author  paulson 
Fri, 11 Jul 1997 13:26:15 +0200  
changeset 3512  9dcb4daa15e8 
parent 3466  30791e5a69c4 
child 3516  470626799511 
permissions  rwrr 
2318  1 
(* Title: HOL/Auth/NS_Public 
2 
ID: $Id$ 

3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

4 
Copyright 1996 University of Cambridge 

5 

6 
Inductive relation "ns_public" for the NeedhamSchroeder PublicKey protocol. 

7 
Version incorporating Lowe's fix (inclusion of B's identify in round 2). 

8 
*) 

9 

10 
open NS_Public; 

11 

12 
proof_timing:=true; 

13 
HOL_quantifiers := false; 

14 

15 
AddIffs [Spy_in_lost]; 

16 

17 
(*Replacing the variable by a constant improves search speed by 50%!*) 

18 
val Says_imp_sees_Spy' = 

19 
read_instantiate_sg (sign_of thy) [("lost","lost")] Says_imp_sees_Spy; 

20 

21 
(*A "possibility property": there are traces that reach the end*) 

22 
goal thy 

23 
"!!A B. A ~= B ==> EX NB. EX evs: ns_public. \ 

3465  24 
\ Says A B (Crypt (pubK B) (Nonce NB)) : set evs"; 
2318  25 
by (REPEAT (resolve_tac [exI,bexI] 1)); 
26 
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; 
2318  28 
result(); 
29 

30 

31 
(**** Inductive proofs about ns_public ****) 

32 

33 
(*Nobody sends themselves messages*) 

3465  34 
goal thy "!!evs. evs : ns_public ==> ALL A X. Says A A X ~: set evs"; 
2318  35 
by (etac ns_public.induct 1); 
36 
by (Auto_tac()); 

37 
qed_spec_mp "not_Says_to_self"; 

38 
Addsimps [not_Says_to_self]; 

39 
AddSEs [not_Says_to_self RSN (2, rev_notE)]; 

40 

41 

42 
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY 

43 
sends messages containing X! **) 

44 

45 
(*Spy never sees another agent's private key! (unless it's lost at start)*) 

46 
goal thy 

47 
"!!evs. evs : ns_public \ 

48 
\ ==> (Key (priK A) : parts (sees lost Spy evs)) = (A : lost)"; 

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by (etac ns_public.induct 1); 
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by (prove_simple_subgoals_tac 1); 
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by (Fake_parts_insert_tac 1); 
2318  52 
qed "Spy_see_priK"; 
53 
Addsimps [Spy_see_priK]; 

54 

55 
goal thy 

56 
"!!evs. evs : ns_public \ 

57 
\ ==> (Key (priK A) : analz (sees lost Spy evs)) = (A : lost)"; 

58 
by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset)); 

59 
qed "Spy_analz_priK"; 

60 
Addsimps [Spy_analz_priK]; 

61 

62 
goal thy "!!A. [ Key (priK A) : parts (sees lost Spy evs); \ 

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\ evs : ns_public ] ==> A:lost"; 

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by (blast_tac (!claset addDs [Spy_see_priK]) 1); 
2318  65 
qed "Spy_see_priK_D"; 
66 

67 
bind_thm ("Spy_analz_priK_D", analz_subset_parts RS subsetD RS Spy_see_priK_D); 

68 
AddSDs [Spy_see_priK_D, Spy_analz_priK_D]; 

69 

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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 addsimps [not_parts_not_analz] 
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setloop split_tac [expand_if])); 
2318  76 

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(**** Authenticity properties obtained from NS2 ****) 

