author | paulson |
Thu, 26 Sep 1996 12:50:48 +0200 | |
changeset 2032 | 1bbf1bdcaf56 |
parent 2026 | 0df5a96bf77e |
child 2045 | ae1030e66745 |
permissions | -rw-r--r-- |
1995 | 1 |
(* Title: HOL/Auth/Yahalom |
1985
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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 "otway" for the Yahalom protocol. |
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7 |
|
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8 |
From page 257 of |
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9 |
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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|
1995 | 13 |
open Yahalom; |
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14 |
|
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proof_timing:=true; |
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HOL_quantifiers := false; |
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|
1995 | 18 |
|
2013 | 19 |
(*Weak liveness: there are traces that reach the end*) |
1995 | 20 |
|
21 |
goal thy |
|
22 |
"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \ |
|
2032 | 23 |
\ ==> EX X NB K. EX evs: yahalom lost. \ |
1995 | 24 |
\ Says A B {|X, Crypt (Nonce NB) K|} : set_of_list evs"; |
25 |
by (REPEAT (resolve_tac [exI,bexI] 1)); |
|
2032 | 26 |
by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS yahalom.YM4) 2); |
1995 | 27 |
by (ALLGOALS (simp_tac (!simpset setsolver safe_solver))); |
28 |
by (ALLGOALS Fast_tac); |
|
2013 | 29 |
result(); |
1995 | 30 |
|
31 |
||
1985
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32 |
(**** Inductive proofs about yahalom ****) |
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33 |
|
2032 | 34 |
(*The Spy can see more than anybody else, except for their initial state*) |
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goal thy |
2032 | 36 |
"!!evs. evs : yahalom lost ==> \ |
37 |
\ sees lost A evs <= initState lost A Un sees lost Spy evs"; |
|
38 |
by (etac yahalom.induct 1); |
|
1985
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39 |
by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD] |
2032 | 40 |
addss (!simpset)))); |
41 |
qed "sees_agent_subset_sees_Spy"; |
|
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42 |
|
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43 |
|
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(*Nobody sends themselves messages*) |
2032 | 45 |
goal thy "!!evs. evs : yahalom lost ==> ALL A X. Says A A X ~: set_of_list evs"; |
46 |
by (etac yahalom.induct 1); |
|
1985
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47 |
by (Auto_tac()); |
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48 |
qed_spec_mp "not_Says_to_self"; |
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Addsimps [not_Says_to_self]; |
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AddSEs [not_Says_to_self RSN (2, rev_notE)]; |
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51 |
|
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52 |
|
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(** For reasoning about the encrypted portion of messages **) |
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54 |
|
1995 | 55 |
(*Lets us treat YM4 using a similar argument as for the Fake case.*) |
56 |
goal thy "!!evs. Says S A {|Crypt Y (shrK A), X|} : set_of_list evs ==> \ |
|
2032 | 57 |
\ X : analz (sees lost Spy evs)"; |
58 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1); |
|
59 |
qed "YM4_analz_sees_Spy"; |
|
1985
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60 |
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1995 | 61 |
goal thy "!!evs. Says S A {|Crypt {|B, K, NA, NB|} (shrK A), X|} \ |
62 |
\ : set_of_list evs ==> \ |
|
2032 | 63 |
\ K : parts (sees lost Spy evs)"; |
1985
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64 |
by (fast_tac (!claset addSEs partsEs |
2032 | 65 |
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1); |
66 |
qed "YM4_parts_sees_Spy"; |
|
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67 |
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69 |
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(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY |
2013 | 71 |
sends messages containing X! **) |
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2032 | 73 |
(*Spy never sees lost another agent's shared key! (unless it is leaked at start)*) |
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74 |
goal thy |
2032 | 75 |
"!!evs. [| evs : yahalom lost; A ~: lost |] \ |
76 |
\ ==> Key (shrK A) ~: parts (sees lost Spy evs)"; |
|
77 |
by (etac yahalom.induct 1); |
|
78 |
by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6); |
|
1995 | 79 |
by (ALLGOALS Asm_simp_tac); |
80 |
by (stac insert_commute 3); |
|
1985
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81 |
by (Auto_tac()); |
1995 | 82 |
(*Fake and YM4 are similar*) |
83 |
by (ALLGOALS (best_tac (!claset addSDs [impOfSubs analz_subset_parts, |
|
2032 | 84 |
impOfSubs Fake_parts_insert]))); |
85 |
qed "Spy_not_see_shrK"; |
|
1985
