Theory Shared

(*  Title:      HOL/Auth/Shared
    ID:         $Id: Shared.thy,v 1.34 2005/06/17 14:13:06 haftmann Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Theory of Shared Keys (common to all symmetric-key protocols)

Shared, long-term keys; initial states of agents
*)

theory Shared imports Event begin

consts
  shrK    :: "agent => key"  (*symmetric keys*);

specification (shrK)
  inj_shrK: "inj shrK"
  --{*No two agents have the same long-term key*}
   apply (rule exI [of _ "agent_case 0 (λn. n + 2) 1"]) 
   apply (simp add: inj_on_def split: agent.split) 
   done

text{*All keys are symmetric*}

defs  all_symmetric_def: "all_symmetric == True"

lemma isSym_keys: "K ∈ symKeys"     
by (simp add: symKeys_def all_symmetric_def invKey_symmetric) 

text{*Server knows all long-term keys; other agents know only their own*}
primrec
  initState_Server:  "initState Server     = Key ` range shrK"
  initState_Friend:  "initState (Friend i) = {Key (shrK (Friend i))}"
  initState_Spy:     "initState Spy        = Key`shrK`bad"


subsection{*Basic properties of shrK*}

(*Injectiveness: Agents' long-term keys are distinct.*)
declare inj_shrK [THEN inj_eq, iff]

lemma invKey_K [simp]: "invKey K = K"
apply (insert isSym_keys)
apply (simp add: symKeys_def) 
done


lemma analz_Decrypt' [dest]:
     "[| Crypt K X ∈ analz H;  Key K  ∈ analz H |] ==> X ∈ analz H"
by auto

text{*Now cancel the @{text dest} attribute given to
 @{text analz.Decrypt} in its declaration.*}
declare analz.Decrypt [rule del]

text{*Rewrites should not refer to  @{term "initState(Friend i)"} because
  that expression is not in normal form.*}

lemma keysFor_parts_initState [simp]: "keysFor (parts (initState C)) = {}"
apply (unfold keysFor_def)
apply (induct_tac "C", auto)
done

(*Specialized to shared-key model: no @{term invKey}*)
lemma keysFor_parts_insert:
     "[| K ∈ keysFor (parts (insert X G));  X ∈ synth (analz H) |]
      ==> K ∈ keysFor (parts (G ∪ H)) | Key K ∈ parts H";
by (force dest: Event.keysFor_parts_insert)  

lemma Crypt_imp_keysFor: "Crypt K X ∈ H ==> K ∈ keysFor H"
by (drule Crypt_imp_invKey_keysFor, simp)


subsection{*Function "knows"*}

(*Spy sees shared keys of agents!*)
lemma Spy_knows_Spy_bad [intro!]: "A: bad ==> Key (shrK A) ∈ knows Spy evs"
apply (induct_tac "evs")
apply (simp_all (no_asm_simp) add: imageI knows_Cons split add: event.split)
done

(*For case analysis on whether or not an agent is compromised*)
lemma Crypt_Spy_analz_bad: "[| Crypt (shrK A) X ∈ analz (knows Spy evs);  A: bad |]  
      ==> X ∈ analz (knows Spy evs)"
apply (force dest!: analz.Decrypt)
done


(** Fresh keys never clash with long-term shared keys **)

(*Agents see their own shared keys!*)
lemma shrK_in_initState [iff]: "Key (shrK A) ∈ initState A"
by (induct_tac "A", auto)

lemma shrK_in_used [iff]: "Key (shrK A) ∈ used evs"
by (rule initState_into_used, blast)

