author | wenzelm |
Fri, 17 Nov 2006 02:20:03 +0100 | |
changeset 21404 | eb85850d3eb7 |
parent 20768 | 1d478c2d621f |
child 23746 | a455e69c31cc |
permissions | -rw-r--r-- |
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(****************************************************************************** |
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date: january 2002 |
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author: Frederic Blanqui |
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email: blanqui@lri.fr |
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webpage: http://www.lri.fr/~blanqui/ |
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University of Cambridge, Computer Laboratory |
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William Gates Building, JJ Thomson Avenue |
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Cambridge CB3 0FD, United Kingdom |
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******************************************************************************) |
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header{*Protocol-Independent Confidentiality Theorem on Nonces*} |
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theory Guard imports Analz Extensions begin |
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(****************************************************************************** |
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messages where all the occurrences of Nonce n are |
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in a sub-message of the form Crypt (invKey K) X with K:Ks |
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******************************************************************************) |
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consts guard :: "nat => key set => msg set" |
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inductive "guard n Ks" |
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intros |
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No_Nonce [intro]: "Nonce n ~:parts {X} ==> X:guard n Ks" |
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Guard_Nonce [intro]: "invKey K:Ks ==> Crypt K X:guard n Ks" |
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Crypt [intro]: "X:guard n Ks ==> Crypt K X:guard n Ks" |
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Pair [intro]: "[| X:guard n Ks; Y:guard n Ks |] ==> {|X,Y|}:guard n Ks" |
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subsection{*basic facts about @{term guard}*} |
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lemma Key_is_guard [iff]: "Key K:guard n Ks" |
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by auto |
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lemma Agent_is_guard [iff]: "Agent A:guard n Ks" |
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by auto |
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lemma Number_is_guard [iff]: "Number r:guard n Ks" |
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by auto |
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lemma Nonce_notin_guard: "X:guard n Ks ==> X ~= Nonce n" |
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by (erule guard.induct, auto) |
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lemma Nonce_notin_guard_iff [iff]: "Nonce n ~:guard n Ks" |
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by (auto dest: Nonce_notin_guard) |
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lemma guard_has_Crypt [rule_format]: "X:guard n Ks ==> Nonce n:parts {X} |
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--> (EX K Y. Crypt K Y:kparts {X} & Nonce n:parts {Y})" |
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by (erule guard.induct, auto) |
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lemma Nonce_notin_kparts_msg: "X:guard n Ks ==> Nonce n ~:kparts {X}" |
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by (erule guard.induct, auto) |
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lemma Nonce_in_kparts_imp_no_guard: "Nonce n:kparts H |
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==> EX X. X:H & X ~:guard n Ks" |
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apply (drule in_kparts, clarify) |
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apply (rule_tac x=X in exI, clarify) |
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by (auto dest: Nonce_notin_kparts_msg) |
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lemma guard_kparts [rule_format]: "X:guard n Ks ==> |
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Y:kparts {X} --> Y:guard n Ks" |
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by (erule guard.induct, auto) |
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lemma guard_Crypt: "[| Crypt K Y:guard n Ks; K ~:invKey`Ks |] ==> Y:guard n Ks" |
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by (ind_cases "Crypt K Y:guard n Ks", auto) |
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lemma guard_MPair [iff]: "({|X,Y|}:guard n Ks) = (X:guard n Ks & Y:guard n Ks)" |
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by (auto, (ind_cases "{|X,Y|}:guard n Ks", auto)+) |
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lemma guard_not_guard [rule_format]: "X:guard n Ks ==> |
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Crypt K Y:kparts {X} --> Nonce n:kparts {Y} --> Y ~:guard n Ks" |
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by (erule guard.induct, auto dest: guard_kparts) |
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lemma guard_extand: "[| X:guard n Ks; Ks <= Ks' |] ==> X:guard n Ks'" |
