src/HOL/Auth/Yahalom2.ML
author paulson
Fri Nov 01 18:28:19 1996 +0100 (1996-11-01)
changeset 2155 dc85854810eb
parent 2111 81c8d46edfa3
child 2160 ad4382e546fc
permissions -rw-r--r--
New version with simpler disambiguation in YM3,
Oops message, and no encryption in YM2
paulson@2111
     1
(*  Title:      HOL/Auth/Yahalom2
paulson@2111
     2
    ID:         $Id$
paulson@2111
     3
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@2111
     4
    Copyright   1996  University of Cambridge
paulson@2111
     5
paulson@2111
     6
Inductive relation "yahalom" for the Yahalom protocol, Variant 2.
paulson@2111
     7
paulson@2111
     8
This version trades encryption of NB for additional explicitness in YM3.
paulson@2111
     9
paulson@2111
    10
From page 259 of
paulson@2111
    11
  Burrows, Abadi and Needham.  A Logic of Authentication.
paulson@2111
    12
  Proc. Royal Soc. 426 (1989)
paulson@2111
    13
*)
paulson@2111
    14
paulson@2111
    15
open Yahalom2;
paulson@2111
    16
paulson@2111
    17
proof_timing:=true;
paulson@2111
    18
HOL_quantifiers := false;
paulson@2111
    19
paulson@2111
    20
paulson@2111
    21
(*Weak liveness: there are traces that reach the end*)
paulson@2111
    22
paulson@2111
    23
goal thy 
paulson@2111
    24
 "!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
paulson@2111
    25
\        ==> EX X NB K. EX evs: yahalom lost.          \
paulson@2111
    26
\               Says A B {|X, Crypt (Nonce NB) K|} : set_of_list evs";
paulson@2111
    27
by (REPEAT (resolve_tac [exI,bexI] 1));
paulson@2111
    28
by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS yahalom.YM4) 2);
paulson@2111
    29
by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
paulson@2111
    30
by (ALLGOALS Fast_tac);
paulson@2111
    31
result();
paulson@2111
    32
paulson@2111
    33
paulson@2111
    34
(**** Inductive proofs about yahalom ****)
paulson@2111
    35
paulson@2111
    36
(*Monotonicity*)
paulson@2111
    37
goal thy "!!evs. lost' <= lost ==> yahalom lost' <= yahalom lost";
paulson@2111
    38
by (rtac subsetI 1);
paulson@2111
    39
by (etac yahalom.induct 1);
paulson@2111
    40
by (REPEAT_FIRST
paulson@2111
    41
    (best_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
paulson@2111
    42
                              :: yahalom.intrs))));
paulson@2111
    43
qed "yahalom_mono";
paulson@2111
    44
paulson@2111
    45
paulson@2111
    46
(*Nobody sends themselves messages*)
paulson@2111
    47
goal thy "!!evs. evs: yahalom lost ==> ALL A X. Says A A X ~: set_of_list evs";
paulson@2111
    48
by (etac yahalom.induct 1);
paulson@2111
    49
by (Auto_tac());
paulson@2111
    50
qed_spec_mp "not_Says_to_self";
paulson@2111
    51
Addsimps [not_Says_to_self];
paulson@2111
    52
AddSEs   [not_Says_to_self RSN (2, rev_notE)];
paulson@2111
    53
paulson@2111
    54
paulson@2111
    55
(** For reasoning about the encrypted portion of messages **)
paulson@2111
    56
paulson@2111
    57
(*Lets us treat YM4 using a similar argument as for the Fake case.*)
paulson@2111
    58
goal thy "!!evs. Says S A {|NB, Crypt Y (shrK A), X|} : set_of_list evs ==> \
paulson@2111
    59
\                X : analz (sees lost Spy evs)";
paulson@2111
    60
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
paulson@2111
    61
qed "YM4_analz_sees_Spy";
paulson@2111
    62
paulson@2111
    63
bind_thm ("YM4_parts_sees_Spy",
paulson@2111
    64
          YM4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
paulson@2111
    65
paulson@2155
    66
(*Relates to both YM4 and Oops*)