(*Used in parts_induct_tac and analz_Fake_tac to distinguish session keys
  from long-term shared keys*)
lemma Key_not_used [simp]: "Key K ∉ used evs ==> K ∉ range shrK"
by blast

lemma shrK_neq [simp]: "Key K ∉ used evs ==> shrK B ≠ K"
by blast

declare shrK_neq [THEN not_sym, simp]


subsection{*Fresh nonces*}

lemma Nonce_notin_initState [iff]: "Nonce N ∉ parts (initState B)"
by (induct_tac "B", auto)

lemma Nonce_notin_used_empty [simp]: "Nonce N ∉ used []"
apply (simp (no_asm) add: used_Nil)
done


subsection{*Supply fresh nonces for possibility theorems.*}

(*In any trace, there is an upper bound N on the greatest nonce in use.*)
lemma Nonce_supply_lemma: "∃N. ALL n. N<=n --> Nonce n ∉ used evs"
apply (induct_tac "evs")
apply (rule_tac x = 0 in exI)
apply (simp_all (no_asm_simp) add: used_Cons split add: event.split)
apply safe
apply (rule msg_Nonce_supply [THEN exE], blast elim!: add_leE)+
done

lemma Nonce_supply1: "∃N. Nonce N ∉ used evs"
by (rule Nonce_supply_lemma [THEN exE], blast)

lemma Nonce_supply2: "∃N N'. Nonce N ∉ used evs & Nonce N' ∉ used evs' & N ≠ N'"
apply (cut_tac evs = evs in Nonce_supply_lemma)
apply (cut_tac evs = "evs'" in Nonce_supply_lemma, clarify)
apply (rule_tac x = N in exI)
apply (rule_tac x = "Suc (N+Na)" in exI)
apply (simp (no_asm_simp) add: less_not_refl3 le_add1 le_add2 less_Suc_eq_le)
done

lemma Nonce_supply3: "∃N N' N''. Nonce N ∉ used evs & Nonce N' ∉ used evs' &  
                    Nonce N'' ∉ used evs'' & N ≠ N' & N' ≠ N'' & N ≠ N''"
apply (cut_tac evs = evs in Nonce_supply_lemma)
apply (cut_tac evs = "evs'" in Nonce_supply_lemma)
apply (cut_tac evs = "evs''" in Nonce_supply_lemma, clarify)
apply (rule_tac x = N in exI)
apply (rule_tac x = "Suc (N+Na)" in exI)
apply (rule_tac x = "Suc (Suc (N+Na+Nb))" in exI)
apply (simp (no_asm_simp) add: less_not_refl3 le_add1 le_add2 less_Suc_eq_le)
done

lemma Nonce_supply: "Nonce (@ N. Nonce N ∉ used evs) ∉ used evs"
apply (rule Nonce_supply_lemma [THEN exE])
apply (rule someI, blast)
done

text{*Unlike the corresponding property of nonces, we cannot prove
    @{term "finite KK ==> ∃K. K ∉ KK & Key K ∉ used evs"}.
    We have infinitely many agents and there is nothing to stop their
    long-term keys from exhausting all the natural numbers.  Instead,
    possibility theorems must assume the existence of a few keys.*}


subsection{*Tactics for possibility theorems*}

ML
{*
val inj_shrK      = thm "inj_shrK";
val isSym_keys    = thm "isSym_keys";
val Nonce_supply = thm "Nonce_supply";
val invKey_K = thm "invKey_K";
val analz_Decrypt' = thm "analz_Decrypt'";
val keysFor_parts_initState = thm "keysFor_parts_initState";
val keysFor_parts_insert = thm "keysFor_parts_insert";
val Crypt_imp_keysFor = thm "Crypt_imp_keysFor";
val Spy_knows_Spy_bad = thm "Spy_knows_Spy_bad";
val Crypt_Spy_analz_bad = thm "Crypt_Spy_analz_bad";
val shrK_in_initState = thm "shrK_in_initState";
val shrK_in_used = thm "shrK_in_used";
val Key_not_used = thm "Key_not_used";
val shrK_neq = thm "shrK_neq";
val Nonce_notin_initState = thm "Nonce_notin_initState";
val Nonce_notin_used_empty = thm "Nonce_notin_used_empty";
val Nonce_supply_lemma = thm "Nonce_supply_lemma";
val Nonce_supply1 = thm "Nonce_supply1";
val Nonce_supply2 = thm "Nonce_supply2";
val Nonce_supply3 = thm "Nonce_supply3";
val Nonce_supply = thm "Nonce_supply";
*}