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by (erule guard.induct, auto) |
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subsection{*guarded sets*} |
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constdefs Guard :: "nat => key set => msg set => bool" |
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"Guard n Ks H == ALL X. X:H --> X:guard n Ks" |
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subsection{*basic facts about @{term Guard}*} |
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lemma Guard_empty [iff]: "Guard n Ks {}" |
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by (simp add: Guard_def) |
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lemma notin_parts_Guard [intro]: "Nonce n ~:parts G ==> Guard n Ks G" |
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apply (unfold Guard_def, clarify) |
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apply (subgoal_tac "Nonce n ~:parts {X}") |
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by (auto dest: parts_sub) |
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lemma Nonce_notin_kparts [simplified]: "Guard n Ks H ==> Nonce n ~:kparts H" |
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by (auto simp: Guard_def dest: in_kparts Nonce_notin_kparts_msg) |
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lemma Guard_must_decrypt: "[| Guard n Ks H; Nonce n:analz H |] ==> |
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EX K Y. Crypt K Y:kparts H & Key (invKey K):kparts H" |
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apply (drule_tac P="%G. Nonce n:G" in analz_pparts_kparts_substD, simp) |
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by (drule must_decrypt, auto dest: Nonce_notin_kparts) |
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lemma Guard_kparts [intro]: "Guard n Ks H ==> Guard n Ks (kparts H)" |
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by (auto simp: Guard_def dest: in_kparts guard_kparts) |
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lemma Guard_mono: "[| Guard n Ks H; G <= H |] ==> Guard n Ks G" |
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by (auto simp: Guard_def) |
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lemma Guard_insert [iff]: "Guard n Ks (insert X H) |
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= (Guard n Ks H & X:guard n Ks)" |
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by (auto simp: Guard_def) |
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lemma Guard_Un [iff]: "Guard n Ks (G Un H) = (Guard n Ks G & Guard n Ks H)" |
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by (auto simp: Guard_def) |
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lemma Guard_synth [intro]: "Guard n Ks G ==> Guard n Ks (synth G)" |
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by (auto simp: Guard_def, erule synth.induct, auto) |
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lemma Guard_analz [intro]: "[| Guard n Ks G; ALL K. K:Ks --> Key K ~:analz G |] |
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==> Guard n Ks (analz G)" |
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apply (auto simp: Guard_def) |
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apply (erule analz.induct, auto) |
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by (ind_cases "Crypt K Xa:guard n Ks", auto) |
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lemma in_Guard [dest]: "[| X:G; Guard n Ks G |] ==> X:guard n Ks" |
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by (auto simp: Guard_def) |
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lemma in_synth_Guard: "[| X:synth G; Guard n Ks G |] ==> X:guard n Ks" |
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by (drule Guard_synth, auto) |
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lemma in_analz_Guard: "[| X:analz G; Guard n Ks G; |
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ALL K. K:Ks --> Key K ~:analz G |] ==> X:guard n Ks" |
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by (drule Guard_analz, auto) |
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lemma Guard_keyset [simp]: "keyset G ==> Guard n Ks G" |
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by (auto simp: Guard_def) |
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lemma Guard_Un_keyset: "[| Guard n Ks G; keyset H |] ==> Guard n Ks (G Un H)" |
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by auto |
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lemma in_Guard_kparts: "[| X:G; Guard n Ks G; Y:kparts {X} |] ==> Y:guard n Ks" |
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by blast |
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lemma in_Guard_kparts_neq: "[| X:G; Guard n Ks G; Nonce n':kparts {X} |] |
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==> n ~= n'" |
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by (blast dest: in_Guard_kparts) |
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lemma in_Guard_kparts_Crypt: "[| X:G; Guard n Ks G; is_MPair X; |
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Crypt K Y:kparts {X}; Nonce n:kparts {Y} |] ==> invKey K:Ks" |
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apply (drule in_Guard, simp) |
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apply (frule guard_not_guard, simp+) |
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apply (drule guard_kparts, simp) |
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by (ind_cases "Crypt K Y:guard n Ks", auto) |
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lemma Guard_extand: "[| Guard n Ks G; Ks <= Ks' |] ==> Guard n Ks' G" |
