author  paulson 
Tue, 16 Dec 1997 15:17:26 +0100  
changeset 4422  21238c9d363e 
parent 4091  771b1f6422a8 
child 4449  df30e75f670f 
permissions  rwrr 
2002  1 
(* Title: HOL/Auth/OtwayRees_Bad 
2 
ID: $Id$ 

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 
Copyright 1996 University of Cambridge 

5 

6 
Inductive relation "otway" for the OtwayRees protocol. 

7 

8 
The FAULTY version omitting encryption of Nonce NB, as suggested on page 247 of 

9 
Burrows, Abadi and Needham. A Logic of Authentication. 

10 
Proc. Royal Soc. 426 (1989) 

11 

12 
This file illustrates the consequences of such errors. We can still prove 

2032  13 
impressivelooking properties such as Spy_not_see_encrypted_key, yet the 
2002  14 
protocol is open to a middleperson attack. Attempting to prove some key lemmas 
15 
indicates the possibility of this attack. 

16 
*) 

17 

18 
open OtwayRees_Bad; 

19 

20 
proof_timing:=true; 

21 
HOL_quantifiers := false; 

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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; A ~= Server; B ~= Server ] \ 

27 
\ ==> EX K. EX NA. EX evs: otway. \ 

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\ Says B A {Nonce NA, Crypt (shrK A) {Nonce NA, Key K}} \ 
3465  29 
\ : set evs"; 
2002  30 
by (REPEAT (resolve_tac [exI,bexI] 1)); 
2032  31 
by (rtac (otway.Nil RS otway.OR1 RS otway.OR2 RS otway.OR3 RS otway.OR4) 2); 
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by possibility_tac; 
2002  33 
result(); 
34 

35 

36 
(**** Inductive proofs about otway ****) 

37 

38 
(*Nobody sends themselves messages*) 

3465  39 
goal thy "!!evs. evs : otway ==> ALL A X. Says A A X ~: set evs"; 
2032  40 
by (etac otway.induct 1); 
2002  41 
by (Auto_tac()); 
42 
qed_spec_mp "not_Says_to_self"; 

43 
Addsimps [not_Says_to_self]; 

44 
AddSEs [not_Says_to_self RSN (2, rev_notE)]; 

45 

46 

47 
(** For reasoning about the encrypted portion of messages **) 

48 

3465  49 
goal thy "!!evs. Says A' B {N, Agent A, Agent B, X} : set evs ==> \ 
3683  50 
\ X : analz (spies evs)"; 
4091  51 
by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1); 
3683  52 
qed "OR2_analz_spies"; 
2002  53 

3465  54 
goal thy "!!evs. Says S' B {N, X, Crypt (shrK B) X'} : set evs ==> \ 
3683  55 
\ X : analz (spies evs)"; 
4091  56 
by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1); 
3683  57 
qed "OR4_analz_spies"; 
2002  58 

3465  59 
goal thy "!!evs. Says Server B {NA, X, Crypt K' {NB,K}} : set evs \ 
3683  60 
\ ==> K : parts (spies evs)"; 
4091  61 
by (blast_tac (claset() addSEs spies_partsEs) 1); 
3683  62 
qed "Oops_parts_spies"; 
2002  63 

64 
(*OR2_analz... and OR4_analz... let us treat those cases using the same 

65 
argument as for the Fake case. This is possible for most, but not all, 

66 
proofs: Fake does not invent new nonces (as in OR2), and of course Fake 

2032  67 
messages originate from the Spy. *) 
2002  68 

3683  69 
bind_thm ("OR2_parts_spies", 
70 
OR2_analz_spies RS (impOfSubs analz_subset_parts)); 

71 
bind_thm ("OR4_parts_spies", 

72 
OR4_analz_spies RS (impOfSubs analz_subset_parts)); 

2052  73 

3683  74 
(*For proving the easier theorems about X ~: parts (spies evs).*) 
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fun parts_induct_tac i = 
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etac otway.induct i THEN 
3683  77 
forward_tac [Oops_parts_spies] (i+6) THEN 
78 
forward_tac [OR4_parts_spies] (i+5) THEN 