78 

79 
(*It is impossible to reuse 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. [ Nonce NA ~: analz (sees lost Spy evs); \ 
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\ Crypt (pubK B) {Nonce NA, Agent A} : parts (sees lost Spy evs); \ 
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\ evs : ns_public ] \ 
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\ ==> Crypt (pubK C) {NA', Nonce NA, Agent D} ~: parts (sees lost Spy 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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(*NS3*) 
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by (blast_tac (!claset addSEs partsEs) 4); 
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(*NS2*) 
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by (blast_tac (!claset addSEs partsEs) 3); 
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(*Fake*) 
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by (blast_tac (!claset addSIs [analz_insertI] 
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addDs [impOfSubs analz_subset_parts, 
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impOfSubs Fake_parts_insert]) 2); 
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(*Base*) 
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by (Blast_tac 1); 
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qed "no_nonce_NS1_NS2"; 
2318  100 

101 

2480  102 
(*Unicity for NS1: nonce NA identifies agents A and B*) 
2318  103 
goal thy 
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"!!evs. [ Nonce NA ~: analz (sees lost Spy 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 (sees lost Spy evs) > \ 
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\ A=A' & B=B'"; 
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by (etac rev_mp 1); 
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by (analz_induct_tac 1); 
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(*NS1*) 
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by (simp_tac (!simpset addsimps [all_conj_distrib]) 3); 
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by (expand_case_tac "NA = ?y" 3 THEN 
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REPEAT (blast_tac (!claset addSEs partsEs) 3)); 
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(*Base*) 
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by (Blast_tac 1); 
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(*Fake*) 
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by (simp_tac (!simpset addsimps [all_conj_distrib, parts_insert_sees]) 1); 
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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 (blast_tac (!claset delrules [conjI] 
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addSDs [impOfSubs Fake_parts_insert] 
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addDs [impOfSubs analz_subset_parts]) 1); 
2318  123 
val lemma = result(); 
124 

125 
goal thy 

126 
"!!evs. [ Crypt(pubK B) {Nonce NA, Agent A} : parts(sees lost Spy evs); \ 

127 
\ Crypt(pubK B') {Nonce NA, Agent A'} : parts(sees lost Spy evs); \ 

128 
\ Nonce NA ~: analz (sees lost Spy evs); \ 

129 
\ evs : ns_public ] \ 

130 
\ ==> A=A' & B=B'"; 

2480  131 
by (prove_unique_tac lemma 1); 
2318  132 
qed "unique_NA"; 
133 

134 

135 
(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure*) 

136 
goal thy 

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"!!evs. [ Says A B (Crypt(pubK B) {Nonce NA, Agent A}) : set evs; \ 
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\ A ~: lost; B ~: lost; evs : ns_public ] \ 
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\ ==> Nonce NA ~: analz (sees lost Spy 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_sees_Spy' 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_sees_Spy' RS parts.Inj, 
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parts.Body, unique_NA]) 3); 
2318  149 
(*NS1*) 
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by (blast_tac (!claset addSEs sees_Spy_partsEs 
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addIs [impOfSubs analz_subset_parts]) 2); 
2318  152 
(*Fake*) 
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by (spy_analz_tac 1); 
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qed "Spy_not_see_NA"; 
2318  155 

156 

157 
(*Authentication for A: if she receives message 2 and has used NA 

158 
to start a run, then B has sent message 2.*) 

159 
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, Agent B}) \ 
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\ : set evs; \ 
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\ A ~: lost; B ~: lost; evs : ns_public ] \ 
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\ ==> Says B A (Crypt(pubK A) {Nonce NA, Nonce NB, Agent B}) \ 
3465  165 
\ : set evs"; 
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by (etac rev_mp 1); 
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(*prepare induction over Crypt (pubK A) {NA,NB,B} : parts H*) 
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by (etac (Says_imp_sees_Spy' RS parts.Inj RS rev_mp) 1); 
2318  169 
by (etac ns_public.induct 1); 
170 
by (ALLGOALS Asm_simp_tac); 

171 
(*NS1*) 

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by (blast_tac (!claset addSEs sees_Spy_partsEs) 2); 
2318  173 
(*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); 
2318  177 
qed "A_trusts_NS2"; 
178 

179 
(*If the encrypted message appears then it originated with Alice in NS1*) 