ML
{*
(*Omitting used_Says makes the tactic much faster: it leaves expressions
    such as  Nonce ?N ∉ used evs that match Nonce_supply*)
fun gen_possibility_tac ss state = state |>
   (REPEAT 
    (ALLGOALS (simp_tac (ss delsimps [used_Says, used_Notes, used_Gets] 
                         setSolver safe_solver))
     THEN
     REPEAT_FIRST (eq_assume_tac ORELSE' 
                   resolve_tac [refl, conjI, Nonce_supply])))

(*Tactic for possibility theorems (ML script version)*)
fun possibility_tac state = gen_possibility_tac (simpset()) state

(*For harder protocols (such as Recur) where we have to set up some
  nonces and keys initially*)
fun basic_possibility_tac st = st |>
    REPEAT 
    (ALLGOALS (asm_simp_tac (simpset() setSolver safe_solver))
     THEN
     REPEAT_FIRST (resolve_tac [refl, conjI]))
*}

subsection{*Specialized Rewriting for Theorems About @{term analz} and Image*}

lemma subset_Compl_range: "A <= - (range shrK) ==> shrK x ∉ A"
by blast

lemma insert_Key_singleton: "insert (Key K) H = Key ` {K} ∪ H"
by blast

lemma insert_Key_image: "insert (Key K) (Key`KK ∪ C) = Key`(insert K KK) ∪ C"
by blast

(** Reverse the normal simplification of "image" to build up (not break down)
    the set of keys.  Use analz_insert_eq with (Un_upper2 RS analz_mono) to
    erase occurrences of forwarded message components (X). **)

lemmas analz_image_freshK_simps =
       simp_thms mem_simps --{*these two allow its use with @{text "only:"}*}
       disj_comms 
       image_insert [THEN sym] image_Un [THEN sym] empty_subsetI insert_subset
       analz_insert_eq Un_upper2 [THEN analz_mono, THEN [2] rev_subsetD]
       insert_Key_singleton subset_Compl_range
       Key_not_used insert_Key_image Un_assoc [THEN sym]

(*Lemma for the trivial direction of the if-and-only-if*)
lemma analz_image_freshK_lemma:
     "(Key K ∈ analz (Key`nE ∪ H)) --> (K ∈ nE | Key K ∈ analz H)  ==>  
         (Key K ∈ analz (Key`nE ∪ H)) = (K ∈ nE | Key K ∈ analz H)"
by (blast intro: analz_mono [THEN [2] rev_subsetD])

ML
{*
val analz_image_freshK_lemma = thm "analz_image_freshK_lemma";

val analz_image_freshK_ss = 
     simpset() delsimps [image_insert, image_Un]
               delsimps [imp_disjL]    (*reduces blow-up*)
               addsimps thms "analz_image_freshK_simps"
*}



(*Lets blast_tac perform this step without needing the simplifier*)
lemma invKey_shrK_iff [iff]:
     "(Key (invKey K) ∈ X) = (Key K ∈ X)"
by auto

(*Specialized methods*)

method_setup analz_freshK = {*
    Method.no_args
     (Method.METHOD
      (fn facts => EVERY [REPEAT_FIRST (resolve_tac [allI, ballI, impI]),
                          REPEAT_FIRST (rtac analz_image_freshK_lemma),
                          ALLGOALS (asm_simp_tac analz_image_freshK_ss)])) *}
    "for proving the Session Key Compromise theorem"

method_setup possibility = {*
    Method.ctxt_args (fn ctxt =>
        Method.METHOD (fn facts =>
            gen_possibility_tac (local_simpset_of ctxt))) *}
    "for proving possibility theorems"

lemma knows_subset_knows_Cons: "knows A evs <= knows A (e # evs)"
by (induct e, auto simp: knows_Cons)

end