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86 |
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2032 | 87 |
bind_thm ("Spy_not_analz_shrK", |
88 |
[analz_subset_parts, Spy_not_see_shrK] MRS contra_subsetD); |
|
1985
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89 |
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2032 | 90 |
Addsimps [Spy_not_see_shrK, Spy_not_analz_shrK]; |
1985
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91 |
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92 |
(*We go to some trouble to preserve R in the 3rd and 4th subgoals |
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93 |
As usual fast_tac cannot be used because it uses the equalities too soon*) |
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94 |
val major::prems = |
2032 | 95 |
goal thy "[| Key (shrK A) : parts (sees lost Spy evs); \ |
96 |
\ evs : yahalom lost; \ |
|
97 |
\ A:lost ==> R \ |
|
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98 |
\ |] ==> R"; |
2032 | 99 |
by (rtac ccontr 1); |
100 |
by (rtac ([major, Spy_not_see_shrK] MRS rev_notE) 1); |
|
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101 |
by (swap_res_tac prems 2); |
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102 |
by (ALLGOALS (fast_tac (!claset addIs prems))); |
2032 | 103 |
qed "Spy_see_shrK_E"; |
1985
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104 |
|
2032 | 105 |
bind_thm ("Spy_analz_shrK_E", |
106 |
analz_subset_parts RS subsetD RS Spy_see_shrK_E); |
|
1985
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107 |
|
2032 | 108 |
AddSEs [Spy_see_shrK_E, Spy_analz_shrK_E]; |
1985
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110 |
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(*** Future keys can't be seen or used! ***) |
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112 |
|
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(*Nobody can have SEEN keys that will be generated in the future. |
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This has to be proved anew for each protocol description, |
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but should go by similar reasoning every time. Hardest case is the |
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116 |
standard Fake rule. |
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The length comparison, and Union over C, are essential for the |
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118 |
induction! *) |
2032 | 119 |
goal thy "!!evs. evs : yahalom lost ==> \ |
1985
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\ length evs <= length evs' --> \ |
2032 | 121 |
\ Key (newK evs') ~: (UN C. parts (sees lost C evs))"; |
122 |
by (etac yahalom.induct 1); |
|
123 |
by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6); |
|
1985
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124 |
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts, |
2032 | 125 |
impOfSubs parts_insert_subset_Un, |
126 |
Suc_leD] |
|
127 |
addss (!simpset)))); |
|
1985
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128 |
val lemma = result(); |
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129 |
|
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(*Variant needed for the main theorem below*) |
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131 |
goal thy |
2032 | 132 |
"!!evs. [| evs : yahalom lost; length evs <= length evs' |] \ |
133 |
\ ==> Key (newK evs') ~: parts (sees lost C evs)"; |
|
1985
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134 |
by (fast_tac (!claset addDs [lemma]) 1); |
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135 |
qed "new_keys_not_seen"; |
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136 |
Addsimps [new_keys_not_seen]; |
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137 |
|
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138 |
(*Another variant: old messages must contain old keys!*) |
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139 |
goal thy |
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140 |
"!!evs. [| Says A B X : set_of_list evs; \ |
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141 |
\ Key (newK evt) : parts {X}; \ |
2032 | 142 |
\ evs : yahalom lost \ |
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143 |
\ |] ==> length evt < length evs"; |
2032 | 144 |
by (rtac ccontr 1); |
145 |
by (dtac leI 1); |
|
146 |
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy] |
|
2013 | 147 |
addIs [impOfSubs parts_mono]) 1); |
1985
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148 |
qed "Says_imp_old_keys"; |
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149 |
|
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150 |
|
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151 |
(*Nobody can have USED keys that will be generated in the future. |
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152 |
...very like new_keys_not_seen*) |
2032 | 153 |
goal thy "!!evs. evs : yahalom lost ==> \ |
1985
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154 |
\ length evs <= length evs' --> \ |
