author  nipkow 
Fri, 24 Nov 2000 16:49:27 +0100  
changeset 10519  ade64af4c57c 
parent 7499  23e090051cb8 
child 10833  c0844a30ea4e 
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 knows_Spy_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)); 
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by (rtac (otway.Nil RS 
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otway.OR1 RS otway.Reception RS 
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otway.OR2 RS otway.Reception RS 
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otway.OR3 RS otway.Reception RS otway.OR4) 2); 
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by possibility_tac; 
2002  34 
result(); 
35 

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Goal "[ Gets B X : set evs; evs : otway ] ==> EX A. Says A B X : set evs"; 
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by (etac rev_mp 1); 
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by (etac otway.induct 1); 
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by Auto_tac; 
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qed"Gets_imp_Says"; 
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(*Must be proved separately for each protocol*) 
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Goal "[ Gets B X : set evs; evs : otway ] ==> X : knows Spy evs"; 
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by (blast_tac (claset() addSDs [Gets_imp_Says, Says_imp_knows_Spy]) 1); 
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qed"Gets_imp_knows_Spy"; 
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AddDs [Gets_imp_knows_Spy RS parts.Inj]; 
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2002  48 

49 
(**** Inductive proofs about otway ****) 

50 

51 

52 
(** For reasoning about the encrypted portion of messages **) 

53 

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Goal "[ Gets B {N, Agent A, Agent B, X} : set evs; evs : otway ] \ 
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\ ==> X : analz (knows Spy evs)"; 
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by (blast_tac (claset() addSDs [Gets_imp_knows_Spy RS analz.Inj]) 1); 
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qed "OR2_analz_knows_Spy"; 
2002  58 

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Goal "[ Gets B {N, X, Crypt (shrK B) X'} : set evs; evs : otway ] \ 
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\ ==> X : analz (knows Spy evs)"; 
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by (blast_tac (claset() addSDs [Gets_imp_knows_Spy RS analz.Inj]) 1); 
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qed "OR4_analz_knows_Spy"; 
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Goal "Says Server B {NA, X, Crypt K' {NB,K}} : set evs \ 
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\ ==> K : parts (knows Spy evs)"; 
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by (Blast_tac 1); 
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qed "Oops_parts_knows_Spy"; 
2002  68 

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bind_thm ("OR2_parts_knows_Spy", 
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OR2_analz_knows_Spy RS (impOfSubs analz_subset_parts)); 
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bind_thm ("OR4_parts_knows_Spy", 
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OR4_analz_knows_Spy RS (impOfSubs analz_subset_parts)); 
2052  73 

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(*For proving the easier theorems about X ~: parts (knows Spy evs).*) 
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fun parts_induct_tac i = 
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etac otway.induct i THEN 
7499  77 
ftac Oops_parts_knows_Spy (i+7) THEN 
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ftac OR4_parts_knows_Spy (i+6) THEN 

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ftac OR2_parts_knows_Spy (i+4) THEN 

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prove_simple_subgoals_tac i; 
2002  81 

82 

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

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(*Spy never sees a good agent's shared key!*) 
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Goal "evs : otway ==> (Key (shrK A) : parts (knows Spy evs)) = (A : bad)"; 
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by (parts_induct_tac 1); 
3961  89 
by (ALLGOALS Blast_tac); 
2131  90 
qed "Spy_see_shrK"; 
91 
Addsimps [Spy_see_shrK]; 

2002  92 

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Goal "evs : otway ==> (Key (shrK A) : analz (knows Spy evs)) = (A : bad)"; 
4091  94 
by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset())); 
2131  95 
qed "Spy_analz_shrK"; 
96 
Addsimps [Spy_analz_shrK]; 

2002  97 

4471  98 
AddSDs [Spy_see_shrK RSN (2, rev_iffD1), 
99 
Spy_analz_shrK RSN (2, rev_iffD1)]; 

2002  100 

101 

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(*Nobody can have used nonexistent keys!*) 
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Goal "evs : otway ==> Key K ~: used evs > K ~: keysFor (parts (knows Spy 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); 
2160  109 
qed_spec_mp "new_keys_not_used"; 
2002  110 

