src/HOL/Auth/Shared.thy
author webertj
Mon Mar 07 19:30:53 2005 +0100 (2005-03-07)
changeset 15584 3478bb4f93ff
parent 15032 02aed07e01bf
child 16417 9bc16273c2d4
permissions -rw-r--r--
refute_params: default value itself=1 added (for type classes)
     1 (*  Title:      HOL/Auth/Shared
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Copyright   1996  University of Cambridge
     5 
     6 Theory of Shared Keys (common to all symmetric-key protocols)
     7 
     8 Shared, long-term keys; initial states of agents
     9 *)
    10 
    11 theory Shared = Event:
    12 
    13 consts
    14   shrK    :: "agent => key"  (*symmetric keys*);
    15 
    16 specification (shrK)
    17   inj_shrK: "inj shrK"
    18   --{*No two agents have the same long-term key*}
    19    apply (rule exI [of _ "agent_case 0 (\<lambda>n. n + 2) 1"]) 
    20    apply (simp add: inj_on_def split: agent.split) 
    21    done
    22 
    23 text{*All keys are symmetric*}
    24 
    25 defs  all_symmetric_def: "all_symmetric == True"
    26 
    27 lemma isSym_keys: "K \<in> symKeys"	
    28 by (simp add: symKeys_def all_symmetric_def invKey_symmetric) 
    29 
    30 text{*Server knows all long-term keys; other agents know only their own*}
    31 primrec
    32   initState_Server:  "initState Server     = Key ` range shrK"
    33   initState_Friend:  "initState (Friend i) = {Key (shrK (Friend i))}"
    34   initState_Spy:     "initState Spy        = Key`shrK`bad"
    35 
    36 
    37 subsection{*Basic properties of shrK*}
    38 
    39 (*Injectiveness: Agents' long-term keys are distinct.*)
    40 declare inj_shrK [THEN inj_eq, iff]
    41 
    42 lemma invKey_K [simp]: "invKey K = K"
    43 apply (insert isSym_keys)
    44 apply (simp add: symKeys_def) 
    45 done
    46 
    47 
    48 lemma analz_Decrypt' [dest]:
    49      "[| Crypt K X \<in> analz H;  Key K  \<in> analz H |] ==> X \<in> analz H"
    50 by auto
    51 
    52 text{*Now cancel the @{text dest} attribute given to
    53  @{text analz.Decrypt} in its declaration.*}
    54 declare analz.Decrypt [rule del]
    55 
    56 text{*Rewrites should not refer to  @{term "initState(Friend i)"} because
    57   that expression is not in normal form.*}
    58 
    59 lemma keysFor_parts_initState [simp]: "keysFor (parts (initState C)) = {}"
    60 apply (unfold keysFor_def)
    61 apply (induct_tac "C", auto)
    62 done
    63 
    64 (*Specialized to shared-key model: no @{term invKey}*)
    65 lemma keysFor_parts_insert:
    66      "[| K \<in> keysFor (parts (insert X G));  X \<in> synth (analz H) |]
    67       ==> K \<in> keysFor (parts (G \<union> H)) | Key K \<in> parts H";
    68 by (force dest: Event.keysFor_parts_insert)  
    69 
    70 lemma Crypt_imp_keysFor: "Crypt K X \<in> H ==> K \<in> keysFor H"
    71 by (drule Crypt_imp_invKey_keysFor, simp)
    72 
    73 
    74 subsection{*Function "knows"*}
    75 
    76 (*Spy sees shared keys of agents!*)
    77 lemma Spy_knows_Spy_bad [intro!]: "A: bad ==> Key (shrK A) \<in> knows Spy evs"
    78 apply (induct_tac "evs")
    79 apply (simp_all (no_asm_simp) add: imageI knows_Cons split add: event.split)
    80 done
    81 
    82 (*For case analysis on whether or not an agent is compromised*)
