Theory GuardK

(*  Title:      HOL/Auth/Guard/GuardK.thy
    Author:     Frederic Blanqui, University of Cambridge Computer Laboratory
    Copyright   2002  University of Cambridge

Very similar to Guard except:
- Guard is replaced by GuardK, guard by guardK, Nonce by Key
- some scripts are slightly modified (+ keyset_in, kparts_parts)
- the hypothesis Key n ~:G (keyset G) is added
*)

sectionprotocol-independent confidentiality theorem on keys

theory GuardK
imports Analz Extensions
begin

(******************************************************************************
messages where all the occurrences of Key n are
in a sub-message of the form Crypt (invKey K) X with K:Ks
******************************************************************************)

inductive_set
  guardK :: "nat => key set => msg set"
  for n :: nat and Ks :: "key set"
where
  No_Key [intro]: "Key n  parts {X}  X  guardK n Ks"
| Guard_Key [intro]: "invKey K  Ks ==> Crypt K X  guardK n Ks"
| Crypt [intro]: "X  guardK n Ks  Crypt K X  guardK n Ks"
| Pair [intro]: "[| X  guardK n Ks; Y  guardK n Ks |] ==> X,Y  guardK n Ks"

subsectionbasic facts about termguardK

lemma Nonce_is_guardK [iff]: "Nonce p  guardK n Ks"
by auto

lemma Agent_is_guardK [iff]: "Agent A  guardK n Ks"
by auto

lemma Number_is_guardK [iff]: "Number r  guardK n Ks"
by auto

lemma Key_notin_guardK: "X  guardK n Ks  X  Key n"
by (erule guardK.induct, auto)

lemma Key_notin_guardK_iff [iff]: "Key n  guardK n Ks"
by (auto dest: Key_notin_guardK)

lemma guardK_has_Crypt [rule_format]: "X  guardK n Ks  Key n  parts {X}
 (K Y. Crypt K Y  kparts {X}  Key n  parts {Y})"
by (erule guardK.induct, auto)

lemma Key_notin_kparts_msg: "X  guardK n Ks  Key n  kparts {X}"
by (erule guardK.induct, auto dest: kparts_parts)

lemma Key_in_kparts_imp_no_guardK: "Key n  kparts H
 X. X  H  X  guardK n Ks"
apply (drule in_kparts, clarify)
apply (rule_tac x=X in exI, clarify)
by (auto dest: Key_notin_kparts_msg)

lemma guardK_kparts [rule_format]: "X  guardK n Ks 
Y  kparts {X}  Y  guardK n Ks"
by (erule guardK.induct, auto dest: kparts_parts parts_sub)

lemma guardK_Crypt: "[| Crypt K Y  guardK n Ks; K  invKey`Ks |] ==> Y  guardK n Ks"
  by (ind_cases "Crypt K Y  guardK n Ks") (auto intro!: image_eqI)

lemma guardK_MPair [iff]: "(X,Y  guardK n Ks)
= (X  guardK n Ks  Y  guardK n Ks)"
by (auto, (ind_cases "X,Y  guardK n Ks", auto)+)

lemma guardK_not_guardK [rule_format]: "X guardK n Ks 
Crypt K Y  kparts {X}  Key n  kparts {Y}  Y  guardK n Ks"
by (erule guardK.induct, auto dest: guardK_kparts)

lemma guardK_extand: "[| X  guardK n Ks; Ks  Ks';
[| K  Ks'; K  Ks |] ==> Key K  parts {X} |] ==> X  guardK n Ks'"
by (erule guardK.induct, auto)

subsectionguarded sets

definition GuardK :: "nat  key set  msg set  bool" where
"GuardK n Ks H  X. X  H  X  guardK n Ks"

subsectionbasic facts about termGuardK

lemma GuardK_empty [iff]: "GuardK n Ks {}"
by (simp add: GuardK_def)

lemma Key_notin_kparts [simplified]: "GuardK n Ks H  Key n  kparts H"
by (auto simp: GuardK_def dest: in_kparts Key_notin_kparts_msg)

lemma GuardK_must_decrypt: "[| GuardK n Ks H; Key n  analz H |] ==>
K Y. Crypt K Y  kparts H  Key (invKey K)  kparts H"
apply (drule_tac P="λG. Key n  G" in analz_pparts_kparts_substD, simp)
by (drule must_decrypt, auto dest: Key_notin_kparts)

lemma GuardK_kparts [intro]: "GuardK n Ks H  GuardK n Ks (kparts H)"
by (auto simp: GuardK_def dest: in_kparts guardK_kparts)

lemma GuardK_mono: "[| GuardK n Ks H; G  H |] ==> GuardK n Ks G"
by (auto simp: GuardK_def)

lemma GuardK_insert [iff]: "GuardK n Ks (insert X H)
= (GuardK n Ks H  X  guardK n Ks)"
by (auto simp: GuardK_def)