79 
forward_tac [OR2_parts_spies] (i+3) THEN 

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prove_simple_subgoals_tac i; 
2002  81 

82 

3683  83 
(** Theorems of the form X ~: parts (spies evs) imply that NOBODY 
2002  84 
sends messages containing X! **) 
85 

3683  86 
(*Spy never sees another agent's shared key! (unless it's bad at start)*) 
2002  87 
goal thy 
3683  88 
"!!evs. evs : otway ==> (Key (shrK 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); 
3961  91 
by (ALLGOALS Blast_tac); 
2131  92 
qed "Spy_see_shrK"; 
93 
Addsimps [Spy_see_shrK]; 

2002  94 

2131  95 
goal thy 
3683  96 
"!!evs. evs : otway ==> (Key (shrK A) : analz (spies evs)) = (A : bad)"; 
4091  97 
by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset())); 
2131  98 
qed "Spy_analz_shrK"; 
99 
Addsimps [Spy_analz_shrK]; 

2002  100 

3683  101 
goal thy "!!A. [ Key (shrK A) : parts (spies evs); evs : otway ] ==> A:bad"; 
4091  102 
by (blast_tac (claset() addDs [Spy_see_shrK]) 1); 
2131  103 
qed "Spy_see_shrK_D"; 
2002  104 

2131  105 
bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D); 
106 
AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D]; 

2002  107 

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(*Nobody can have used nonexistent keys!*) 
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goal thy "!!evs. evs : otway ==> \ 
3683  111 
\ Key K ~: used evs > K ~: keysFor (parts (spies evs))"; 
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by (parts_induct_tac 1); 
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(*Fake*) 
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by (best_tac 
4091  115 
(claset() addSDs [impOfSubs (parts_insert_subset_Un RS keysFor_mono)] 
3961  116 
addIs [impOfSubs analz_subset_parts] 
117 
addDs [impOfSubs (analz_subset_parts RS keysFor_mono)] 

4091  118 
addss (simpset())) 1); 
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(*OR13*) 
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by (ALLGOALS Blast_tac); 
2160  121 
qed_spec_mp "new_keys_not_used"; 
2002  122 

123 
bind_thm ("new_keys_not_analzd", 

2032  124 
[analz_subset_parts RS keysFor_mono, 
125 
new_keys_not_used] MRS contra_subsetD); 

2002  126 

127 
Addsimps [new_keys_not_used, new_keys_not_analzd]; 

128 

129 

2131  130 

131 
(*** Proofs involving analz ***) 

132 

133 
(*Describes the form of K and NA when the Server sends this message. Also 

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for Oops case.*) 

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goal thy 

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"!!evs. [ Says Server B \ 
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\ {NA, X, Crypt (shrK B) {NB, Key K}} : set evs; \ 
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\ evs : otway ] \ 
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\ ==> K ~: range shrK & (EX i. NA = Nonce i) & (EX j. NB = Nonce j)"; 
2131  140 
by (etac rev_mp 1); 
141 
by (etac otway.induct 1); 

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by (prove_simple_subgoals_tac 1); 
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by (Blast_tac 1); 
2131  144 
qed "Says_Server_message_form"; 
145 

146 

147 
(*For proofs involving analz.*) 

3683  148 
val analz_spies_tac = 
149 
dtac OR2_analz_spies 4 THEN 

150 
dtac OR4_analz_spies 6 THEN 

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forward_tac [Says_Server_message_form] 7 THEN assume_tac 7 THEN 
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REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 7); 
2002  153 

154 

155 
(**** 

156 
The following is to prove theorems of the form 

157 

3683  158 
Key K : analz (insert (Key KAB) (spies evs)) ==> 
159 
Key K : analz (spies evs) 

2002  160 

161 
A more general formula must be proved inductively. 

162 
****) 

163 

164 

165 
(** Session keys are not used to encrypt other session keys **) 

166 

167 
(*The equality makes the induction hypothesis easier to apply*) 