    83 lemma Crypt_Spy_analz_bad: "[| Crypt (shrK A) X \<in> analz (knows Spy evs);  A: bad |]  
    84       ==> X \<in> analz (knows Spy evs)"
    85 apply (force dest!: analz.Decrypt)
    86 done
    87 
    88 
    89 (** Fresh keys never clash with long-term shared keys **)
    90 
    91 (*Agents see their own shared keys!*)
    92 lemma shrK_in_initState [iff]: "Key (shrK A) \<in> initState A"
    93 by (induct_tac "A", auto)
    94 
    95 lemma shrK_in_used [iff]: "Key (shrK A) \<in> used evs"
    96 by (rule initState_into_used, blast)
    97 
    98 (*Used in parts_induct_tac and analz_Fake_tac to distinguish session keys
    99   from long-term shared keys*)
   100 lemma Key_not_used [simp]: "Key K \<notin> used evs ==> K \<notin> range shrK"
   101 by blast
   102 
   103 lemma shrK_neq [simp]: "Key K \<notin> used evs ==> shrK B \<noteq> K"
   104 by blast
   105 
   106 declare shrK_neq [THEN not_sym, simp]
   107 
   108 
   109 subsection{*Fresh nonces*}
   110 
   111 lemma Nonce_notin_initState [iff]: "Nonce N \<notin> parts (initState B)"
   112 by (induct_tac "B", auto)
   113 
   114 lemma Nonce_notin_used_empty [simp]: "Nonce N \<notin> used []"
   115 apply (simp (no_asm) add: used_Nil)
   116 done
   117 
   118 
   119 subsection{*Supply fresh nonces for possibility theorems.*}
   120 
   121 (*In any trace, there is an upper bound N on the greatest nonce in use.*)
   122 lemma Nonce_supply_lemma: "\<exists>N. ALL n. N<=n --> Nonce n \<notin> used evs"
   123 apply (induct_tac "evs")
   124 apply (rule_tac x = 0 in exI)
   125 apply (simp_all (no_asm_simp) add: used_Cons split add: event.split)
   126 apply safe
   127 apply (rule msg_Nonce_supply [THEN exE], blast elim!: add_leE)+
   128 done
   129 
   130 lemma Nonce_supply1: "\<exists>N. Nonce N \<notin> used evs"
   131 by (rule Nonce_supply_lemma [THEN exE], blast)
   132 
   133 lemma Nonce_supply2: "\<exists>N N'. Nonce N \<notin> used evs & Nonce N' \<notin> used evs' & N \<noteq> N'"
   134 apply (cut_tac evs = evs in Nonce_supply_lemma)
   135 apply (cut_tac evs = "evs'" in Nonce_supply_lemma, clarify)
   136 apply (rule_tac x = N in exI)
   137 apply (rule_tac x = "Suc (N+Na)" in exI)
   138 apply (simp (no_asm_simp) add: less_not_refl3 le_add1 le_add2 less_Suc_eq_le)
   139 done
   140 
   141 lemma Nonce_supply3: "\<exists>N N' N''. Nonce N \<notin> used evs & Nonce N' \<notin> used evs' &  
   142                     Nonce N'' \<notin> used evs'' & N \<noteq> N' & N' \<noteq> N'' & N \<noteq> N''"
   143 apply (cut_tac evs = evs in Nonce_supply_lemma)
   144 apply (cut_tac evs = "evs'" in Nonce_supply_lemma)
   145 apply (cut_tac evs = "evs''" in Nonce_supply_lemma, clarify)
   146 apply (rule_tac x = N in exI)
   147 apply (rule_tac x = "Suc (N+Na)" in exI)
   148 apply (rule_tac x = "Suc (Suc (N+Na+Nb))" in exI)
   149 apply (simp (no_asm_simp) add: less_not_refl3 le_add1 le_add2 less_Suc_eq_le)
   150 done
   151 
   152 lemma Nonce_supply: "Nonce (@ N. Nonce N \<notin> used evs) \<notin> used evs"
   153 apply (rule Nonce_supply_lemma [THEN exE])
   154 apply (rule someI, blast)
   155 done
   156 
   157 text{*Unlike the corresponding property of nonces, we cannot prove
   158     @{term "finite KK ==> \<exists>K. K \<notin> KK & Key K \<notin> used evs"}.
