src/HOL/Auth/Public.ML
author nipkow
Tue Apr 08 10:48:42 1997 +0200 (1997-04-08)
changeset 2919 953a47dc0519
parent 2637 e9b203f854ae
child 3121 cbb6c0c1c58a
permissions -rw-r--r--
Dep. on Provers/nat_transitive
     1 (*  Title:      HOL/Auth/Public
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1996  University of Cambridge
     5 
     6 Theory of Public Keys (common to all symmetric-key protocols)
     7 
     8 Server keys; initial states of agents; new nonces and keys; function "sees" 
     9 *)
    10 
    11 
    12 open Public;
    13 
    14 (*Injectiveness of new nonces*)
    15 AddIffs [inj_newN RS inj_eq];
    16 
    17 (*Holds because Friend is injective: thus cannot prove for all f*)
    18 goal thy "(Friend x : Friend``A) = (x:A)";
    19 by (Auto_tac());
    20 qed "Friend_image_eq";
    21 Addsimps [Friend_image_eq];
    22 
    23 Addsimps [Un_insert_left, Un_insert_right];
    24 
    25 (*By default only o_apply is built-in.  But in the presence of eta-expansion
    26   this means that some terms displayed as (f o g) will be rewritten, and others
    27   will not!*)
    28 Addsimps [o_def];
    29 
    30 goalw thy [keysFor_def] "keysFor (parts (initState lost C)) = {}";
    31 by (agent.induct_tac "C" 1);
    32 by (auto_tac (!claset addIs [range_eqI], !simpset));
    33 qed "keysFor_parts_initState";
    34 Addsimps [keysFor_parts_initState];
    35 
    36 goalw thy [keysFor_def] "keysFor (Key``E) = {}";
    37 by (Auto_tac ());
    38 qed "keysFor_image_Key";
    39 Addsimps [keysFor_image_Key];
    40 
    41 
    42 (*** Function "sees" ***)
    43 
    44 goal thy
    45     "!!evs. lost' <= lost ==> sees lost' A evs <= sees lost A evs";
    46 by (list.induct_tac "evs" 1);
    47 by (agent.induct_tac "A" 1);
    48 by (event.induct_tac "a" 2);
    49 by (Auto_tac ());
    50 qed "sees_mono";
    51 
    52 (*** Basic properties of pubK & priK ***)
    53 
    54 AddIffs [inj_pubK RS inj_eq];
    55 Addsimps [priK_neq_pubK, priK_neq_pubK RS not_sym];
    56 
    57 (*Agents see their own private keys!*)
    58 goal thy "A ~= Spy --> Key (priK A) : sees lost A evs";
    59 by (list.induct_tac "evs" 1);
    60 by (agent.induct_tac "A" 1);
    61 by (Auto_tac ());
    62 qed_spec_mp "sees_own_priK";
    63 
    64 (*All public keys are visible*)
    65 goal thy "Key (pubK A) : sees lost A evs";
    66 by (list.induct_tac "evs" 1);
    67 by (agent.induct_tac "A" 1);
    68 by (Auto_tac ());
    69 qed_spec_mp "sees_pubK";
    70 
    71 (*Spy sees private keys of lost agents!*)
    72 goal thy "!!A. A: lost ==> Key (priK A) : sees lost Spy evs";
    73 by (list.induct_tac "evs" 1);
    74 by (Auto_tac());
    75 qed "Spy_sees_lost";
    76 
    77 AddIffs [sees_pubK];
    78 AddSIs  [sees_own_priK, Spy_sees_lost];
    79 
    80 (*Added for Yahalom/lost_tac*)
    81 goal thy "!!A. [| Crypt (pubK A) X : analz (sees lost Spy evs);  A: lost |] \
    82 \              ==> X : analz (sees lost Spy evs)";
    83 by (fast_tac (!claset addSDs [analz.Decrypt] addss (!simpset)) 1);
    84 qed "Crypt_Spy_analz_lost";
    85 
    86 (** Specialized rewrite rules for (sees lost A (Says...#evs)) **)
    87 
    88 goal thy "sees lost B (Says A B X # evs) = insert X (sees lost B evs)";
    89 by (Simp_tac 1);
    90 qed "sees_own";
    91 
    92 goal thy "!!A. Server ~= B ==> \
    93 \          sees lost Server (Says A B X # evs) = sees lost Server evs";
    94 by (Asm_simp_tac 1);
    95 qed "sees_Server";
    96 
    97 goal thy "!!A. Friend i ~= B ==> \
    98 \          sees lost (Friend i) (Says A B X # evs) = sees lost (Friend i) evs";
    99 by (Asm_simp_tac 1);
   100 qed "sees_Friend";
   101 
   102 goal thy "sees lost Spy (Says A B X # evs) = insert X (sees lost Spy evs)";
   103 by (Simp_tac 1);
   104 qed "sees_Spy";
   105 
   106 goal thy "sees lost A (Says A' B X # evs) <= insert X (sees lost A evs)";
   107 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
   108 by (Fast_tac 1);
   109 qed "sees_Says_subset_insert";
   110 
   111 goal thy "sees lost A evs <= sees lost A (Says A' B X # evs)";
   112 by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
   113 by (Fast_tac 1);
   114 qed "sees_subset_sees_Says";
   115 
   116 (*Pushing Unions into parts.  One of the agents A is B, and thus sees Y.
