src/HOL/Auth/NS_Public_Bad.ML
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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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AddEs spies_partsEs;
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AddDs [impOfSubs analz_subset_parts];
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AddDs [impOfSubs Fake_parts_insert];
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AddIffs [Spy_in_bad];
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Addsimps [image_eq_UN];  (*accelerates proofs involving nested images*)
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(*A "possibility property": there are traces that reach the end*)
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Goal
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  "EX NB. EX evs: ns_public. 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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(*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 "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 (Blast_tac 1);
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qed "Spy_see_priK";
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Addsimps [Spy_see_priK];
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Goal "evs: ns_public ==> (Key (priK A) : analz (spies evs)) = (A : bad)";
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by Auto_tac;
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qed "Spy_analz_priK";
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Addsimps [Spy_analz_priK];
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AddSDs [Spy_see_priK RSN (2, rev_iffD1), 
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	Spy_analz_priK RSN (2, rev_iffD1)];
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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 "[| 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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by (ALLGOALS Blast_tac);
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qed "no_nonce_NS1_NS2";
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(*Adding it to the claset slows down proofs...*)
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val nonce_NS1_NS2_E = no_nonce_NS1_NS2 RSN (2, rev_notE);
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(*Unicity for NS1: nonce NA identifies agents A and B*)
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Goal "[| 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 (asm_simp_tac (simpset() addsimps [all_conj_distrib])));
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(*NS1*)
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by (expand_case_tac "NA = ?y" 2 THEN Blast_tac 2);
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(*Fake*)
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by (Clarify_tac 1);
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by (Blast_tac 1);
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val lemma = result();
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Goal "[| 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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(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure*)
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Goal "[| 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() addEs [nonce_NS1_NS2_E]) 4);
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(*NS2*)
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by (blast_tac (claset() addDs [unique_NA]) 3);
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(*NS1*)
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by (Blast_tac 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 "[| 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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by (ALLGOALS Clarify_tac);
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(*NS2*)
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by (blast_tac (claset() addDs [Spy_not_see_NA, unique_NA]) 3);
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(*NS1*)
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by (Blast_tac 2);
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(*Fake*)
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by (blast_tac (claset() addDs [Spy_not_see_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 "[| 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 (Blast_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 nonce NA and agent A
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  [proof closely follows that for unique_NA] *)
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Goal "[| 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|} : parts (spies evs)          \
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\      -->  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 (asm_simp_tac (simpset() addsimps [all_conj_distrib])));
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(*NS2*)
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by (expand_case_tac "NB = ?y" 2 THEN Blast_tac 2);
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(*Fake*)
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by (Blast_tac 1);
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val lemma = result();
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Goal "[| 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 "[| 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 (ALLGOALS Clarify_tac);
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(*NS3: because NB determines A*)
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by (blast_tac (claset() addDs [unique_NB]) 4);
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(*NS2: by freshness and unicity of NB*)
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by (blast_tac (claset() addEs [nonce_NS1_NS2_E]) 3);
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(*NS1: by freshness*)
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by (Blast_tac 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 "[| 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 (ALLGOALS Clarify_tac);
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(*NS3: because NB determines A (this use of unique_NB is more robust) *)
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by (blast_tac (claset() addDs [Spy_not_see_NB]
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			addIs [unique_NB RS conjunct1]) 3);
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(*NS1: by freshness*)
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by (Blast_tac 2);
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(*Fake*)
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by (blast_tac (claset() addDs [Spy_not_see_NB]) 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 "[| 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 (ALLGOALS Clarify_tac);
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   237
(*NS2: by freshness and unicity of NB*)
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by (blast_tac (claset() addEs [nonce_NS1_NS2_E]) 3);
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   239
(*NS1: by freshness*)
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by (Blast_tac 2);
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   241
(*Fake*)
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by (spy_analz_tac 1);
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   243
(*NS3: unicity of NB identifies A and NA, but not B*)
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   244
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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   251
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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*)