Theory Public

(*  Title:      HOL/Auth/Public
    ID:         $Id: Public.thy,v 1.24 2005/06/17 14:13:06 haftmann Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Theory of Public Keys (common to all public-key protocols)

Private and public keys; initial states of agents
*)

theory Public imports Event begin

lemma invKey_K: "K ∈ symKeys ==> invKey K = K"
by (simp add: symKeys_def)

subsection{*Asymmetric Keys*}

consts
  (*the bool is TRUE if a signing key*)
  publicKey :: "[bool,agent] => key"

syntax
  pubEK :: "agent => key"
  pubSK :: "agent => key"

  privateKey :: "[bool,agent] => key"
  priEK :: "agent => key"
  priSK :: "agent => key"

translations
  "pubEK"  == "publicKey False"
  "pubSK"  == "publicKey True"

  (*BEWARE!! priEK, priSK DON'T WORK with inj, range, image, etc.*)
  "privateKey b A" == "invKey (publicKey b A)"
  "priEK A"  == "privateKey False A"
  "priSK A"  == "privateKey True A"


text{*These translations give backward compatibility.  They represent the
simple situation where the signature and encryption keys are the same.*}
syntax
  pubK :: "agent => key"
  priK :: "agent => key"

translations
  "pubK A" == "pubEK A"
  "priK A" == "invKey (pubEK A)"


text{*By freeness of agents, no two agents have the same key.  Since
  @{term "True≠False"}, no agent has identical signing and encryption keys*}
specification (publicKey)
  injective_publicKey:
    "publicKey b A = publicKey c A' ==> b=c & A=A'"
   apply (rule exI [of _ "%b A. 2 * agent_case 0 (λn. n + 2) 1 A + (if b then 1 else 0)"])
   apply (auto simp add: inj_on_def split: agent.split, presburger+)
   done                       


axioms
  (*No private key equals any public key (essential to ensure that private
    keys are private!) *)
  privateKey_neq_publicKey [iff]: "privateKey b A ≠ publicKey c A'"

declare privateKey_neq_publicKey [THEN not_sym, iff]


subsection{*Basic properties of @{term pubK} and @{term priK}*}

lemma [iff]: "(publicKey b A = publicKey c A') = (b=c & A=A')"
by (blast dest!: injective_publicKey) 

lemma not_symKeys_pubK [iff]: "publicKey b A ∉ symKeys"
by (simp add: symKeys_def)

lemma not_symKeys_priK [iff]: "privateKey b A ∉ symKeys"
by (simp add: symKeys_def)

lemma symKey_neq_priEK: "K ∈ symKeys ==> K ≠ priEK A"
by auto

lemma symKeys_neq_imp_neq: "(K ∈ symKeys) ≠ (K' ∈ symKeys) ==> K ≠ K'"
by blast

lemma symKeys_invKey_iff [iff]: "(invKey K ∈ symKeys) = (K ∈ symKeys)"
by (unfold symKeys_def, auto)

lemma analz_symKeys_Decrypt:
     "[| Crypt K X ∈ analz H;  K ∈ symKeys;  Key K ∈ analz H |]  
      ==> X ∈ analz H"
by (auto simp add: symKeys_def)



subsection{*"Image" equations that hold for injective functions*}

lemma invKey_image_eq [simp]: "(invKey x ∈ invKey`A) = (x ∈ A)"
by auto

(*holds because invKey is injective*)
lemma publicKey_image_eq [simp]:
     "(publicKey b x ∈ publicKey c ` AA) = (b=c & x ∈ AA)"
by auto

lemma privateKey_notin_image_publicKey [simp]: "privateKey b x ∉ publicKey c ` AA"
by auto

lemma privateKey_image_eq [simp]:
     "(privateKey b A ∈ invKey ` publicKey c ` AS) = (b=c & A∈AS)"
by auto

lemma publicKey_notin_image_privateKey [simp]: "publicKey b A ∉ invKey ` publicKey c ` AS"
by auto