111 
bind_thm ("new_keys_not_analzd", 

2032  112 
[analz_subset_parts RS keysFor_mono, 
113 
new_keys_not_used] MRS contra_subsetD); 

2002  114 

115 
Addsimps [new_keys_not_used, new_keys_not_analzd]; 

116 

117 

2131  118 

119 
(*** Proofs involving analz ***) 

120 

121 
(*Describes the form of K and NA when the Server sends this message. Also 

122 
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  126 
by (etac rev_mp 1); 
127 
by (etac otway.induct 1); 

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by (ALLGOALS Simp_tac); 
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by (ALLGOALS Blast_tac); 
2131  130 
qed "Says_Server_message_form"; 
131 

132 

133 
(*For proofs involving analz.*) 

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val analz_knows_Spy_tac = 
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dtac OR2_analz_knows_Spy 5 THEN assume_tac 5 THEN 
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dtac OR4_analz_knows_Spy 7 THEN assume_tac 7 THEN 
7499  137 
ftac Says_Server_message_form 8 THEN assume_tac 8 THEN 
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REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 8); 
2002  139 

140 

141 
(**** 

142 
The following is to prove theorems of the form 

143 

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Key K : analz (insert (Key KAB) (knows Spy evs)) ==> 
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Key K : analz (knows Spy evs) 
2002  146 

147 
A more general formula must be proved inductively. 

148 
****) 

149 

150 

151 
(** Session keys are not used to encrypt other session keys **) 

152 

153 
(*The equality makes the induction hypothesis easier to apply*) 

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Goal "evs : otway ==> \ 
5492  155 
\ ALL K KK. KK <=  (range shrK) > \ 
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\ (Key K : analz (Key``KK Un (knows Spy evs))) = \ 
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\ (K : KK  Key K : analz (knows Spy evs))"; 
2032  158 
by (etac otway.induct 1); 
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by analz_knows_Spy_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  166 

167 

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Goal "[ evs : otway; KAB ~: range shrK ] ==> \ 
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\ Key K : analz (insert (Key KAB) (knows Spy evs)) = \ 
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\ (K = KAB  Key K : analz (knows Spy 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  173 

174 

2131  175 
(*** The Key K uniquely identifies the Server's message. **) 
2002  176 

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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  180 
\ B=B' & NA=NA' & NB=NB' & X=X'"; 
2032  181 
by (etac otway.induct 1); 
4091  182 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib]))); 
3730  183 
by (ALLGOALS Clarify_tac); 
2002  184 
(*Remaining cases: OR3 and OR4*) 
185 
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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188 
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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190 
by (blast_tac (claset() addSEs knows_Spy_partsEs) 1); 
2002  191 
val lemma = result(); 
192 

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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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195 
\ evs : otway ] ==> X=X' & B=B' & NA=NA' & NB=NB'"; 
2417  196 
by (prove_unique_tac lemma 1); 
2002  197 
qed "unique_session_keys"; 
198 

199 

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(** Crucial secrecy property: Spy does not see the keys sent in msg OR3 
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201 
Does not in itself guarantee security: an attack could violate 
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202 
the premises, e.g. by having A=Spy **) 
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203 

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Goal "[ A ~: bad; B ~: bad; evs : otway ] \ 
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205 
\ ==> Says Server B \ 
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206 
\ {NA, Crypt (shrK A) {NA, Key K}, \ 
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207 
\ 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 (knows Spy evs)"; 
2131  210 
by (etac otway.induct 1); 
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by analz_knows_Spy_tac; 
2131  212 
by (ALLGOALS 
4091  213 
(asm_simp_tac (simpset() addcongs [conj_cong] 
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addsimps [analz_insert_eq, analz_insert_freshK] 
5535  215 
@ pushes @ split_ifs))); 
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(*Oops*) 
4091  217 
by (blast_tac (claset() addSDs [unique_session_keys]) 4); 
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(*OR4*) 
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by (Blast_tac 3); 
2131  220 
(*OR3*) 
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by (Blast_tac 2); 
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(*Fake*) 
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by (spy_analz_tac 1); 
2131  224 
val lemma = result() RS mp RS mp RSN(2,rev_notE); 
225 