2032 | 155 |
\ newK evs' ~: keysFor (UN C. parts (sees lost C evs))"; |
156 |
by (etac yahalom.induct 1); |
|
157 |
by (forward_tac [YM4_parts_sees_Spy] 6); |
|
158 |
by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6); |
|
1995 | 159 |
by (ALLGOALS Asm_full_simp_tac); |
160 |
(*YM1, YM2 and YM3*) |
|
161 |
by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,3,2])); |
|
162 |
(*Fake and YM4: these messages send unknown (X) components*) |
|
163 |
by (stac insert_commute 2); |
|
164 |
by (Simp_tac 2); |
|
165 |
(*YM4: the only way K could have been used is if it had been seen, |
|
166 |
contradicting new_keys_not_seen*) |
|
167 |
by (ALLGOALS |
|
1985
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168 |
(best_tac |
2026
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169 |
(!claset addDs [impOfSubs analz_subset_parts, |
2032 | 170 |
impOfSubs (analz_subset_parts RS keysFor_mono), |
171 |
impOfSubs (parts_insert_subset_Un RS keysFor_mono), |
|
172 |
Suc_leD] |
|
173 |
addEs [new_keys_not_seen RSN(2,rev_notE)] |
|
174 |
addss (!simpset)))); |
|
1985
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175 |
val lemma = result(); |
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176 |
|
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177 |
goal thy |
2032 | 178 |
"!!evs. [| evs : yahalom lost; length evs <= length evs' |] \ |
179 |
\ ==> newK evs' ~: keysFor (parts (sees lost C evs))"; |
|
1985
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180 |
by (fast_tac (!claset addSDs [lemma] addss (!simpset)) 1); |
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181 |
qed "new_keys_not_used"; |
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182 |
|
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183 |
bind_thm ("new_keys_not_analzd", |
2032 | 184 |
[analz_subset_parts RS keysFor_mono, |
185 |
new_keys_not_used] MRS contra_subsetD); |
|
1985
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186 |
|
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187 |
Addsimps [new_keys_not_used, new_keys_not_analzd]; |
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188 |
|
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189 |
|
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190 |
(** Lemmas concerning the form of items passed in messages **) |
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191 |
|
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192 |
|
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193 |
(**** |
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194 |
The following is to prove theorems of the form |
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195 |
|
2032 | 196 |
Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==> |
197 |
Key K : analz (sees lost Spy evs) |
|
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198 |
|
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199 |
A more general formula must be proved inductively. |
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200 |
|
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201 |
****) |
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202 |
|
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203 |
|
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204 |
(*NOT useful in this form, but it says that session keys are not used |
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205 |
to encrypt messages containing other keys, in the actual protocol. |
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206 |
We require that agents should behave like this subsequently also.*) |
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207 |
goal thy |
2032 | 208 |
"!!evs. evs : yahalom lost ==> \ |
209 |
\ (Crypt X (newK evt)) : parts (sees lost Spy evs) & \ |
|
210 |
\ Key K : parts {X} --> Key K : parts (sees lost Spy evs)"; |
|
211 |
by (etac yahalom.induct 1); |
|
212 |
by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6); |
|
1985
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213 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps pushes))); |
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214 |
(*Deals with Faked messages*) |
1995 | 215 |
by (EVERY |
216 |
(map (best_tac (!claset addSEs partsEs |
|
2032 | 217 |
addDs [impOfSubs parts_insert_subset_Un] |
218 |
addss (!simpset))) |
|
1995 | 219 |
[3,2])); |
220 |
(*Base case*) |
|
1985
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221 |
by (Auto_tac()); |
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222 |
result(); |
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223 |
|
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224 |
|
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225 |
(** Specialized rewriting for this proof **) |
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226 |
|
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227 |
Delsimps [image_insert]; |
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228 |
Addsimps [image_insert RS sym]; |
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229 |
|
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230 |