subsection{*Symmetric Keys*}

text{*For some protocols, it is convenient to equip agents with symmetric as
well as asymmetric keys.  The theory @{text Shared} assumes that all keys
are symmetric.*}

consts
  shrK    :: "agent => key"    --{*long-term shared 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

axioms
  sym_shrK [iff]: "shrK X ∈ symKeys" --{*All shared keys are symmetric*}

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

lemma invKey_shrK [simp]: "invKey (shrK A) = shrK A"
by (simp add: invKey_K) 

lemma analz_shrK_Decrypt:
     "[| Crypt (shrK A) X ∈ analz H; Key(shrK A) ∈ analz H |] ==> X ∈ analz H"
by auto

lemma analz_Decrypt':
     "[| Crypt K X ∈ analz H; K ∈ symKeys; Key K ∈ analz H |] ==> X ∈ analz H"
by (auto simp add: invKey_K)

lemma priK_neq_shrK [iff]: "shrK A ≠ privateKey b C"
by (simp add: symKeys_neq_imp_neq)

declare priK_neq_shrK [THEN not_sym, simp]

lemma pubK_neq_shrK [iff]: "shrK A ≠ publicKey b C"
by (simp add: symKeys_neq_imp_neq)

declare pubK_neq_shrK [THEN not_sym, simp]

lemma priEK_noteq_shrK [simp]: "priEK A ≠ shrK B" 
by auto

lemma publicKey_notin_image_shrK [simp]: "publicKey b x ∉ shrK ` AA"
by auto

lemma privateKey_notin_image_shrK [simp]: "privateKey b x ∉ shrK ` AA"
by auto

lemma shrK_notin_image_publicKey [simp]: "shrK x ∉ publicKey b ` AA"
by auto

lemma shrK_notin_image_privateKey [simp]: "shrK x ∉ invKey ` publicKey b ` AA" 
by auto

lemma shrK_image_eq [simp]: "(shrK x ∈ shrK ` AA) = (x ∈ AA)"
by auto

text{*For some reason, moving this up can make some proofs loop!*}
declare invKey_K [simp]


subsection{*Initial States of Agents*}

text{*Note: for all practical purposes, all that matters is the initial
knowledge of the Spy.  All other agents are automata, merely following the
protocol.*}

primrec
        (*Agents know their private key and all public keys*)
  initState_Server:
    "initState Server     =    
       {Key (priEK Server), Key (priSK Server)} ∪ 
       (Key ` range pubEK) ∪ (Key ` range pubSK) ∪ (Key ` range shrK)"

  initState_Friend:
    "initState (Friend i) =    
       {Key (priEK(Friend i)), Key (priSK(Friend i)), Key (shrK(Friend i))} ∪ 
       (Key ` range pubEK) ∪ (Key ` range pubSK)"

  initState_Spy:
    "initState Spy        =    
       (Key ` invKey ` pubEK ` bad) ∪ (Key ` invKey ` pubSK ` bad) ∪ 
       (Key ` shrK ` bad) ∪ 
       (Key ` range pubEK) ∪ (Key ` range pubSK)"


text{*These lemmas allow reasoning about @{term "used evs"} rather than
   @{term "knows Spy evs"}, which is useful when there are private Notes. 
   Because they depend upon the definition of @{term initState}, they cannot
   be moved up.*}

lemma used_parts_subset_parts [rule_format]:
     "∀X ∈ used evs. parts {X} ⊆ used evs"
apply (induct evs) 
 prefer 2
 apply (simp add: used_Cons)
 apply (rule ballI)  
 apply (case_tac a, auto)  
apply (auto dest!: parts_cut) 
txt{*Base case*}
apply (simp add: used_Nil) 
done

lemma MPair_used_D: "{|X,Y|} ∈ used H ==> X ∈ used H & Y ∈ used H"
by (drule used_parts_subset_parts, simp, blast)

lemma MPair_used [elim!]:
     "[| {|X,Y|} ∈ used H;
         [| X ∈ used H; Y ∈ used H |] ==> P |] 
      ==> P"
by (blast dest: MPair_used_D) 