paulson@2111
    67
goal thy "!!evs. Says S A {|NB, Crypt {|B, K, NA|} (shrK A), X|} \
paulson@2111
    68
\                  : set_of_list evs ==> \
paulson@2111
    69
\                K : parts (sees lost Spy evs)";
paulson@2111
    70
by (fast_tac (!claset addSEs partsEs
paulson@2111
    71
                      addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
paulson@2111
    72
qed "YM4_Key_parts_sees_Spy";
paulson@2111
    73
paulson@2111
    74
(*We instantiate the variable to "lost".  Leaving it as a Var makes proofs
paulson@2111
    75
  harder: the simplifier does less.*)
paulson@2111
    76
val parts_Fake_tac = 
paulson@2111
    77
    forw_inst_tac [("lost","lost")] YM4_parts_sees_Spy 6 THEN
paulson@2111
    78
    forw_inst_tac [("lost","lost")] YM4_Key_parts_sees_Spy 7;
paulson@2111
    79
paulson@2111
    80
(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
paulson@2111
    81
fun parts_induct_tac i = SELECT_GOAL
paulson@2111
    82
    (DETERM (etac yahalom.induct 1 THEN parts_Fake_tac THEN
paulson@2111
    83
	     (*Fake message*)
paulson@2111
    84
	     TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
paulson@2111
    85
					   impOfSubs Fake_parts_insert]
paulson@2111
    86
                                    addss (!simpset)) 2)) THEN
paulson@2111
    87
     (*Base case*)
paulson@2111
    88
     fast_tac (!claset addss (!simpset)) 1 THEN
paulson@2111
    89
     ALLGOALS Asm_simp_tac) i;
paulson@2111
    90
paulson@2111
    91
paulson@2111
    92
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
paulson@2111
    93
    sends messages containing X! **)
paulson@2111
    94
paulson@2111
    95
(*Spy never sees another agent's shared key! (unless it is leaked at start)*)
paulson@2111
    96
goal thy 
paulson@2111
    97
 "!!evs. [| evs : yahalom lost;  A ~: lost |]    \
paulson@2111
    98
\        ==> Key (shrK A) ~: parts (sees lost Spy evs)";
paulson@2111
    99
by (parts_induct_tac 1);
paulson@2111
   100
by (Auto_tac());
paulson@2111
   101
qed "Spy_not_see_shrK";
paulson@2111
   102
paulson@2111
   103
bind_thm ("Spy_not_analz_shrK",
paulson@2111
   104
          [analz_subset_parts, Spy_not_see_shrK] MRS contra_subsetD);
paulson@2111
   105
paulson@2111
   106
Addsimps [Spy_not_see_shrK, Spy_not_analz_shrK];
paulson@2111
   107
paulson@2111
   108
(*We go to some trouble to preserve R in the 3rd and 4th subgoals
paulson@2111
   109
  As usual fast_tac cannot be used because it uses the equalities too soon*)
paulson@2111
   110
val major::prems = 
paulson@2111
   111
goal thy  "[| Key (shrK A) : parts (sees lost Spy evs);       \
paulson@2111
   112
\             evs : yahalom lost;                               \
paulson@2111
   113
\             A:lost ==> R                                  \
paulson@2111
   114
\           |] ==> R";
paulson@2111
   115
by (rtac ccontr 1);
paulson@2111
   116
by (rtac ([major, Spy_not_see_shrK] MRS rev_notE) 1);
paulson@2111
   117
by (swap_res_tac prems 2);
paulson@2111
   118
by (ALLGOALS (fast_tac (!claset addIs prems)));
paulson@2111
   119
qed "Spy_see_shrK_E";
paulson@2111
   120
paulson@2111
   121
bind_thm ("Spy_analz_shrK_E", 
paulson@2111
   122
          analz_subset_parts RS subsetD RS Spy_see_shrK_E);
paulson@2111
   123
paulson@2111
   124
AddSEs [Spy_see_shrK_E, Spy_analz_shrK_E];
paulson@2111
   125
paulson@2111
   126
paulson@2111
   127
(*** Future keys can't be seen or used! ***)
paulson@2111
   128
paulson@2111
   129
(*Nobody can have SEEN keys that will be generated in the future.