   159     We have infinitely many agents and there is nothing to stop their
   160     long-term keys from exhausting all the natural numbers.  Instead,
   161     possibility theorems must assume the existence of a few keys.*}
   162 
   163 
   164 subsection{*Tactics for possibility theorems*}
   165 
   166 ML
   167 {*
   168 val inj_shrK      = thm "inj_shrK";
   169 val isSym_keys    = thm "isSym_keys";
   170 val Nonce_supply = thm "Nonce_supply";
   171 val invKey_K = thm "invKey_K";
   172 val analz_Decrypt' = thm "analz_Decrypt'";
   173 val keysFor_parts_initState = thm "keysFor_parts_initState";
   174 val keysFor_parts_insert = thm "keysFor_parts_insert";
   175 val Crypt_imp_keysFor = thm "Crypt_imp_keysFor";
   176 val Spy_knows_Spy_bad = thm "Spy_knows_Spy_bad";
   177 val Crypt_Spy_analz_bad = thm "Crypt_Spy_analz_bad";
   178 val shrK_in_initState = thm "shrK_in_initState";
   179 val shrK_in_used = thm "shrK_in_used";
   180 val Key_not_used = thm "Key_not_used";
   181 val shrK_neq = thm "shrK_neq";
   182 val Nonce_notin_initState = thm "Nonce_notin_initState";
   183 val Nonce_notin_used_empty = thm "Nonce_notin_used_empty";
   184 val Nonce_supply_lemma = thm "Nonce_supply_lemma";
   185 val Nonce_supply1 = thm "Nonce_supply1";
   186 val Nonce_supply2 = thm "Nonce_supply2";
   187 val Nonce_supply3 = thm "Nonce_supply3";
   188 val Nonce_supply = thm "Nonce_supply";
   189 *}
   190 
   191 
   192 ML
   193 {*
   194 (*Omitting used_Says makes the tactic much faster: it leaves expressions
   195     such as  Nonce ?N \<notin> used evs that match Nonce_supply*)
   196 fun gen_possibility_tac ss state = state |>
   197    (REPEAT 
   198     (ALLGOALS (simp_tac (ss delsimps [used_Says, used_Notes, used_Gets] 
   199                          setSolver safe_solver))
   200      THEN
   201      REPEAT_FIRST (eq_assume_tac ORELSE' 
   202                    resolve_tac [refl, conjI, Nonce_supply])))
   203 
   204 (*Tactic for possibility theorems (ML script version)*)
   205 fun possibility_tac state = gen_possibility_tac (simpset()) state
   206 
   207 (*For harder protocols (such as Recur) where we have to set up some
   208   nonces and keys initially*)
   209 fun basic_possibility_tac st = st |>
   210     REPEAT 
   211     (ALLGOALS (asm_simp_tac (simpset() setSolver safe_solver))
   212      THEN
   213      REPEAT_FIRST (resolve_tac [refl, conjI]))
   214 *}
   215 
   216 subsection{*Specialized Rewriting for Theorems About @{term analz} and Image*}
   217 
   218 lemma subset_Compl_range: "A <= - (range shrK) ==> shrK x \<notin> A"
   219 by blast
   220 
   221 lemma insert_Key_singleton: "insert (Key K) H = Key ` {K} \<union> H"
   222 by blast
   223 
   224 lemma insert_Key_image: "insert (Key K) (Key`KK \<union> C) = Key`(insert K KK) \<union> C"
   225 by blast
   226 
   227 (** Reverse the normal simplification of "image" to build up (not break down)
   228     the set of keys.  Use analz_insert_eq with (Un_upper2 RS analz_mono) to
   229     erase occurrences of forwarded message components (X). **)
   230 
   231 lemmas analz_image_freshK_simps =
   232        simp_thms mem_simps --{*these two allow its use with @{text "only:"}*}
   233        disj_comms 
   234        image_insert [THEN sym] image_Un [THEN sym] empty_subsetI insert_subset
   235        analz_insert_eq Un_upper2 [THEN analz_mono, THEN [2] rev_subsetD]
   236        insert_Key_singleton subset_Compl_range
   237        Key_not_used insert_Key_image Un_assoc [THEN sym]
   238 
   239 (*Lemma for the trivial direction of the if-and-only-if*)
   240 lemma analz_image_freshK_lemma:
   241      "(Key K \<in> analz (Key`nE \<union> H)) --> (K \<in> nE | Key K \<in> analz H)  ==>  
   242          (Key K \<in> analz (Key`nE \<union> H)) = (K \<in> nE | Key K \<in> analz H)"
   243 by (blast intro: analz_mono [THEN [2] rev_subsetD])
   244 
   245 ML
   246 {*
   247 val analz_image_freshK_lemma = thm "analz_image_freshK_lemma";
   248 
   249 val analz_image_freshK_ss = 
   250      simpset() delsimps [image_insert, image_Un]
   251 	       delsimps [imp_disjL]    (*reduces blow-up*)
   252 	       addsimps thms "analz_image_freshK_simps"
   253 *}
   254 
   255 
   256 
   257 (*Lets blast_tac perform this step without needing the simplifier*)
   258 lemma invKey_shrK_iff [iff]:
   259      "(Key (invKey K) \<in> X) = (Key K \<in> X)"
   260 by auto
   261 
   262 (*Specialized methods*)
   263 
   264 method_setup analz_freshK = {*
   265     Method.no_args
   266      (Method.METHOD
   267       (fn facts => EVERY [REPEAT_FIRST (resolve_tac [allI, ballI, impI]),
   268                           REPEAT_FIRST (rtac analz_image_freshK_lemma),
   269                           ALLGOALS (asm_simp_tac analz_image_freshK_ss)])) *}
   270     "for proving the Session Key Compromise theorem"
   271 
   272 method_setup possibility = {*
   273     Method.ctxt_args (fn ctxt =>
   274         Method.METHOD (fn facts =>
   275             gen_possibility_tac (local_simpset_of ctxt))) *}
   276     "for proving possibility theorems"
   277 
   278 lemma knows_subset_knows_Cons: "knows A evs <= knows A (e # evs)"
   279 by (induct e, auto simp: knows_Cons)
   280 
   281 end