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")
apply (auto intro: range_eqI)
done

lemma Crypt_notin_initState: "Crypt K X ∉ parts (initState B)"
by (induct B, auto)

lemma Crypt_notin_used_empty [simp]: "Crypt K X ∉ used []"
by (simp add: Crypt_notin_initState used_Nil)

(*** Basic properties of shrK ***)

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

lemma shrK_in_knows [iff]: "Key (shrK A) ∈ knows A evs"
by (simp add: initState_subset_knows [THEN subsetD])

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


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

(*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: "Key K ∉ used evs ==> shrK B ≠ K"
by blast

declare shrK_neq [THEN not_sym, simp]


subsection{*Function @{term spies} *}

text{*Agents see their own private keys!*}
lemma priK_in_initState [iff]: "Key (privateKey b A) ∈ initState A"
by (induct_tac "A", auto)

text{*Agents see all public keys!*}
lemma publicKey_in_initState [iff]: "Key (publicKey b A) ∈ initState B"
by (case_tac "B", auto)

text{*All public keys are visible*}
lemma spies_pubK [iff]: "Key (publicKey b A) ∈ spies evs"
apply (induct_tac "evs")
apply (simp_all add: imageI knows_Cons split add: event.split)
done

declare spies_pubK [THEN analz.Inj, iff]

text{*Spy sees private keys of bad agents!*}
lemma Spy_spies_bad_privateKey [intro!]:
     "A ∈ bad ==> Key (privateKey b A) ∈ spies evs"
apply (induct_tac "evs")
apply (simp_all add: imageI knows_Cons split add: event.split)
done

text{*Spy sees long-term shared keys of bad agents!*}
lemma Spy_spies_bad_shrK [intro!]:
     "A ∈ bad ==> Key (shrK A) ∈ spies evs"
apply (induct_tac "evs")
apply (simp_all add: imageI knows_Cons split add: event.split)
done

lemma publicKey_into_used [iff] :"Key (publicKey b A) ∈ used evs"
apply (rule initState_into_used)
apply (rule publicKey_in_initState [THEN parts.Inj])
done

lemma privateKey_into_used [iff]: "Key (privateKey b A) ∈ used evs"
apply(rule initState_into_used)
apply(rule priK_in_initState [THEN parts.Inj])
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)"
by force


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 []"
by (simp add: used_Nil)


subsection{*Supply fresh nonces for possibility theorems*}

text{*In any trace, there is an upper bound N on the greatest nonce in use*}
lemma Nonce_supply_lemma: "EX 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: "EX N. Nonce N ∉ used evs"
by (rule Nonce_supply_lemma [THEN exE], blast)

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

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

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

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

ML
{*
val Key_not_used = thm "Key_not_used";
val insert_Key_singleton = thm "insert_Key_singleton";
val insert_Key_image = thm "insert_Key_image";
*}


lemma Crypt_imp_keysFor :"[|Crypt K X ∈ H; K ∈ symKeys|] ==> K ∈ keysFor H"
by (drule Crypt_imp_invKey_keysFor, simp)

text{*Lemma for the trivial direction of the if-and-only-if of the 
Session Key Compromise Theorem*}
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])

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 subsetD]
       insert_Key_singleton 
       Key_not_used insert_Key_image Un_assoc [THEN sym]

ML
{*
val analz_image_freshK_lemma = thm "analz_image_freshK_lemma";
val analz_image_freshK_simps = thms "analz_image_freshK_simps";

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

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"

subsection{*Specialized Methods for Possibility Theorems*}

ML
{*
val Nonce_supply = thm "Nonce_supply";

(*Tactic for possibility theorems (Isar interface)*)
fun gen_possibility_tac ss state = state |>
    REPEAT (*omit used_Says so that Nonces start from different traces!*)
    (ALLGOALS (simp_tac (ss delsimps [used_Says]))
     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]))
*}

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

end