180 
goal thy 

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"!!evs. [ Crypt (pubK B) {Nonce NA, Agent A} : parts (sees lost Spy evs); \ 
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\ Nonce NA ~: analz (sees lost Spy evs); \ 
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\ evs : ns_public ] \ 
3465  184 
\ ==> 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); 
2480  187 
by (analz_induct_tac 1); 
2318  188 
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by (blast_tac (!claset addSDs [impOfSubs Fake_parts_insert] 
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addIs [analz_insertI] 
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addDs [impOfSubs analz_subset_parts]) 2); 
2318  192 
(*Base*) 
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by (Blast_tac 1); 
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qed "B_trusts_NS1"; 
2318  195 

196 

197 

198 
(**** Authenticity properties obtained from NS2 ****) 

199 

2480  200 
(*Unicity for NS2: nonce NB identifies nonce NA and agents A, B 
2318  201 
[unicity of B makes Lowe's fix work] 
202 
[proof closely follows that for unique_NA] *) 

203 
goal thy 

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"!!evs. [ Nonce NB ~: analz (sees lost Spy evs); evs : ns_public ] \ 
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\ ==> EX A' NA' B'. ALL A NA B. \ 
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\ Crypt (pubK A) {Nonce NA, Nonce NB, Agent B} \ 
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\ : parts (sees lost Spy evs) > A=A' & NA=NA' & B=B'"; 
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by (etac rev_mp 1); 
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by (analz_induct_tac 1); 
2318  210 
(*NS2*) 
2497  211 
by (simp_tac (!simpset addsimps [all_conj_distrib]) 3); 
2318  212 
by (expand_case_tac "NB = ?y" 3 THEN 
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REPEAT (blast_tac (!claset addSEs partsEs) 3)); 
2318  214 
(*Base*) 
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by (Blast_tac 1); 
2318  216 
(*Fake*) 
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by (simp_tac (!simpset addsimps [all_conj_distrib, parts_insert_sees]) 1); 
2374  218 
by (step_tac (!claset addSIs [analz_insertI]) 1); 
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by (ex_strip_tac 1); 
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by (blast_tac (!claset delrules [conjI] 
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addSDs [impOfSubs Fake_parts_insert] 
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addDs [impOfSubs analz_subset_parts]) 1); 
2318  223 
val lemma = result(); 
224 

225 
goal thy 

226 
"!!evs. [ Crypt(pubK A) {Nonce NA, Nonce NB, Agent B} \ 

227 
\ : parts(sees lost Spy evs); \ 

228 
\ Crypt(pubK A') {Nonce NA', Nonce NB, Agent B'} \ 

229 
\ : parts(sees lost Spy evs); \ 

230 
\ Nonce NB ~: analz (sees lost Spy evs); \ 

231 
\ evs : ns_public ] \ 

232 
\ ==> A=A' & NA=NA' & B=B'"; 

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by (prove_unique_tac lemma 1); 
2318  234 
qed "unique_NB"; 
235 

236 

237 
(*Secrecy: Spy does not see the nonce sent in msg NS2 if A and B are secure*) 

238 
goal thy 

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"!!evs. [ Says B A (Crypt (pubK A) {Nonce NA, Nonce NB, Agent B}) \ 
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\ : set evs; \ 
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\ A ~: lost; B ~: lost; evs : ns_public ] \ 
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\ ==> Nonce NB ~: analz (sees lost Spy evs)"; 
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by (etac rev_mp 1); 
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by (analz_induct_tac 1); 
2318  245 
(*NS3*) 
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by (blast_tac (!claset addDs [Says_imp_sees_Spy' RS parts.Inj, 
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247 
unique_NB]) 4); 
2318  248 
(*NS1*) 
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by (blast_tac (!claset addSEs sees_Spy_partsEs) 2); 
2318  250 
(*Fake*) 
2497  251 
by (spy_analz_tac 1); 
2318  252 
(*NS2*) 
253 
by (Step_tac 1); 