paulson@2111
   130
  This has to be proved anew for each protocol description,
paulson@2111
   131
  but should go by similar reasoning every time.  Hardest case is the
paulson@2111
   132
  standard Fake rule.  
paulson@2111
   133
      The Union over C is essential for the induction! *)
paulson@2111
   134
goal thy "!!evs. evs : yahalom lost ==> \
paulson@2111
   135
\                length evs <= length evs' --> \
paulson@2111
   136
\                          Key (newK evs') ~: (UN C. parts (sees lost C evs))";
paulson@2111
   137
by (parts_induct_tac 1);
paulson@2111
   138
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts,
paulson@2111
   139
                                           impOfSubs parts_insert_subset_Un,
paulson@2111
   140
                                           Suc_leD]
paulson@2111
   141
                                    addss (!simpset))));
paulson@2111
   142
val lemma = result();
paulson@2111
   143
paulson@2111
   144
(*Variant needed for the main theorem below*)
paulson@2111
   145
goal thy 
paulson@2111
   146
 "!!evs. [| evs : yahalom lost;  length evs <= length evs' |]    \
paulson@2111
   147
\        ==> Key (newK evs') ~: parts (sees lost C evs)";
paulson@2111
   148
by (fast_tac (!claset addDs [lemma]) 1);
paulson@2111
   149
qed "new_keys_not_seen";
paulson@2111
   150
Addsimps [new_keys_not_seen];
paulson@2111
   151
paulson@2111
   152
(*Another variant: old messages must contain old keys!*)
paulson@2111
   153
goal thy 
paulson@2111
   154
 "!!evs. [| Says A B X : set_of_list evs;  \
paulson@2111
   155
\           Key (newK evt) : parts {X};    \
paulson@2111
   156
\           evs : yahalom lost                 \
paulson@2111
   157
\        |] ==> length evt < length evs";
paulson@2111
   158
by (rtac ccontr 1);
paulson@2111
   159
by (dtac leI 1);
paulson@2111
   160
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
paulson@2111
   161
                      addIs  [impOfSubs parts_mono]) 1);
paulson@2111
   162
qed "Says_imp_old_keys";
paulson@2111
   163
paulson@2111
   164
paulson@2111
   165
(*Nobody can have USED keys that will be generated in the future.
paulson@2111
   166
  ...very like new_keys_not_seen*)
paulson@2111
   167
goal thy "!!evs. evs : yahalom lost ==> \
paulson@2111
   168
\                length evs <= length evs' --> \
paulson@2111
   169
\                newK evs' ~: keysFor (UN C. parts (sees lost C evs))";
paulson@2111
   170
by (parts_induct_tac 1);
paulson@2111
   171
by (dresolve_tac [YM4_Key_parts_sees_Spy] 5);
paulson@2111
   172
(*YM1, YM2 and YM3*)
paulson@2111
   173
by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,3,2]));
paulson@2111
   174
(*Fake and YM4: these messages send unknown (X) components*)
paulson@2111
   175
by (stac insert_commute 2);
paulson@2111
   176
by (Simp_tac 2);
paulson@2111
   177
(*YM4: the only way K could have been used is if it had been seen,
paulson@2111