   117   Once used to prove new_keys_not_seen; now obsolete.*)
   118 goal thy "(UN A. parts (sees lost A (Says B C Y # evs))) = \
   119 \         parts {Y} Un (UN A. parts (sees lost A evs))";
   120 by (Step_tac 1);
   121 by (etac rev_mp 1);     (*split_tac does not work on assumptions*)
   122 by (ALLGOALS
   123     (fast_tac (!claset addss (!simpset addsimps [parts_Un, sees_Cons] 
   124                        setloop split_tac [expand_if]))));
   125 qed "UN_parts_sees_Says";
   126 
   127 goal thy "Says A B X : set_of_list evs --> X : sees lost Spy evs";
   128 by (list.induct_tac "evs" 1);
   129 by (Auto_tac ());
   130 qed_spec_mp "Says_imp_sees_Spy";
   131 
   132 (*Use with addSEs to derive contradictions from old Says events containing
   133   items known to be fresh*)
   134 val sees_Spy_partsEs = make_elim (Says_imp_sees_Spy RS parts.Inj):: partsEs;
   135 
   136 goal thy  
   137  "!!evs. [| Says A B (Crypt (pubK C) X) : set_of_list evs;  C : lost |] \
   138 \        ==> X : analz (sees lost Spy evs)";
   139 by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
   140                       addss (!simpset)) 1);
   141 qed "Says_Crypt_lost";
   142 
   143 goal thy  
   144  "!!evs. [| Says A B (Crypt (pubK C) X) : set_of_list evs;        \
   145 \           X ~: analz (sees lost Spy evs) |]                     \
   146 \        ==> C ~: lost";
   147 by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
   148                       addss (!simpset)) 1);
   149 qed "Says_Crypt_not_lost";
   150 
   151 Addsimps [sees_own, sees_Server, sees_Friend, sees_Spy];
   152 Delsimps [sees_Cons];   (**** NOTE REMOVAL -- laws above are cleaner ****)
   153 
   154 
   155 (*** Fresh nonces ***)
   156 
   157 goal thy "Nonce N ~: parts (initState lost B)";
   158 by (agent.induct_tac "B" 1);
   159 by (Auto_tac ());
   160 qed "Nonce_notin_initState";
   161 
   162 AddIffs [Nonce_notin_initState];
   163 
   164 goalw thy [used_def] "!!X. X: parts (sees lost B evs) ==> X: used evs";
   165 by (etac (impOfSubs parts_mono) 1);
   166 by (Fast_tac 1);
   167 qed "usedI";
   168 
   169 AddIs [usedI];
   170 
   171 (** A supply of fresh nonces for possibility theorems. **)
   172 
   173 goalw thy [used_def] "EX N. ALL n. N<=n --> Nonce n ~: used evs";
   174 by (list.induct_tac "evs" 1);
   175 by (res_inst_tac [("x","0")] exI 1);
   176 by (Step_tac 1);
   177 by (Full_simp_tac 1);
   178 (*Inductive step*)
   179 by (event.induct_tac "a" 1);
   180 by (full_simp_tac (!simpset addsimps [UN_parts_sees_Says]) 1);
   181 by (msg.induct_tac "msg" 1);
   182 by (ALLGOALS (asm_simp_tac (!simpset addsimps [exI, parts_insert2])));
   183 by (Step_tac 1);
   184 (*MPair case*)
   185 by (res_inst_tac [("x","Na+Nb")] exI 2);
   186 by (fast_tac (!claset addSEs [add_leE]) 2);
   187 (*Nonce case*)
   188 by (res_inst_tac [("x","N + Suc nat")] exI 1);
   189 by (fast_tac (!claset addSEs [add_leE] addaltern trans_tac) 1);
   190 val lemma = result();
   191 
   192 goal thy "Nonce (@ N. Nonce N ~: used evs) ~: used evs";
   193 by (rtac (lemma RS exE) 1);
   194 by (rtac selectI 1);
   195 by (Fast_tac 1);
   196 qed "Nonce_supply";
   197 
   198 (*Tactic for possibility theorems*)
   199 val possibility_tac =
   200     REPEAT 
   201     (ALLGOALS (simp_tac (!simpset setSolver safe_solver))
   202      THEN
   203      REPEAT_FIRST (eq_assume_tac ORELSE' 
   204                    resolve_tac [refl, conjI, Nonce_supply]));
   205 
   206 (** Power of the Spy **)
   207 
   208 (*The Spy can see more than anybody else, except for their initial state*)
   209 goal thy "sees lost A evs <= initState lost A Un sees lost Spy evs";
   210 by (list.induct_tac "evs" 1);
   211 by (event.induct_tac "a" 2);
   212 by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD] 
   213                                 addss (!simpset))));
   214 qed "sees_agent_subset_sees_Spy";
   215 
   216 (*The Spy can see more than anybody else who's lost their key!*)
   217 goal thy "A: lost --> A ~= Server --> sees lost A evs <= sees lost Spy evs";
   218 by (list.induct_tac "evs" 1);
   219 by (event.induct_tac "a" 2);
   220 by (agent.induct_tac "A" 1);
   221 by (auto_tac (!claset addDs [sees_Says_subset_insert RS subsetD], (!simpset)));
   222 qed_spec_mp "sees_lost_agent_subset_sees_Spy";
   223 
   224 
   225 (** Simplifying   parts (insert X (sees lost A evs))
   226       = parts {X} Un parts (sees lost A evs) -- since general case loops*)
   227 
   228 val parts_insert_sees = 
   229     parts_insert |> read_instantiate_sg (sign_of thy)
   230                                         [("H", "sees lost A evs")]
   231                  |> standard;