168 
goal thy 

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"!!evs. evs : otway ==> \ 
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\ ALL K KK. KK <= Compl (range shrK) > \ 
3683  171 
\ (Key K : analz (Key``KK Un (spies evs))) = \ 
172 
\ (K : KK  Key K : analz (spies evs))"; 

2032  173 
by (etac otway.induct 1); 
3683  174 
by analz_spies_tac; 
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by (REPEAT_FIRST (resolve_tac [allI, impI])); 
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by (REPEAT_FIRST (rtac analz_image_freshK_lemma )); 
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by (ALLGOALS (asm_simp_tac analz_image_freshK_ss)); 
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(*Fake*) 
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by (spy_analz_tac 1); 
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qed_spec_mp "analz_image_freshK"; 
2002  181 

182 

183 
goal thy 

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"!!evs. [ evs : otway; KAB ~: range shrK ] ==> \ 
3683  185 
\ Key K : analz (insert (Key KAB) (spies evs)) = \ 
186 
\ (K = KAB  Key K : analz (spies evs))"; 

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by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1); 
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qed "analz_insert_freshK"; 
2002  189 

190 

2131  191 
(*** The Key K uniquely identifies the Server's message. **) 
2002  192 

193 
goal thy 

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"!!evs. evs : otway ==> \ 
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\ EX B' NA' NB' X'. ALL B NA NB X. \ 
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\ Says Server B {NA, X, Crypt (shrK B) {NB, K}} : set evs > \ 
2131  197 
\ B=B' & NA=NA' & NB=NB' & X=X'"; 
2032  198 
by (etac otway.induct 1); 
4091  199 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib]))); 
3730  200 
by (ALLGOALS Clarify_tac); 
2002  201 
(*Remaining cases: OR3 and OR4*) 
202 
by (ex_strip_tac 2); 

3102  203 
by (Blast_tac 2); 
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by (expand_case_tac "K = ?y" 1); 
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by (REPEAT (ares_tac [refl,exI,impI,conjI] 2)); 
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(*...we assume X is a recent message, and handle this case by contradiction*) 
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by (blast_tac (claset() addSEs spies_partsEs 
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delrules [conjI] (*no splitup to 4 subgoals*)) 1); 
2002  209 
val lemma = result(); 
210 

211 
goal thy 

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"!!evs. [ Says Server B {NA, X, Crypt (shrK B) {NB, K}} : set evs; \ 
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\ Says Server B' {NA',X',Crypt (shrK B') {NB',K}} : set evs; \ 
2131  214 
\ evs : otway ] ==> X=X' & B=B' & NA=NA' & NB=NB'"; 
2417  215 
by (prove_unique_tac lemma 1); 
2002  216 
qed "unique_session_keys"; 
217 

218 

2131  219 
(*Crucial security property, but not itself enough to guarantee correctness!*) 
220 
goal thy 

3683  221 
"!!evs. [ A ~: bad; B ~: bad; evs : otway ] \ 
2166  222 
\ ==> Says Server B \ 
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\ {NA, Crypt (shrK A) {NA, Key K}, \ 
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\ Crypt (shrK B) {NB, Key K}} : set evs > \ 
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\ Says B Spy {NA, NB, Key K} ~: set evs > \ 
3683  226 
\ Key K ~: analz (spies evs)"; 
2131  227 
by (etac otway.induct 1); 
3683  228 
by analz_spies_tac; 
2131  229 
by (ALLGOALS 
4091  230 
(asm_simp_tac (simpset() addcongs [conj_cong] 
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addsimps [analz_insert_eq, analz_insert_freshK] 
3961  232 
addsimps (pushes@expand_ifs)))); 
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(*Oops*) 
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by (blast_tac (claset() addSDs [unique_session_keys]) 4); 
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(*OR4*) 
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by (Blast_tac 3); 
2131  237 
(*OR3*) 
4091  238 
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); 
2131  242 
val lemma = result() RS mp RS mp RSN(2,rev_notE); 
243 