lemma GuardK_Un [iff]: "GuardK n Ks (G Un H) = (GuardK n Ks G & GuardK n Ks H)"
by (auto simp: GuardK_def)

lemma GuardK_synth [intro]: "GuardK n Ks G  GuardK n Ks (synth G)"
by (auto simp: GuardK_def, erule synth.induct, auto)

lemma GuardK_analz [intro]: "[| GuardK n Ks G; K. K  Ks  Key K  analz G |]
==> GuardK n Ks (analz G)"
apply (auto simp: GuardK_def)
apply (erule analz.induct, auto)
by (ind_cases "Crypt K Xa  guardK n Ks" for K Xa, auto)

lemma in_GuardK [dest]: "[| X  G; GuardK n Ks G |] ==> X  guardK n Ks"
by (auto simp: GuardK_def)

lemma in_synth_GuardK: "[| X  synth G; GuardK n Ks G |] ==> X  guardK n Ks"
by (drule GuardK_synth, auto)

lemma in_analz_GuardK: "[| X  analz G; GuardK n Ks G;
K. K  Ks  Key K  analz G |] ==> X  guardK n Ks"
by (drule GuardK_analz, auto)

lemma GuardK_keyset [simp]: "[| keyset G; Key n  G |] ==> GuardK n Ks G"
by (simp only: GuardK_def, clarify, drule keyset_in, auto)

lemma GuardK_Un_keyset: "[| GuardK n Ks G; keyset H; Key n  H |]
==> GuardK n Ks (G Un H)"
by auto

lemma in_GuardK_kparts: "[| X  G; GuardK n Ks G; Y  kparts {X} |] ==> Y  guardK n Ks"
by blast

lemma in_GuardK_kparts_neq: "[| X  G; GuardK n Ks G; Key n'  kparts {X} |]
==> n  n'"
by (blast dest: in_GuardK_kparts)

lemma in_GuardK_kparts_Crypt: "[| X  G; GuardK n Ks G; is_MPair X;
Crypt K Y  kparts {X}; Key n  kparts {Y} |] ==> invKey K  Ks"
apply (drule in_GuardK, simp)
apply (frule guardK_not_guardK, simp+)
apply (drule guardK_kparts, simp)
by (ind_cases "Crypt K Y  guardK n Ks", auto)

lemma GuardK_extand: "[| GuardK n Ks G; Ks  Ks';
[| K  Ks'; K  Ks |] ==> Key K  parts G |] ==> GuardK n Ks' G"
by (auto simp: GuardK_def dest: guardK_extand parts_sub)

subsectionset obtained by decrypting a message

abbreviation (input)
  decrypt :: "msg set  key  msg  msg set" where
  "decrypt H K Y  insert Y (H - {Crypt K Y})"

lemma analz_decrypt: "[| Crypt K Y  H; Key (invKey K)  H; Key n  analz H |]
==> Key n  analz (decrypt H K Y)"
apply (drule_tac P="λH. Key n  analz H" in ssubst [OF insert_Diff])
apply assumption 
apply (simp only: analz_Crypt_if, simp)
done

lemma parts_decrypt: "[| Crypt K Y  H; X  parts (decrypt H K Y) |] ==> X  parts H"
by (erule parts.induct, auto intro: parts.Fst parts.Snd parts.Body)

subsectionnumber of Crypt's in a message

fun crypt_nb :: "msg => nat" where
"crypt_nb (Crypt K X) = Suc (crypt_nb X)" |
"crypt_nb X,Y = crypt_nb X + crypt_nb Y" |
"crypt_nb X = 0" (* otherwise *)

subsectionbasic facts about termcrypt_nb

lemma non_empty_crypt_msg: "Crypt K Y  parts {X}  crypt_nb X  0"
by (induct X, simp_all, safe, simp_all)

subsectionnumber of Crypt's in a message list

primrec cnb :: "msg list => nat" where
"cnb [] = 0" |
"cnb (X#l) = crypt_nb X + cnb l"

subsectionbasic facts about termcnb

lemma cnb_app [simp]: "cnb (l @ l') = cnb l + cnb l'"
by (induct l, auto)

lemma mem_cnb_minus: "x  set l ==> cnb l = crypt_nb x + (cnb l - crypt_nb x)"
by (induct l, auto)

lemmas mem_cnb_minus_substI = mem_cnb_minus [THEN ssubst]

lemma cnb_minus [simp]: "x  set l ==> cnb (remove l x) = cnb l - crypt_nb x"
apply (induct l, auto)
by (erule_tac l=l and x=x in mem_cnb_minus_substI, simp)

lemma parts_cnb: "Z  parts (set l) 
cnb l = (cnb l - crypt_nb Z) + crypt_nb Z"
by (erule parts.induct, auto simp: in_set_conv_decomp)

lemma non_empty_crypt: "Crypt K Y  parts (set l)  cnb l  0"
by (induct l, auto dest: non_empty_crypt_msg parts_insert_substD)