   178
  contradicting new_keys_not_seen*)
paulson@2111
   179
by (REPEAT
paulson@2111
   180
     (best_tac
paulson@2111
   181
      (!claset addDs [impOfSubs analz_subset_parts,
paulson@2111
   182
                      impOfSubs (analz_subset_parts RS keysFor_mono),
paulson@2111
   183
                      impOfSubs (parts_insert_subset_Un RS keysFor_mono),
paulson@2111
   184
                      Suc_leD]
paulson@2111
   185
               addEs [new_keys_not_seen RSN(2,rev_notE)]
paulson@2111
   186
               addss (!simpset)) 1));
paulson@2111
   187
val lemma = result();
paulson@2111
   188
paulson@2111
   189
goal thy 
paulson@2111
   190
 "!!evs. [| evs : yahalom lost;  length evs <= length evs' |]    \
paulson@2111
   191
\        ==> newK evs' ~: keysFor (parts (sees lost C evs))";
paulson@2111
   192
by (fast_tac (!claset addSDs [lemma] addss (!simpset)) 1);
paulson@2111
   193
qed "new_keys_not_used";
paulson@2111
   194
paulson@2111
   195
bind_thm ("new_keys_not_analzd",
paulson@2111
   196
          [analz_subset_parts RS keysFor_mono,
paulson@2111
   197
           new_keys_not_used] MRS contra_subsetD);
paulson@2111
   198
paulson@2111
   199
Addsimps [new_keys_not_used, new_keys_not_analzd];
paulson@2111
   200
paulson@2111
   201
paulson@2155
   202
(*Describes the form of K when the Server sends this message.  Useful for
paulson@2155
   203
  Oops as well as main secrecy property.*)
paulson@2111
   204
goal thy 
paulson@2155
   205
 "!!evs. [| Says Server A {|NB', Crypt {|Agent B, K, NA|} (shrK A), X|} \
paulson@2155
   206
\            : set_of_list evs;                                         \
paulson@2155
   207
\           evs : yahalom lost |]                                       \
paulson@2155
   208
\        ==> (EX evt: yahalom lost. K = Key(newK evt)) & A ~= B";
paulson@2155
   209
by (etac rev_mp 1);
paulson@2155
   210
by (etac yahalom.induct 1);
paulson@2155
   211
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
paulson@2155
   212
qed "Says_Server_message_form";
paulson@2111
   213
paulson@2111
   214
paulson@2111
   215
(*For proofs involving analz.  We again instantiate the variable to "lost".*)
paulson@2111
   216
val analz_Fake_tac = 
paulson@2111
   217
    dres_inst_tac [("lost","lost")] YM4_analz_sees_Spy 6 THEN
paulson@2155
   218
    forw_inst_tac [("lost","lost")] Says_Server_message_form 7 THEN
paulson@2155
   219
    assume_tac 7 THEN Full_simp_tac 7 THEN
paulson@2155
   220
    REPEAT ((eresolve_tac [bexE,conjE] ORELSE' hyp_subst_tac) 7);
paulson@2111
   221
paulson@2111
   222
paulson@2111
   223
(****
paulson@2111
   224
 The following is to prove theorems of the form
paulson@2111
   225
paulson@2111
   226
          Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==>
paulson@2111
   227
          Key K : analz (sees lost Spy evs)
paulson@2111
   228
paulson@2111
   229
 A more general formula must be proved inductively.