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by (auto simp: Guard_def dest: guard_extand) |
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lemma guard_invKey [rule_format]: "[| X:guard n Ks; Nonce n:kparts {Y} |] ==> |
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Crypt K Y:kparts {X} --> invKey K:Ks" |
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by (erule guard.induct, auto) |
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lemma Crypt_guard_invKey [rule_format]: "[| Crypt K Y:guard n Ks; |
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Nonce n:kparts {Y} |] ==> invKey K:Ks" |
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by (auto dest: guard_invKey) |
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subsection{*set obtained by decrypting a message*} |
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abbreviation (input) |
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decrypt :: "msg set => key => msg => msg set" where |
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"decrypt H K Y == insert Y (H - {Crypt K Y})" |
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lemma analz_decrypt: "[| Crypt K Y:H; Key (invKey K):H; Nonce n:analz H |] |
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==> Nonce n:analz (decrypt H K Y)" |
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apply (drule_tac P="%H. Nonce n:analz H" in ssubst [OF insert_Diff]) |
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apply assumption |
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apply (simp only: analz_Crypt_if, simp) |
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done |
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lemma parts_decrypt: "[| Crypt K Y:H; X:parts (decrypt H K Y) |] ==> X:parts H" |
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by (erule parts.induct, auto intro: parts.Fst parts.Snd parts.Body) |
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subsection{*number of Crypt's in a message*} |
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consts crypt_nb :: "msg => nat" |
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recdef crypt_nb "measure size" |
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"crypt_nb (Crypt K X) = Suc (crypt_nb X)" |
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"crypt_nb {|X,Y|} = crypt_nb X + crypt_nb Y" |
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"crypt_nb X = 0" (* otherwise *) |
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subsection{*basic facts about @{term crypt_nb}*} |
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lemma non_empty_crypt_msg: "Crypt K Y:parts {X} ==> 0 < crypt_nb X" |
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by (induct X, simp_all, safe, simp_all) |
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subsection{*number of Crypt's in a message list*} |
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consts cnb :: "msg list => nat" |
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recdef cnb "measure size" |
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"cnb [] = 0" |
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"cnb (X#l) = crypt_nb X + cnb l" |
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subsection{*basic facts about @{term cnb}*} |
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lemma cnb_app [simp]: "cnb (l @ l') = cnb l + cnb l'" |
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by (induct l, auto) |
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lemma mem_cnb_minus: "x mem l ==> cnb l = crypt_nb x + (cnb l - crypt_nb x)" |
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by (induct l, auto) |
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lemmas mem_cnb_minus_substI = mem_cnb_minus [THEN ssubst] |
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lemma cnb_minus [simp]: "x mem l ==> cnb (remove l x) = cnb l - crypt_nb x" |
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apply (induct l, auto) |
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by (erule_tac l1=l and x1=x in mem_cnb_minus_substI, simp) |
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lemma parts_cnb: "Z:parts (set l) ==> |
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cnb l = (cnb l - crypt_nb Z) + crypt_nb Z" |
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by (erule parts.induct, auto simp: in_set_conv_decomp) |
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lemma non_empty_crypt: "Crypt K Y:parts (set l) ==> 0 < cnb l" |
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by (induct l, auto dest: non_empty_crypt_msg parts_insert_substD) |
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subsection{*list of kparts*} |
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lemma kparts_msg_set: "EX l. kparts {X} = set l & cnb l = crypt_nb X" |
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apply (induct X, simp_all) |
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apply (rule_tac x="[Agent agent]" in exI, simp) |
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apply (rule_tac x="[Number nat]" in exI, simp) |
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apply (rule_tac x="[Nonce nat]" in exI, simp) |
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apply (rule_tac x="[Key nat]" in exI, simp) |
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apply (rule_tac x="[Hash X]" in exI, simp) |
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apply (clarify, rule_tac x="l@la" in exI, simp) |
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by (clarify, rule_tac x="[Crypt nat X]" in exI, simp) |