Delsimps [image_Un]; |
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231 |
Addsimps [image_Un RS sym]; |
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232 |
|
2032 | 233 |
goal thy "insert (Key (newK x)) (sees lost A evs) = \ |
234 |
\ Key `` (newK``{x}) Un (sees lost A evs)"; |
|
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235 |
by (Fast_tac 1); |
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236 |
val insert_Key_singleton = result(); |
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237 |
|
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238 |
goal thy "insert (Key (f x)) (Key``(f``E) Un C) = \ |
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239 |
\ Key `` (f `` (insert x E)) Un C"; |
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240 |
by (Fast_tac 1); |
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241 |
val insert_Key_image = result(); |
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242 |
|
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243 |
|
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244 |
(*This lets us avoid analyzing the new message -- unless we have to!*) |
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245 |
(*NEEDED??*) |
2032 | 246 |
goal thy "synth (analz (sees lost Spy evs)) <= \ |
247 |
\ synth (analz (sees lost Spy (Says A B X # evs)))"; |
|
1985
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248 |
by (Simp_tac 1); |
2032 | 249 |
by (rtac (subset_insertI RS analz_mono RS synth_mono) 1); |
1985
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250 |
qed "synth_analz_thin"; |
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251 |
|
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252 |
AddIs [impOfSubs synth_analz_thin]; |
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253 |
|
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254 |
|
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255 |
|
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256 |
(** Session keys are not used to encrypt other session keys **) |
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257 |
|
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258 |
(*Lemma for the trivial direction of the if-and-only-if*) |
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259 |
goal thy |
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260 |
"!!evs. (Key K : analz (Key``nE Un sEe)) --> \ |
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261 |
\ (K : nE | Key K : analz sEe) ==> \ |
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262 |
\ (Key K : analz (Key``nE Un sEe)) = (K : nE | Key K : analz sEe)"; |
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263 |
by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 1); |
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264 |
val lemma = result(); |
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265 |
|
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266 |
|
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267 |
goal thy |
2032 | 268 |
"!!evs. evs : yahalom lost ==> \ |
269 |
\ ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \ |
|
270 |
\ (K : newK``E | Key K : analz (sees lost Spy evs))"; |
|
271 |
by (etac yahalom.induct 1); |
|
272 |
by (dtac YM4_analz_sees_Spy 6); |
|
1985
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273 |
by (REPEAT_FIRST (resolve_tac [allI, lemma])); |
1995 | 274 |
by (ALLGOALS |
1985
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275 |
(asm_simp_tac |
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276 |
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK] |
2032 | 277 |
@ pushes) |
1985
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278 |
setloop split_tac [expand_if]))); |
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279 |
(*YM4*) |
2032 | 280 |
by (spy_analz_tac 4); |
1985
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281 |
(*YM3*) |
1995 | 282 |
by (Fast_tac 3); |
283 |
(*Fake case*) |
|
2032 | 284 |
by (spy_analz_tac 2); |
1985
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285 |
(*Base case*) |
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286 |
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1); |
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287 |
qed_spec_mp "analz_image_newK"; |
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288 |
|
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289 |
goal thy |
2032 | 290 |
"!!evs. evs : yahalom lost ==> \ |
291 |
\ Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \ |
|
292 |
\ (K = newK evt | Key K : analz (sees lost Spy evs))"; |
|
1985
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293 |
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK, |
2032 | 294 |
insert_Key_singleton]) 1); |
1985
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295 |
by (Fast_tac 1); |
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296 |
qed "analz_insert_Key_newK"; |
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297 |
|
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298 |