paulson@2111
   230
paulson@2111
   231
****)
paulson@2111
   232
paulson@2111
   233
(** Session keys are not used to encrypt other session keys **)
paulson@2111
   234
paulson@2111
   235
goal thy  
paulson@2111
   236
 "!!evs. evs : yahalom lost ==> \
paulson@2111
   237
\  ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
paulson@2111
   238
\           (K : newK``E | Key K : analz (sees lost Spy evs))";
paulson@2111
   239
by (etac yahalom.induct 1);
paulson@2111
   240
by analz_Fake_tac;
paulson@2111
   241
by (REPEAT_FIRST (resolve_tac [allI, analz_image_newK_lemma]));
paulson@2111
   242
by (ALLGOALS  (*Takes 26 secs*)
paulson@2111
   243
    (asm_simp_tac 
paulson@2111
   244
     (!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
paulson@2111
   245
                         @ pushes)
paulson@2111
   246
               setloop split_tac [expand_if])));
paulson@2155
   247
(*YM4, Fake*) 
paulson@2155
   248
by (EVERY (map spy_analz_tac [4, 2]));
paulson@2155
   249
(*Oops, YM3, Base*)
paulson@2111
   250
by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
paulson@2111
   251
qed_spec_mp "analz_image_newK";
paulson@2111
   252
paulson@2111
   253
goal thy
paulson@2111
   254
 "!!evs. evs : yahalom lost ==>                               \
paulson@2111
   255
\        Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \
paulson@2111
   256
\        (K = newK evt | Key K : analz (sees lost Spy evs))";
paulson@2111
   257
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK, 
paulson@2111
   258
                                   insert_Key_singleton]) 1);
paulson@2111
   259
by (Fast_tac 1);
paulson@2111
   260
qed "analz_insert_Key_newK";
paulson@2111
   261
paulson@2111
   262
paulson@2111
   263
(*** The Key K uniquely identifies the Server's  message. **)
paulson@2111
   264
paulson@2111
   265
goal thy 
paulson@2111
   266
 "!!evs. evs : yahalom lost ==>                                     \
paulson@2155
   267
\      EX A' B' NA' NB' X'. ALL A B NA NB X.                        \
paulson@2111
   268
\          Says Server A                                            \
paulson@2155
   269
\           {|NB, Crypt {|Agent B, Key K, NA|} (shrK A), X|}        \
paulson@2155
   270
\          : set_of_list evs --> A=A' & B=B' & NA=NA' & NB=NB' & X=X'";
paulson@2111
   271
by (etac yahalom.induct 1);
paulson@2111
   272
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
paulson@2111
   273
by (Step_tac 1);
paulson@2111
   274
(*Remaining case: YM3*)
paulson@2111
   275
by (expand_case_tac "K = ?y" 1);
paulson@2111
   276
by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
paulson@2111
   277
(*...we assume X is a very new message, and handle this case by contradiction*)
paulson@2111
   278
by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
paulson@2111
   279
                      delrules [conjI]    (*prevent split-up into 4 subgoals*)
paulson@2111
   280
                      addss (!simpset addsimps [parts_insertI])) 1);
paulson@2111
   281
val lemma = result();
paulson@2111
   282
paulson@2111
   283
goal thy 
paulson@2111
   284
"!!evs. [| Says Server A                                            \
paulson@2155
   285
\           {|NB, Crypt {|Agent B, Key K, NA|} (shrK A), X|}        \
paulson@2111
   286
\           : set_of_list evs;                                      \
paulson@2111
   287
\          Says Server A'                                           \
paulson@2155
   288
\           {|NB', Crypt {|Agent B', Key K, NA'|} (shrK A'), X'|}   \
paulson@2111
   289
\           : set_of_list evs;                                      \
paulson@2111
   290
\          evs : yahalom lost |]                                    \
paulson@2111
   291
\       ==> A=A' & B=B' & NA=NA' & NB=NB'";
paulson@2111
   292
by (dtac lemma 1);
paulson@2111
   293
by (REPEAT (etac exE 1));
paulson@2111