subsectionlist of kparts

lemma kparts_msg_set: "l. kparts {X} = set l  cnb l = crypt_nb X"
apply (induct X, simp_all)
apply (rename_tac agent, rule_tac x="[Agent agent]" in exI, simp)
apply (rename_tac nat, rule_tac x="[Number nat]" in exI, simp)
apply (rename_tac nat, rule_tac x="[Nonce nat]" in exI, simp)
apply (rename_tac nat, rule_tac x="[Key nat]" in exI, simp)
apply (rule_tac x="[Hash X]" in exI, simp)
apply (clarify, rule_tac x="l@la" in exI, simp)
by (clarify, rename_tac nat X y, rule_tac x="[Crypt nat X]" in exI, simp)

lemma kparts_set: "l'. kparts (set l) = set l' & cnb l' = cnb l"
apply (induct l)
apply (rule_tac x="[]" in exI, simp, clarsimp)
apply (rename_tac a b l')
apply (subgoal_tac "l''.  kparts {a} = set l'' & cnb l'' = crypt_nb a", clarify)
apply (rule_tac x="l''@l'" in exI, simp)
apply (rule kparts_insert_substI, simp)
by (rule kparts_msg_set)

subsectionlist corresponding to "decrypt"

definition decrypt' :: "msg list => key => msg => msg list" where
"decrypt' l K Y == Y # remove l (Crypt K Y)"

declare decrypt'_def [simp]

subsectionbasic facts about termdecrypt'

lemma decrypt_minus: "decrypt (set l) K Y <= set (decrypt' l K Y)"
by (induct l, auto)

textif the analysis of a finite guarded set gives n then it must also give
one of the keys of Ks

lemma GuardK_invKey_by_list [rule_format]: "l. cnb l = p
 GuardK n Ks (set l)  Key n  analz (set l)
 (K. K  Ks  Key K  analz (set l))"
apply (induct p)
(* case p=0 *)
apply (clarify, drule GuardK_must_decrypt, simp, clarify)
apply (drule kparts_parts, drule non_empty_crypt, simp)
(* case p>0 *)
apply (clarify, frule GuardK_must_decrypt, simp, clarify)
apply (drule_tac P="λG. Key n  G" in analz_pparts_kparts_substD, simp)
apply (frule analz_decrypt, simp_all)
apply (subgoal_tac "l'. kparts (set l) = set l'  cnb l' = cnb l", clarsimp)
apply (drule_tac G="insert Y (set l' - {Crypt K Y})"
and H="set (decrypt' l' K Y)" in analz_sub, rule decrypt_minus)
apply (rule_tac analz_pparts_kparts_substI, simp)
apply (case_tac "K  invKey`Ks")
(* K:invKey`Ks *)
apply (clarsimp, blast)
(* K ~:invKey`Ks *)
apply (subgoal_tac "GuardK n Ks (set (decrypt' l' K Y))")
apply (drule_tac x="decrypt' l' K Y" in spec, simp)
apply (subgoal_tac "Crypt K Y  parts (set l)")
apply (drule parts_cnb, rotate_tac -1, simp)
apply (clarify, drule_tac X="Key Ka" and H="insert Y (set l')" in analz_sub)
apply (rule insert_mono, rule set_remove)
apply (simp add: analz_insertD, blast)
(* Crypt K Y:parts (set l) *)
apply (blast dest: kparts_parts)
(* GuardK n Ks (set (decrypt' l' K Y)) *)
apply (rule_tac H="insert Y (set l')" in GuardK_mono)
apply (subgoal_tac "GuardK n Ks (set l')", simp)
apply (rule_tac K=K in guardK_Crypt, simp add: GuardK_def, simp)
apply (drule_tac t="set l'" in sym, simp)
apply (rule GuardK_kparts, simp, simp)
apply (rule_tac B="set l'" in subset_trans, rule set_remove, blast)
by (rule kparts_set)

lemma GuardK_invKey_finite: "[| Key n  analz G; GuardK n Ks G; finite G |]
==> K. K  Ks  Key K  analz G"
apply (drule finite_list, clarify)
by (rule GuardK_invKey_by_list, auto)

lemma GuardK_invKey: "[| Key n  analz G; GuardK n Ks G |]
==> K. K  Ks  Key K  analz G"
by (auto dest: analz_needs_only_finite GuardK_invKey_finite)

textif the analyse of a finite guarded set and a (possibly infinite) set of
keys gives n then it must also gives Ks

lemma GuardK_invKey_keyset: "[| Key n  analz (G  H); GuardK n Ks G; finite G;
keyset H; Key n  H |] ==> K. K  Ks  Key K  analz (G  H)"
apply (frule_tac P="λG. Key n  G" and G=G in analz_keyset_substD, simp_all)
apply (drule_tac G="G Un (H Int keysfor G)" in GuardK_invKey_finite)
apply (auto simp: GuardK_def intro: analz_sub)
by (drule keyset_in, auto)

end