(*  Title:      HOL/Auth/Public

    ID:         $Id$

    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

(*>*)

text {*

The function

@{text pubK} maps agents to their public keys.  The function

@{text priK} maps agents to their private keys.  It is merely

an abbreviation (cf.\ \S\ref{sec:abbreviations}) defined in terms of

@{text invKey} and @{text pubK}.

*}

consts pubK :: "agent \<Rightarrow> key"

abbreviation priK :: "agent \<Rightarrow> key"

where "priK x  \<equiv>  invKey(pubK x)"

(*<*)

primrec

        (*Agents know their private key and all public keys*)

  initState_Server:  "initState Server     =    

 		         insert (Key (priK Server)) (Key ` range pubK)"

  initState_Friend:  "initState (Friend i) =    

 		         insert (Key (priK (Friend i))) (Key ` range pubK)"

  initState_Spy:     "initState Spy        =    

 		         (Key`invKey`pubK`bad) Un (Key ` range pubK)"

(*>*)

text {*

\noindent

The set @{text bad} consists of those agents whose private keys are known to

the spy.

Two axioms are asserted about the public-key cryptosystem. 

No two agents have the same public key, and no private key equals

any public key.

*}

axioms

  inj_pubK:        "inj pubK"

  priK_neq_pubK:   "priK A \<noteq> pubK B"

(*<*)

lemmas [iff] = inj_pubK [THEN inj_eq]

lemma priK_inj_eq[iff]: "(priK A = priK B) = (A=B)"

  apply safe

  apply (drule_tac f=invKey in arg_cong)

  apply simp

  done

lemmas [iff] = priK_neq_pubK priK_neq_pubK [THEN not_sym]

lemma not_symKeys_pubK[iff]: "pubK A \<notin> symKeys"

  by (simp add: symKeys_def)

lemma not_symKeys_priK[iff]: "priK A \<notin> symKeys"

  by (simp add: symKeys_def)

lemma symKeys_neq_imp_neq: "(K \<in> symKeys) \<noteq> (K' \<in> symKeys) \<Longrightarrow> K \<noteq> K'"

  by blast

lemma analz_symKeys_Decrypt: "[| Crypt K X \<in> analz H;  K \<in> symKeys;  Key K \<in> analz H |]

     ==> X \<in> analz H"

  by (auto simp add: symKeys_def)

(** "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 pubK_image_eq[simp]: "(pubK x : pubK`A) = (x:A)"

  by auto

lemma priK_pubK_image_eq[simp]: "(priK x ~: pubK`A)"

  by auto

(** Rewrites should not refer to  initState(Friend i) 

    -- not in normal form! **)

lemma keysFor_parts_initState[simp]: "keysFor (parts (initState C)) = {}"

  apply (unfold keysFor_def)

  apply (induct C)

  apply (auto intro: range_eqI)

  done

(*** Function "spies" ***)

(*Agents see their own private keys!*)

lemma priK_in_initState[iff]: "Key (priK A) : initState A"

  by (induct A) auto

(*All public keys are visible*)

lemma spies_pubK[iff]: "Key (pubK A) : spies evs"

  by (induct evs) (simp_all add: imageI knows_Cons split: event.split)

(*Spy sees private keys of bad agents!*)

lemma Spy_spies_bad[intro!]: "A: bad ==> Key (priK A) : spies evs"

  by (induct evs) (simp_all add: imageI knows_Cons split: event.split)

lemmas [iff] = spies_pubK [THEN analz.Inj]

(*** Fresh nonces ***)

lemma Nonce_notin_initState[iff]: "Nonce N ~: parts (initState B)"

  by (induct B) auto

lemma Nonce_notin_used_empty[simp]: "Nonce N ~: used []"

  by (simp add: used_Nil)

(*** 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: "EX N. ALL n. N<=n --> Nonce n \<notin> 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 \<notin> used evs"

by (rule Nonce_supply_lemma [THEN exE], blast)

lemma Nonce_supply: "Nonce (@ N. Nonce N \<notin> used evs) \<notin> used evs"

apply (rule Nonce_supply_lemma [THEN exE])

apply (rule someI, fast)

done

(*** Specialized rewriting for the analz_image_... theorems ***)

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

  by blast

lemma insert_Key_image: "insert (Key K) (Key`KK Un C) = Key ` (insert K KK) Un C"

  by blast

(*Specialized methods*)

(*Tactic for possibility theorems*)

ML {*

fun possibility_tac st = st |>

    REPEAT (*omit used_Says so that Nonces start from different traces!*)

    (ALLGOALS (simp_tac (@{simpset} delsimps [used_Says]))

     THEN

     REPEAT_FIRST (eq_assume_tac ORELSE' 

                   resolve_tac [refl, conjI, @{thm Nonce_supply}]));

*}

method_setup possibility = {*

    Method.no_args (Method.METHOD (fn facts => possibility_tac)) *}

    "for proving possibility theorems"

end

(*>*)