|
2013 | 299 |
(*Describes the form of K when the Server sends this message.*) |
1985
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300 |
goal thy |
1995 | 301 |
"!!evs. [| Says Server A \ |
302 |
\ {|Crypt {|Agent B, K, NA, NB|} (shrK A), \ |
|
303 |
\ Crypt {|Agent A, K|} (shrK B)|} : set_of_list evs; \ |
|
2032 | 304 |
\ evs : yahalom lost |] \ |
305 |
\ ==> (EX evt: yahalom lost. K = Key(newK evt))"; |
|
306 |
by (etac rev_mp 1); |
|
307 |
by (etac yahalom.induct 1); |
|
2013 | 308 |
by (ALLGOALS (fast_tac (!claset addss (!simpset)))); |
1985
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|
309 |
qed "Says_Server_message_form"; |
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310 |
|
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|
311 |
|
2032 | 312 |
(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 |
2013 | 313 |
As with Otway-Rees, proof does not need uniqueness of session keys. **) |
314 |
||
315 |
goal thy |
|
2032 | 316 |
"!!evs. [| A ~: lost; B ~: lost; evs : yahalom lost; evt : yahalom lost |] \ |
2013 | 317 |
\ ==> Says Server A \ |
318 |
\ {|Crypt {|Agent B, Key(newK evt), NA, NB|} (shrK A), \ |
|
319 |
\ Crypt {|Agent A, Key(newK evt)|} (shrK B)|} \ |
|
320 |
\ : set_of_list evs --> \ |
|
2032 | 321 |
\ Key(newK evt) ~: analz (sees lost Spy evs)"; |
322 |
by (etac yahalom.induct 1); |
|
323 |
by (dtac YM4_analz_sees_Spy 6); |
|
2013 | 324 |
by (ALLGOALS |
325 |
(asm_simp_tac |
|
326 |
(!simpset addsimps ([analz_subset_parts RS contra_subsetD, |
|
2032 | 327 |
analz_insert_Key_newK] @ pushes) |
2013 | 328 |
setloop split_tac [expand_if]))); |
329 |
(*YM4*) |
|
2032 | 330 |
by (spy_analz_tac 3); |
2013 | 331 |
(*YM3*) |
332 |
by (fast_tac (!claset addIs [parts_insertI] |
|
2032 | 333 |
addEs [Says_imp_old_keys RS less_irrefl] |
334 |
addss (!simpset)) 2); |
|
2013 | 335 |
(*Fake*) (** LEVEL 10 **) |
2032 | 336 |
by (spy_analz_tac 1); |
2013 | 337 |
val lemma = result() RS mp RSN(2,rev_notE); |
338 |
||
339 |
||
340 |
(*Final version: Server's message in the most abstract form*) |
|
1985
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|
341 |
goal thy |
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|
342 |
"!!evs. [| Says Server A \ |
1995 | 343 |
\ {|Crypt {|Agent B, K, NA, NB|} (shrK A), \ |
344 |
\ Crypt {|Agent A, K|} (shrK B)|} : set_of_list evs; \ |
|
2032 | 345 |
\ A ~: lost; B ~: lost; evs : yahalom lost |] ==> \ |
346 |
\ K ~: analz (sees lost Spy evs)"; |
|
2013 | 347 |
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1); |
348 |
by (fast_tac (!claset addSEs [lemma]) 1); |
|
2032 | 349 |
qed "Spy_not_see_encrypted_key"; |
2001 | 350 |
|
351 |
||
2013 | 352 |
(** Towards proofs of stronger authenticity properties **) |
353 |
||
2001 | 354 |
goal thy |
2032 | 355 |
"!!evs. [| Crypt {|Agent A, Key K|} (shrK B) : parts (sees lost Spy evs); \ |
356 |
\ B ~: lost; evs : yahalom lost |] \ |
|
2001 | 357 |
\ ==> EX NA NB. Says Server A \ |
2013 | 358 |
\ {|Crypt {|Agent B, Key K, \ |
359 |
\ Nonce NA, Nonce NB|} (shrK A), \ |
|
360 |
\ Crypt {|Agent A, Key K|} (shrK B)|} \ |
|
361 |
\ : set_of_list evs"; |
|
2032 | 362 |
by (etac rev_mp 1); |
363 |
by (etac yahalom.induct 1); |
|
364 |
by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6); |
|
2013 | 365 |
by (ALLGOALS Asm_simp_tac); |
2001 | 366 |
(*YM3*) |
367 |
by (Fast_tac 3); |
|
368 |
(*Base case*) |
|
369 |
by (fast_tac (!claset addss (!simpset)) 1); |
|
2013 | 370 |
(*Prepare YM4*) |
371 |
by (stac insert_commute 2 THEN Simp_tac 2); |
|
372 |
(*Fake and YM4 are similar*) |
|
373 |
by (ALLGOALS (best_tac (!claset addSDs [impOfSubs analz_subset_parts, |
|
2032 | 374 |
impOfSubs Fake_parts_insert]))); |
2013 | 375 |
qed "Crypt_imp_Server_msg"; |
2001 | 376 |
|
377 |
||
378 |
(*What can B deduce from receipt of YM4? |
|
379 |
NOT THAT THE NONCES AGREE (in this version). But what does the Nonce |
|
2032 | 380 |
give us??*) |
2001 | 381 |
goal thy |
382 |
"!!evs. [| Says A' B {|Crypt {|Agent A, Key K|} (shrK B), \ |
|
383 |
\ Crypt (Nonce NB) K|} : set_of_list evs; \ |
|
2032 | 384 |
\ B ~: lost; evs : yahalom lost |] \ |
2001 | 385 |
\ ==> EX NA NB. Says Server A \ |
386 |
\ {|Crypt {|Agent B, Key K, \ |
|
387 |
\ Nonce NA, Nonce NB|} (shrK A), \ |
|
388 |
\ Crypt {|Agent A, Key K|} (shrK B)|} \ |
|
389 |
\ : set_of_list evs"; |
|
2032 | 390 |
by (etac rev_mp 1); |
391 |
by (etac yahalom.induct 1); |
|
392 |
by (dtac YM4_analz_sees_Spy 6); |
|
2001 | 393 |
by (ALLGOALS Asm_simp_tac); |
2013 | 394 |
by (ALLGOALS (fast_tac (!claset addSDs [impOfSubs analz_subset_parts RS |
2032 | 395 |
Crypt_imp_Server_msg]))); |
2001 | 396 |
qed "YM4_imp_Says_Server_A"; |
397 |
||
398 |
||
399 |
||
400 |
goal thy |
|
401 |
"!!evs. [| Says A' B {|Crypt {|Agent A, Key K|} (shrK B), \ |
|
402 |
\ Crypt (Nonce NB) K|} : set_of_list evs; \ |
|
2032 | 403 |
\ A ~: lost; B ~: lost; evs : yahalom lost |] \ |
404 |
\ ==> Key K ~: analz (sees lost Spy evs)"; |
|
2001 | 405 |
by (fast_tac (!claset addSDs [YM4_imp_Says_Server_A, |
2032 | 406 |
Spy_not_see_encrypted_key]) 1); |
2001 | 407 |
qed "B_gets_secure_key"; |