author  paulson 
Tue, 15 Sep 1998 15:10:38 +0200  
changeset 5492  d9fc3457554e 
parent 5434  9b4bed3f394c 
child 5535  678999604ee9 
permissions  rwrr 
2002  1 
(* Title: HOL/Auth/OtwayRees_Bad 
2 
ID: $Id$ 

3 
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 

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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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2002  22 

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(*A "possibility property": there are traces that reach the end*) 
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Goal "[ A ~= B; B ~= Server ] \ 
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\ ==> EX K. EX NA. EX evs: otway. \ 
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\ Says B A {Nonce NA, Crypt (shrK A) {Nonce NA, Key K}} \ 
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\ : set evs"; 
2002  28 
by (REPEAT (resolve_tac [exI,bexI] 1)); 
2032  29 
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  31 
result(); 
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33 

34 
(**** Inductive proofs about otway ****) 

35 

36 

37 
(** For reasoning about the encrypted portion of messages **) 

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Goal "Says A' B {N, Agent A, Agent B, X} : set evs \ 
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\ ==> X : analz (spies evs)"; 
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by (dtac (Says_imp_spies RS analz.Inj) 1); 
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by (Blast_tac 1); 
3683  43 
qed "OR2_analz_spies"; 
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Goal "Says S' B {N, X, Crypt (shrK B) X'} : set evs \ 
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\ ==> X : analz (spies evs)"; 
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by (dtac (Says_imp_spies RS analz.Inj) 1); 
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by (Blast_tac 1); 
3683  49 
qed "OR4_analz_spies"; 
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Goal "Says Server B {NA, X, Crypt K' {NB,K}} : set evs \ 
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\ ==> K : parts (spies evs)"; 
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by (Blast_tac 1); 
3683  54 
qed "Oops_parts_spies"; 
2002  55 

3683  56 
bind_thm ("OR2_parts_spies", 
57 
OR2_analz_spies RS (impOfSubs analz_subset_parts)); 

58 
bind_thm ("OR4_parts_spies", 

59 
OR4_analz_spies RS (impOfSubs analz_subset_parts)); 

2052  60 

3683  61 
(*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  64 
forward_tac [Oops_parts_spies] (i+6) THEN 
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forward_tac [OR4_parts_spies] (i+5) THEN 

66 
forward_tac [OR2_parts_spies] (i+3) THEN 

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prove_simple_subgoals_tac i; 
2002  68 

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3683  70 
(** Theorems of the form X ~: parts (spies evs) imply that NOBODY 
2002  71 
sends messages containing X! **) 
72 

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(*Spy never sees a good agent's shared key!*) 
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Goal "evs : otway ==> (Key (shrK A) : parts (spies evs)) = (A : bad)"; 
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by (parts_induct_tac 1); 
3961  76 
by (ALLGOALS Blast_tac); 
2131  77 
qed "Spy_see_shrK"; 
78 
Addsimps [Spy_see_shrK]; 

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Goal "evs : otway ==> (Key (shrK A) : analz (spies evs)) = (A : bad)"; 
4091  81 
by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset())); 
2131  82 
qed "Spy_analz_shrK"; 
83 
Addsimps [Spy_analz_shrK]; 

2002  84 

4471  85 
AddSDs [Spy_see_shrK RSN (2, rev_iffD1), 
86 
Spy_analz_shrK RSN (2, rev_iffD1)]; 

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(*Nobody can have used nonexistent keys!*) 
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Goal "evs : otway ==> 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 (blast_tac (claset() addSDs [keysFor_parts_insert]) 1); 
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(*OR2, OR3*) 
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by (ALLGOALS Blast_tac); 
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qed_spec_mp "new_keys_not_used"; 
2002  97 

98 
bind_thm ("new_keys_not_analzd", 

2032  99 
[analz_subset_parts RS keysFor_mono, 
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new_keys_not_used] MRS contra_subsetD); 

2002  101 

102 
Addsimps [new_keys_not_used, new_keys_not_analzd]; 

103 

104 

2131  105 

106 
(*** Proofs involving analz ***) 

107 

108 
(*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 "[ Says Server B {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  113 
by (etac rev_mp 1); 
114 
by (etac otway.induct 1); 

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by (ALLGOALS Simp_tac); 
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by (ALLGOALS Blast_tac); 
2131  117 
qed "Says_Server_message_form"; 
118 

119 

120 
(*For proofs involving analz.*) 

3683  121 
val analz_spies_tac = 
122 
dtac OR2_analz_spies 4 THEN 

123 
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  126 

127 

128 
(**** 

129 
The following is to prove theorems of the form 

130 

3683  131 
Key K : analz (insert (Key KAB) (spies evs)) ==> 
132 
Key K : analz (spies evs) 

2002  133 

134 
A more general formula must be proved inductively. 

135 
****) 

136 

137 

138 
(** Session keys are not used to encrypt other session keys **) 

139 

140 
(*The equality makes the induction hypothesis easier to apply*) 

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Goal "evs : otway ==> \ 
5492  142 
\ ALL K KK. KK <=  (range shrK) > \ 
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\ (Key K : analz (Key``KK Un (spies evs))) = \ 
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\ (K : KK  Key K : analz (spies evs))"; 
2032  145 
by (etac otway.induct 1); 
3683  146 
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  153 