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by (blast_tac (!claset addSEs sees_Spy_partsEs) 3); 
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by (blast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj] 
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addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 2); 
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by (blast_tac (!claset addSIs [impOfSubs analz_subset_parts]) 1); 
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qed "Spy_not_see_NB"; 
2318  259 

260 

261 
(*Authentication for B: if he receives message 3 and has used NB 

262 
in message 2, then A has sent message 3.*) 

263 
goal thy 

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"!!evs. [ Says B A (Crypt (pubK A) {Nonce NA, Nonce NB, Agent B}) \ 
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\ : set evs; \ 
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\ Says A' B (Crypt (pubK B) (Nonce NB)): set evs; \ 
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\ A ~: lost; B ~: lost; evs : ns_public ] \ 
3465  268 
\ ==> Says A B (Crypt (pubK B) (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_sees_Spy' RS parts.Inj RS rev_mp) 1); 
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by (analz_induct_tac 1); 
2318  273 
(*NS1*) 
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by (blast_tac (!claset addSEs sees_Spy_partsEs) 2); 
2318  275 
(*Fake*) 
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by (blast_tac (!claset addSDs [impOfSubs Fake_parts_insert] 
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277 
addDs [Spy_not_see_NB, 
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278 
impOfSubs analz_subset_parts]) 1); 
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279 
(*NS3; not clear why blast_tac needs to be preceeded by Step_tac*) 
2318  280 
by (Step_tac 1); 
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281 
by (blast_tac (!claset addDs [Says_imp_sees_Spy' RS parts.Inj, 
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282 
Spy_not_see_NB, unique_NB]) 1); 
2318  283 
qed "B_trusts_NS3"; 
284 

285 

286 
(**** Overall guarantee for B*) 

287 

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(*Matches only NS2, not NS1 (or NS3)*) 
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289 
val Says_imp_sees_Spy'' = 
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read_instantiate [("X","Crypt ?K {?XX,?YY,?ZZ}")] Says_imp_sees_Spy'; 
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291 

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292 

2318  293 
(*If B receives NS3 and the nonce NB agrees with the nonce he joined with 
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294 
NA, then A initiated the run using NA. SAME proof as B_trusts_NS3!*) 
2318  295 
goal thy 
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296 
"!!evs. [ Says B A (Crypt (pubK A) {Nonce NA, Nonce NB, Agent B}) \ 
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297 
\ : set evs; \ 
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298 
\ Says A' B (Crypt (pubK B) (Nonce NB)): set evs; \ 
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299 
\ A ~: lost; B ~: lost; evs : ns_public ] \ 
3465  300 
\ ==> Says A B (Crypt (pubK B) {Nonce NA, Agent A}) : set evs"; 
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301 
by (etac rev_mp 1); 
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302 
(*prepare induction over Crypt (pubK B) {NB} : parts H*) 
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303 
by (etac (Says_imp_sees_Spy' RS parts.Inj RS rev_mp) 1); 
2318  304 
by (etac ns_public.induct 1); 
305 
by (ALLGOALS Asm_simp_tac); 

306 
(*Fake, NS2, NS3*) 

307 
(*NS1*) 

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308 
by (blast_tac (!claset addSEs sees_Spy_partsEs) 2); 
2318  309 
(*Fake*) 
310 
by (REPEAT_FIRST (resolve_tac [impI, conjI])); 

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311 
by (Blast_tac 1); 
2318  312 
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1)); 
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313 
by (blast_tac (!claset addSIs [disjI2] 
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314 
addDs [impOfSubs analz_subset_parts, 
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315 
impOfSubs Fake_parts_insert]) 1); 
2318  316 
(*NS3*) 
317 
by (Step_tac 1); 

318 
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1)); 

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319 
by (blast_tac (!claset addSDs [Says_imp_sees_Spy'' RS parts.Inj] 
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320 
addDs [unique_NB]) 1); 
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321 
qed "B_trusts_protocol"; 
2318  322 