (pubK A) {|Nonce NA, Nonce NB|}                       \
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\        : parts (spies evs)  -->  A=A' & NA=NA'";
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by (etac rev_mp 1);
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by (parts_induct_tac 1);
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by (ALLGOALS
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    (asm_simp_tac (!simpset addsimps [all_conj_distrib, parts_insert_spies])));
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(*NS2*)
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by (expand_case_tac "NB = ?y" 2 THEN blast_tac (!claset addSEs partsEs) 2);
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(*Fake*)
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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 (Fake_parts_insert_tac 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(spies evs); \
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\           Crypt(pubK A'){|Nonce NA', Nonce NB|} : parts(spies evs); \
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\           Nonce NB ~: analz (spies 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.[| Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs;  \
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\          ALL C. Says A C (Crypt (pubK C) (Nonce NB)) ~: set evs;      \
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\          A ~: bad;  B ~: bad;  evs : ns_public |]                     \
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\       ==> Nonce NB ~: analz (spies evs)";
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by (etac rev_mp 1);
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by (etac rev_mp 1);
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by (analz_induct_tac 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
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by clarify_tac;
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(*NS3: because NB determines A*)
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by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj, unique_NB]) 4);
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(*NS2: by freshness and unicity of NB*)
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by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj]
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                       addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]
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                       addEs partsEs
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		       addIs [impOfSubs analz_subset_parts]) 3);
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(*NS1: by freshness*)
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by (blast_tac (!claset addSEs spies_partsEs) 2);
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(*Fake*)
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by (spy_analz_tac 1);
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qed "Spy_not_see_NB";
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(*Authentication for B: if he receives message 3 and has used NB
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  in message 2, then A has sent message 3--to somebody....*)
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goal thy 
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 "!!evs. [| Says B A  (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs; \
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\           Says A' B (Crypt (pubK B) (Nonce NB)): set evs;              \
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\           A ~: bad;  B ~: bad;  evs : ns_public |]                   \
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\        ==> EX C. Says A C (Crypt (pubK C) (Nonce NB)) : set evs";
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by (etac rev_mp 1);
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(*prepare induction over Crypt (pubK B) NB : parts H*)
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by (etac (Says_imp_spies RS parts.Inj RS rev_mp) 1);
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by (parts_induct_tac 1);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
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by clarify_tac;
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(*NS3: because NB determines A*)
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by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj, 
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			      Spy_not_see_NB, unique_NB]) 3);
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(*NS1: by freshness*)
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by (blast_tac (!claset addSEs spies_partsEs) 2);
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(*Fake*)
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by (blast_tac (!claset addSDs [impOfSubs Fake_parts_insert]
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                       addDs  [Spy_not_see_NB, 
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			       impOfSubs analz_subset_parts]) 1);
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qed "B_trusts_NS3";
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(*Can we strengthen the secrecy theorem?  NO*)
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goal thy 
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 "!!evs. [| A ~: bad;  B ~: bad;  evs : ns_public |]           \
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\ ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs \
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\     --> Nonce NB ~: analz (spies evs)";
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by (analz_induct_tac 1);
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by clarify_tac;
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(*NS2: by freshness and unicity of NB*)
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by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj]
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                       addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]
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                       addEs partsEs
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		       addIs [impOfSubs analz_subset_parts]) 3);
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(*NS1: by freshness*)
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by (blast_tac (!claset addSEs spies_partsEs) 2);
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(*Fake*)
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by (spy_analz_tac 1);
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(*NS3: unicity of NB identifies A and NA, but not B*)
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by (forw_inst_tac [("A'","A")] (Says_imp_spies RS parts.Inj RS unique_NB) 1
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    THEN REPEAT (eresolve_tac [asm_rl, Says_imp_spies RS parts.Inj] 1));
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by (Auto_tac());
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by (rename_tac "C B' evs3" 1);
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(*
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THIS IS THE ATTACK!
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Level 8
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!!evs. [| A ~: bad; B ~: bad; evs : ns_public |]
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       ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs -->
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           Nonce NB ~: analz (spies evs)
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 1. !!C B' evs3.
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       [| A ~: bad; B ~: bad; evs3 : ns_public;
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          Says A C (Crypt (pubK C) {|Nonce NA, Agent A|}) : set evs3;
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          Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs3; C : bad;
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          Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set evs3;
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          Nonce NB ~: analz (spies evs3) |]
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       ==> False
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*)