   294
(*Duplicate the assumption*)
paulson@2111
   295
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
paulson@2111
   296
by (fast_tac (!claset addSDs [spec]) 1);
paulson@2111
   297
qed "unique_session_keys";
paulson@2111
   298
paulson@2111
   299
paulson@2111
   300
(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
paulson@2111
   301
paulson@2111
   302
goal thy 
paulson@2155
   303
 "!!evs. [| A ~: lost;  B ~: lost;  A ~= B;                          \
paulson@2155
   304
\           evs : yahalom lost |]            \
paulson@2111
   305
\        ==> Says Server A                                           \
paulson@2111
   306
\              {|NB, Crypt {|Agent B, Key K, NA|} (shrK A),           \
paulson@2155
   307
\                    Crypt {|NB, Key K, Agent A|} (shrK B)|}          \
paulson@2111
   308
\             : set_of_list evs -->                               \
paulson@2155
   309
\            Says A Spy {|NA, NB, Key K|} ~: set_of_list evs -->  \
paulson@2111
   310
\            Key K ~: analz (sees lost Spy evs)";
paulson@2111
   311
by (etac yahalom.induct 1);
paulson@2111
   312
by analz_Fake_tac;
paulson@2111
   313
by (ALLGOALS
paulson@2111
   314
    (asm_simp_tac 
paulson@2111
   315
     (!simpset addsimps ([analz_subset_parts RS contra_subsetD,
paulson@2111
   316
                          analz_insert_Key_newK] @ pushes)
paulson@2111
   317
               setloop split_tac [expand_if])));
paulson@2111
   318
(*YM3*)
paulson@2111
   319
by (fast_tac (!claset addIs [parts_insertI]
paulson@2111
   320
                      addEs [Says_imp_old_keys RS less_irrefl]
paulson@2111
   321
                      addss (!simpset)) 2);
paulson@2155
   322
(*OR4, Fake*) 
paulson@2111
   323
by (REPEAT_FIRST (resolve_tac [conjI, impI] ORELSE' spy_analz_tac));
paulson@2155
   324
(*Oops*)
paulson@2111
   325
by (fast_tac (!claset delrules [disjE] 
paulson@2155
   326
		      addDs [unique_session_keys]
paulson@2111
   327
	              addss (!simpset)) 1);
paulson@2111
   328
val lemma = result() RS mp RS mp RSN(2,rev_notE);
paulson@2111
   329
paulson@2111
   330
paulson@2111
   331
(*Final version: Server's message in the most abstract form*)
paulson@2111
   332
goal thy 
paulson@2111
   333
 "!!evs. [| Says Server A                                         \
paulson@2111
   334
\              {|NB, Crypt {|Agent B, K, NA|} (shrK A),           \
paulson@2155
   335
\                    Crypt {|NB, K, Agent A|} (shrK B)|}          \
paulson@2111
   336
\           : set_of_list evs;                                    \
paulson@2155
   337
\           Says A Spy {|NA, NB, K|} ~: set_of_list evs;          \
paulson@2155
   338
\           A ~: lost;  B ~: lost;  evs : yahalom lost |]         \
paulson@2155
   339
\        ==> K ~: analz (sees lost Spy evs)";
paulson@2111
   340
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
paulson@2111
   341
by (fast_tac (!claset addSEs [lemma]) 1);
paulson@2111
   342
qed "Spy_not_see_encrypted_key";
paulson@2111
   343
paulson@2111
   344
paulson@2111
   345
goal thy 
paulson@2155
   346
 "!!evs. [| C ~: {A,B,Server};                                    \
paulson@2111
   347
\           Says Server A                                         \
paulson@2111
   348
\              {|NB, Crypt {|Agent B, K, NA|} (shrK A),           \
paulson@2155
   349
\                    Crypt {|NB, K, Agent A|} (shrK B)|}          \
paulson@2111
   350
\           : set_of_list evs;                                    \
paulson@2155
   351
\           Says A Spy {|NA, NB, K|} ~: set_of_list evs;          \
paulson@2155
   352
\           A ~: lost;  B ~: lost;  evs : yahalom lost |]         \
paulson@2155
   353
\        ==> K ~: analz (sees lost C evs)";
paulson@2111
   354
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
paulson@2111
   355