244 
goal thy 

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"!!evs. [ Says Server B \ 
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\ {NA, Crypt (shrK A) {NA, Key K}, \ 
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\ Crypt (shrK B) {NB, Key K}} : set evs; \ 
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\ Says B Spy {NA, NB, Key K} ~: set evs; \ 
3683  249 
\ A ~: bad; B ~: bad; evs : otway ] \ 
250 
\ ==> Key K ~: analz (spies evs)"; 

2131  251 
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1); 
4091  252 
by (blast_tac (claset() addSEs [lemma]) 1); 
2131  253 
qed "Spy_not_see_encrypted_key"; 
254 

255 

256 
(*** Attempting to prove stronger properties ***) 

257 

2052  258 
(*Only OR1 can have caused such a part of a message to appear. 
259 
I'm not sure why A ~= B premise is needed: OtwayRees.ML doesn't need it. 

260 
Perhaps it's because OR2 has two similarlooking encrypted messages in 

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this version.*) 
2002  262 
goal thy 
3683  263 
"!!evs. [ A ~: bad; A ~= B; evs : otway ] \ 
264 
\ ==> Crypt (shrK A) {NA, Agent A, Agent B} : parts (spies evs) > \ 

2131  265 
\ Says A B {NA, Agent A, Agent B, \ 
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\ Crypt (shrK A) {NA, Agent A, Agent B}} : set evs"; 
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by (parts_induct_tac 1); 
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by (Fake_parts_insert_tac 1); 
3102  269 
by (Blast_tac 1); 
2002  270 
qed_spec_mp "Crypt_imp_OR1"; 
271 

272 

2131  273 
(*Crucial property: If the encrypted message appears, and A has used NA 
274 
to start a run, then it originated with the Server!*) 

275 
(*Only it is FALSE. Somebody could make a fake message to Server 

2002  276 
substituting some other nonce NA' for NB.*) 
277 
goal thy 

3683  278 
"!!evs. [ A ~: bad; A ~= Spy; evs : otway ] \ 
279 
\ ==> Crypt (shrK A) {NA, Key K} : parts (spies evs) > \ 

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\ Says A B {NA, Agent A, Agent B, \ 
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\ Crypt (shrK A) {NA, Agent A, Agent B}} \ 
ab0a9fbed4c0
Changing "lost" from a parameter of protocol definitions to a constant.
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\ : set evs > \ 
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Changing "lost" from a parameter of protocol definitions to a constant.
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\ (EX B NB. Says Server B \ 
ab0a9fbed4c0
Changing "lost" from a parameter of protocol definitions to a constant.
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\ {NA, \ 
ab0a9fbed4c0
Changing "lost" from a parameter of protocol definitions to a constant.
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\ Crypt (shrK A) {NA, Key K}, \ 
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Corrected indentations and margins after the renaming of "set_of_list"
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\ Crypt (shrK B) {NB, Key K}} : set evs)"; 
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by (parts_induct_tac 1); 
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by (Fake_parts_insert_tac 1); 
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(*OR1: it cannot be a new Nonce, contradiction.*) 
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by (blast_tac (claset() addSIs [parts_insertI] 
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addSEs spies_partsEs) 1); 
292 
(*OR3 and OR4*) 

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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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(*OR4*) 
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by (blast_tac (claset() addSIs [Crypt_imp_OR1] 
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addEs spies_partsEs) 2); 
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(*OR3*) (** LEVEL 5 **) 
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(*The hypotheses at this point suggest an attack in which nonce NB is used 
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in two different roles: 
301 
Says B' Server 

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{Nonce NA, Agent Aa, Agent A, 
303 
Crypt (shrK Aa) {Nonce NA, Agent Aa, Agent A}, Nonce NB, 

304 
Crypt (shrK A) {Nonce NA, Agent Aa, Agent A}} 

305 
: set evs3; 

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Says A B 
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{Nonce NB, Agent A, Agent B, 
308 
Crypt (shrK A) {Nonce NB, Agent A, Agent B}} 

309 
: set evs3; 

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*) 
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writeln "GIVE UP! on NA_Crypt_imp_Server_msg"; 
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313 

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(*Thus the key property A_can_trust probably fails too.*) 