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(* Title: HOL/Auth/Message
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1996 University of Cambridge
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Datatypes of agents and messages;
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Inductive relations "parts", "analz" and "synth"
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*)
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16417
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theory Message imports Main
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uses ("Message_lemmas.ML") begin
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(*Needed occasionally with spy_analz_tac, e.g. in analz_insert_Key_newK*)
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lemma [simp] : "A Un (B Un A) = B Un A"
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by blast
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types
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key = nat
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consts
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invKey :: "key=>key"
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axioms
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invKey [simp] : "invKey (invKey K) = K"
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(*The inverse of a symmetric key is itself;
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that of a public key is the private key and vice versa*)
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constdefs
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symKeys :: "key set"
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"symKeys == {K. invKey K = K}"
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datatype (*We allow any number of friendly agents*)
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agent = Server | Friend nat | Spy
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datatype
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msg = Agent agent (*Agent names*)
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| Number nat (*Ordinary integers, timestamps, ...*)
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| Nonce nat (*Unguessable nonces*)
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| Key key (*Crypto keys*)
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| Hash msg (*Hashing*)
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| MPair msg msg (*Compound messages*)
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| Crypt key msg (*Encryption, public- or shared-key*)
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(*Concrete syntax: messages appear as {|A,B,NA|}, etc...*)
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syntax
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"@MTuple" :: "['a, args] => 'a * 'b" ("(2{|_,/ _|})")
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syntax (xsymbols)
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"@MTuple" :: "['a, args] => 'a * 'b" ("(2\<lbrace>_,/ _\<rbrace>)")
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translations
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"{|x, y, z|}" == "{|x, {|y, z|}|}"
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"{|x, y|}" == "MPair x y"
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constdefs
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(*Message Y, paired with a MAC computed with the help of X*)
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HPair :: "[msg,msg] => msg" ("(4Hash[_] /_)" [0, 1000])
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"Hash[X] Y == {| Hash{|X,Y|}, Y|}"
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(*Keys useful to decrypt elements of a message set*)
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keysFor :: "msg set => key set"
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"keysFor H == invKey ` {K. \<exists>X. Crypt K X \<in> H}"
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(** Inductive definition of all "parts" of a message. **)
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consts parts :: "msg set => msg set"
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inductive "parts H"
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intros
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Inj [intro]: "X \<in> H ==> X \<in> parts H"
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Fst: "{|X,Y|} \<in> parts H ==> X \<in> parts H"
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Snd: "{|X,Y|} \<in> parts H ==> Y \<in> parts H"
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Body: "Crypt K X \<in> parts H ==> X \<in> parts H"
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(*Monotonicity*)
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lemma parts_mono: "G<=H ==> parts(G) <= parts(H)"
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apply auto
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apply (erule parts.induct)
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apply (auto dest: Fst Snd Body)
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done
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(** Inductive definition of "analz" -- what can be broken down from a set of
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messages, including keys. A form of downward closure. Pairs can
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be taken apart; messages decrypted with known keys. **)
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consts analz :: "msg set => msg set"
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inductive "analz H"
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intros
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Inj [intro,simp] : "X \<in> H ==> X \<in> analz H"
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Fst: "{|X,Y|} \<in> analz H ==> X \<in> analz H"
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Snd: "{|X,Y|} \<in> analz H ==> Y \<in> analz H"
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Decrypt [dest]:
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"[|Crypt K X \<in> analz H; Key(invKey K) \<in> analz H|] ==> X \<in> analz H"
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(*Monotonicity; Lemma 1 of Lowe's paper*)
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lemma analz_mono: "G<=H ==> analz(G) <= analz(H)"
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apply auto
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apply (erule analz.induct)
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apply (auto dest: Fst Snd)
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done
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(** Inductive definition of "synth" -- what can be built up from a set of
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messages. A form of upward closure. Pairs can be built, messages
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encrypted with known keys. Agent names are public domain.
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Numbers can be guessed, but Nonces cannot be. **)
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consts synth :: "msg set => msg set"
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inductive "synth H"
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intros
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Inj [intro]: "X \<in> H ==> X \<in> synth H"
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Agent [intro]: "Agent agt \<in> synth H"
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Number [intro]: "Number n \<in> synth H"
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Hash [intro]: "X \<in> synth H ==> Hash X \<in> synth H"
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MPair [intro]: "[|X \<in> synth H; Y \<in> synth H|] ==> {|X,Y|} \<in> synth H"
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Crypt [intro]: "[|X \<in> synth H; Key(K) \<in> H|] ==> Crypt K X \<in> synth H"
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(*Monotonicity*)
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lemma synth_mono: "G<=H ==> synth(G) <= synth(H)"
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apply auto
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apply (erule synth.induct)
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apply (auto dest: Fst Snd Body)
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done
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(*NO Agent_synth, as any Agent name can be synthesized. Ditto for Number*)
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inductive_cases Nonce_synth [elim!]: "Nonce n \<in> synth H"
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inductive_cases Key_synth [elim!]: "Key K \<in> synth H"
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inductive_cases Hash_synth [elim!]: "Hash X \<in> synth H"
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inductive_cases MPair_synth [elim!]: "{|X,Y|} \<in> synth H"
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inductive_cases Crypt_synth [elim!]: "Crypt K X \<in> synth H"
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use "Message_lemmas.ML"
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lemma Fake_parts_insert_in_Un:
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"[|Z \<in> parts (insert X H); X: synth (analz H)|]
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==> Z \<in> synth (analz H) \<union> parts H";
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by (blast dest: Fake_parts_insert [THEN subsetD, dest])
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method_setup spy_analz = {*
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Method.no_args (Method.METHOD (fn facts => spy_analz_tac 1)) *}
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"for proving the Fake case when analz is involved"
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end
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