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
paulson@2111
   356
by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
paulson@2111
   357
by (REPEAT_FIRST (fast_tac (!claset addIs [yahalom_mono RS subsetD])));
paulson@2111
   358
qed "Agent_not_see_encrypted_key";
paulson@2111
   359
paulson@2111
   360
paulson@2155
   361
(*** Security Guarantees for A and B ***)
paulson@2155
   362
paulson@2155
   363
(*If the encrypted message appears then it originated with the Server.*)
paulson@2155
   364
goal thy
paulson@2155
   365
 "!!evs. [| Crypt {|Agent B, Key K, NA|} (shrK A)                \
paulson@2155
   366
\            : parts (sees lost Spy evs);                              \
paulson@2155
   367
\           A ~: lost;  evs : yahalom lost |]                          \
paulson@2155
   368
\         ==> EX NB. Says Server A                                     \
paulson@2155
   369
\                      {|NB, Crypt {|Agent B, Key K, NA|} (shrK A),    \
paulson@2155
   370
\                            Crypt {|NB, Key K, Agent A|} (shrK B)|}   \
paulson@2155
   371
\                    : set_of_list evs";
paulson@2155
   372
by (etac rev_mp 1);
paulson@2155
   373
by (parts_induct_tac 1);
paulson@2155
   374
(*The nested conjunctions are entirely useless*)
paulson@2155
   375
by (REPEAT_FIRST (rtac conjI ORELSE' fast_tac (!claset delrules [conjI])));
paulson@2155
   376
qed "A_trust_YM3";
paulson@2155
   377
paulson@2111
   378
paulson@2111
   379
(*B knows, by the first part of A's message, that the Server distributed 
paulson@2155
   380
  the key for A and B. *)
paulson@2111
   381
goal thy 
paulson@2155
   382
 "!!evs. [| Crypt {|Nonce NB, Key K, Agent A|} (shrK B)              \
paulson@2155
   383
\            : parts (sees lost Spy evs);                            \
paulson@2155
   384
\           B ~: lost;  evs : yahalom lost |]                        \
paulson@2111
   385
\        ==> EX NA. Says Server A                                    \
paulson@2155
   386
\                    {|Nonce NB,                                     \
paulson@2155
   387
\                      Crypt {|Agent B, Key K, Nonce NA|} (shrK A),  \
paulson@2155
   388
\                      Crypt {|Nonce NB, Key K, Agent A|} (shrK B)|} \
paulson@2111
   389
\                       : set_of_list evs";
paulson@2111
   390
by (etac rev_mp 1);
paulson@2111
   391
by (parts_induct_tac 1);
paulson@2111
   392
(*YM3*)
paulson@2111
   393
by (Fast_tac 1);
paulson@2111
   394
qed "B_trusts_YM4_shrK";
paulson@2111
   395
paulson@2111
   396
(*With this variant we don't bother to use the 2nd part of YM4 at all, since
paulson@2111
   397
  Nonce NB is available in the first part.  However the 2nd part does assure B
paulson@2111
   398
  of A's existence.*)
paulson@2111
   399
paulson@2155
   400
(*What can B deduce from receipt of YM4?  Stronger and simpler than Yahalom
paulson@2155
   401
  because we do not have to show that NB is secret. *)
paulson@2111
   402
goal thy 
paulson@2155
   403
 "!!evs. [| Says A' B {|Crypt {|Nonce NB, Key K, Agent A|} (shrK B),    \
paulson@2111
   404
\                       Crypt (Nonce NB) K|} : set_of_list evs;         \
paulson@2111
   405
\           A ~: lost;  B ~: lost;  evs : yahalom lost |]               \
paulson@2111
   406
\        ==> EX NA. Says Server A                                       \
paulson@2155
   407
\                    {|Nonce NB,                                        \
paulson@2155
   408
\                      Crypt {|Agent B, Key K, Nonce NA|} (shrK A),     \
paulson@2155
   409
\                      Crypt {|Nonce NB, Key K, Agent A|} (shrK B)|}    \
paulson@2111
   410
\                   : set_of_list evs";
paulson@2111
   411
be (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1;
paulson@2111
   412
by (fast_tac (!claset addSDs [B_trusts_YM4_shrK]) 1);
paulson@2111
   413
qed "B_trust_YM4";