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226 
Goal "[ Says Server B \ 
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227 
\ {NA, Crypt (shrK A) {NA, Key K}, \ 
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228 
\ Crypt (shrK B) {NB, Key K}} : set evs; \ 
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\ Notes Spy {NA, NB, Key K} ~: set evs; \ 
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230 
\ A ~: bad; B ~: bad; evs : otway ] \ 
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\ ==> Key K ~: analz (knows Spy evs)"; 
7499  232 
by (ftac Says_Server_message_form 1 THEN assume_tac 1); 
4091  233 
by (blast_tac (claset() addSEs [lemma]) 1); 
2131  234 
qed "Spy_not_see_encrypted_key"; 
235 

236 

237 
(*** Attempting to prove stronger properties ***) 

238 

2052  239 
(*Only OR1 can have caused such a part of a message to appear. 
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240 
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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242 
Goal "[ A ~: bad; A ~= B; evs : otway ] \ 
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243 
\ ==> Crypt (shrK A) {NA, Agent A, Agent B} : parts (knows Spy evs) > \ 
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244 
\ Says A B {NA, Agent A, Agent B, \ 
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245 
\ Crypt (shrK A) {NA, Agent A, Agent B}} : set evs"; 
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246 
by (parts_induct_tac 1); 
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by (ALLGOALS Blast_tac); 
2002  248 
qed_spec_mp "Crypt_imp_OR1"; 
249 

250 

2131  251 
(*Crucial property: If the encrypted message appears, and A has used NA 
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252 
to start a run, then it originated with the Server! 
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253 
The premise A ~= B allows use of Crypt_imp_OR1*) 
2131  254 
(*Only it is FALSE. Somebody could make a fake message to Server 
2002  255 
substituting some other nonce NA' for NB.*) 
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256 
Goal "[ A ~: bad; A ~= B; evs : otway ] \ 
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257 
\ ==> Crypt (shrK A) {NA, Key K} : parts (knows Spy evs) > \ 
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258 
\ Says A B {NA, Agent A, Agent B, \ 
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259 
\ Crypt (shrK A) {NA, Agent A, Agent B}} \ 
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260 
\ : set evs > \ 
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261 
\ (EX B NB. Says Server B \ 
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262 
\ {NA, \ 
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263 
\ Crypt (shrK A) {NA, Key K}, \ 
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264 
\ Crypt (shrK B) {NB, Key K}} : set evs)"; 
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265 
by (parts_induct_tac 1); 
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266 
(*Fake*) 
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by (Blast_tac 1); 
2002  268 
(*OR1: it cannot be a new Nonce, contradiction.*) 
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by (Blast_tac 1); 
3730  270 
(*OR3 and OR4*) 
4091  271 
by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib]))); 
3730  272 
by (ALLGOALS Clarify_tac); 
2002  273 
(*OR4*) 
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274 
by (blast_tac (claset() addSIs [Crypt_imp_OR1]) 2); 
2131  275 
(*OR3*) (** LEVEL 5 **) 
3730  276 
(*The hypotheses at this point suggest an attack in which nonce NB is used 
2052  277 
in two different roles: 
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Gets Server 
3730  279 
{Nonce NA, Agent Aa, Agent A, 
280 
Crypt (shrK Aa) {Nonce NA, Agent Aa, Agent A}, Nonce NB, 

281 
Crypt (shrK A) {Nonce NA, Agent Aa, Agent A}} 

282 
: set evs3; 

2052  283 
Says A B 
3730  284 
{Nonce NB, Agent A, Agent B, 
285 
Crypt (shrK A) {Nonce NB, Agent A, Agent B}} 

286 
: set evs3; 

2052  287 
*) 
2131  288 
writeln "GIVE UP! on NA_Crypt_imp_Server_msg"; 
2002  289 

290 

2052  291 
(*Thus the key property A_can_trust probably fails too.*) 