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Goal "[ evs : otway; KAB ~: range shrK ] ==> \ 
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\ Key K : analz (insert (Key KAB) (spies evs)) = \ 
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\ (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"; 
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161 

2131  162 
(*** The Key K uniquely identifies the Server's message. **) 
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Goal "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  167 
\ B=B' & NA=NA' & NB=NB' & X=X'"; 
2032  168 
by (etac otway.induct 1); 
4091  169 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib]))); 
3730  170 
by (ALLGOALS Clarify_tac); 
2002  171 
(*Remaining cases: OR3 and OR4*) 
172 
by (ex_strip_tac 2); 

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by (Best_tac 2); (*Blast_tac's too slow (in reconstruction)*) 
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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) 1); 
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val lemma = result(); 
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Goal "[ 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; \ 
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\ evs : otway ] ==> X=X' & B=B' & NA=NA' & NB=NB'"; 
2417  183 
by (prove_unique_tac lemma 1); 
2002  184 
qed "unique_session_keys"; 
185 

186 

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(** Crucial secrecy property: Spy does not see the keys sent in msg OR3 
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Does not in itself guarantee security: an attack could violate 
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the premises, e.g. by having A=Spy **) 
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Goal "[ A ~: bad; B ~: bad; evs : otway ] \ 
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\ ==> 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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\ Notes Spy {NA, NB, Key K} ~: set evs > \ 
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\ Key K ~: analz (spies evs)"; 
2131  197 
by (etac otway.induct 1); 
3683  198 
by analz_spies_tac; 
2131  199 
by (ALLGOALS 
4091  200 
(asm_simp_tac (simpset() addcongs [conj_cong] 
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addsimps [analz_insert_eq, analz_insert_freshK] 
4831  202 
addsimps (pushes@split_ifs)))); 
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(*Oops*) 
4091  204 
by (blast_tac (claset() addSDs [unique_session_keys]) 4); 
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(*OR4*) 
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by (Blast_tac 3); 
2131  207 
(*OR3*) 
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by (Blast_tac 2); 
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(*Fake*) 
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by (spy_analz_tac 1); 
2131  211 
val lemma = result() RS mp RS mp RSN(2,rev_notE); 
212 

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Goal "[ 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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\ Notes Spy {NA, NB, Key K} ~: set evs; \ 
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\ A ~: bad; B ~: bad; evs : otway ] \ 
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\ ==> Key K ~: analz (spies evs)"; 
2131  219 
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1); 
4091  220 
by (blast_tac (claset() addSEs [lemma]) 1); 
2131  221 
qed "Spy_not_see_encrypted_key"; 
222 

223 

224 
(*** Attempting to prove stronger properties ***) 

225 

2052  226 
(*Only OR1 can have caused such a part of a message to appear. 
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The premise A ~= B prevents OR2's similarlooking cryptogram from being 
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picked up. Original OtwayRees doesn't need it.*) 
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Goal "[ A ~: bad; A ~= B; evs : otway ] \ 
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\ ==> Crypt (shrK A) {NA, Agent A, Agent B} : 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}} : set evs"; 
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by (parts_induct_tac 1); 
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by (ALLGOALS Blast_tac); 
2002  235 
qed_spec_mp "Crypt_imp_OR1"; 
236 

237 

2131  238 
(*Crucial property: If the encrypted message appears, and A has used NA 
239 
to start a run, then it originated with the Server!*) 

240 
(*Only it is FALSE. Somebody could make a fake message to Server 

2002  241 
substituting some other nonce NA' for NB.*) 
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Goal "[ A ~: bad; evs : otway ] \ 
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\ ==> 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}} \ 
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\ : set evs > \ 
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\ (EX B NB. Says Server B \ 
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\ {NA, \ 
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\ Crypt (shrK A) {NA, Key K}, \ 
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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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(*Fake*) 
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by (Blast_tac 1); 
2002  254 
(*OR1: it cannot be a new Nonce, contradiction.*) 
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by (Blast_tac 1); 
3730  256 
(*OR3 and OR4*) 
4091  257 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib]))); 
3730  258 
by (ALLGOALS Clarify_tac); 
2002  259 
(*OR4*) 
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by (blast_tac (claset() addSIs [Crypt_imp_OR1]) 2); 
2131  261 
(*OR3*) (** LEVEL 5 **) 
3730  262 
(*The hypotheses at this point suggest an attack in which nonce NB is used 
2052  263 
in two different roles: 
264 
Says B' Server 

3730  265 
{Nonce NA, Agent Aa, Agent A, 
266 
Crypt (shrK Aa) {Nonce NA, Agent Aa, Agent A}, Nonce NB, 

267 
Crypt (shrK A) {Nonce NA, Agent Aa, Agent A}} 

268 
: set evs3; 

2052  269 
Says A B 
3730  270 
{Nonce NB, Agent A, Agent B, 
271 
Crypt (shrK A) {Nonce NB, Agent A, Agent B}} 

272 
: set evs3; 

2052  273 
*) 
2131  274 
writeln "GIVE UP! on NA_Crypt_imp_Server_msg"; 
2002  275 

276 

2052  277 
(*Thus the key property A_can_trust probably fails too.*) 