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lemma kparts_set: "EX l'. kparts (set l) = set l' & cnb l' = cnb l" |
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apply (induct l) |
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apply (rule_tac x="[]" in exI, simp, clarsimp) |
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apply (subgoal_tac "EX l''. kparts {a} = set l'' & cnb l'' = crypt_nb a", clarify) |
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apply (rule_tac x="l''@l'" in exI, simp) |
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apply (rule kparts_insert_substI, simp) |
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by (rule kparts_msg_set) |
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subsection{*list corresponding to "decrypt"*} |
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constdefs decrypt' :: "msg list => key => msg => msg list" |
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"decrypt' l K Y == Y # remove l (Crypt K Y)" |
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declare decrypt'_def [simp] |
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subsection{*basic facts about @{term decrypt'}*} |
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lemma decrypt_minus: "decrypt (set l) K Y <= set (decrypt' l K Y)" |
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by (induct l, auto) |
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subsection{*if the analyse of a finite guarded set gives n then it must also gives |
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one of the keys of Ks*} |
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lemma Guard_invKey_by_list [rule_format]: "ALL l. cnb l = p |
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--> Guard n Ks (set l) --> Nonce n:analz (set l) |
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--> (EX K. K:Ks & Key K:analz (set l))" |
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apply (induct p) |
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(* case p=0 *) |
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apply (clarify, drule Guard_must_decrypt, simp, clarify) |
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apply (drule kparts_parts, drule non_empty_crypt, simp) |
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(* case p>0 *) |
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apply (clarify, frule Guard_must_decrypt, simp, clarify) |
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apply (drule_tac P="%G. Nonce n:G" in analz_pparts_kparts_substD, simp) |
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apply (frule analz_decrypt, simp_all) |
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apply (subgoal_tac "EX l'. kparts (set l) = set l' & cnb l' = cnb l", clarsimp) |
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apply (drule_tac G="insert Y (set l' - {Crypt K Y})" |
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and H="set (decrypt' l' K Y)" in analz_sub, rule decrypt_minus) |
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apply (rule_tac analz_pparts_kparts_substI, simp) |
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apply (case_tac "K:invKey`Ks") |
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(* K:invKey`Ks *) |
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apply (clarsimp, blast) |
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(* K ~:invKey`Ks *) |
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apply (subgoal_tac "Guard n Ks (set (decrypt' l' K Y))") |
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apply (drule_tac x="decrypt' l' K Y" in spec, simp add: mem_iff) |
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apply (subgoal_tac "Crypt K Y:parts (set l)") |
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apply (drule parts_cnb, rotate_tac -1, simp) |
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apply (clarify, drule_tac X="Key Ka" and H="insert Y (set l')" in analz_sub) |
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apply (rule insert_mono, rule set_remove) |
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apply (simp add: analz_insertD, blast) |
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(* Crypt K Y:parts (set l) *) |
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apply (blast dest: kparts_parts) |
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(* Guard n Ks (set (decrypt' l' K Y)) *) |
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apply (rule_tac H="insert Y (set l')" in Guard_mono) |
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apply (subgoal_tac "Guard n Ks (set l')", simp) |
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apply (rule_tac K=K in guard_Crypt, simp add: Guard_def, simp) |
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apply (drule_tac t="set l'" in sym, simp) |
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apply (rule Guard_kparts, simp, simp) |
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apply (rule_tac B="set l'" in subset_trans, rule set_remove, blast) |
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by (rule kparts_set) |
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lemma Guard_invKey_finite: "[| Nonce n:analz G; Guard n Ks G; finite G |] |
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==> EX K. K:Ks & Key K:analz G" |
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apply (drule finite_list, clarify) |
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by (rule Guard_invKey_by_list, auto) |
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lemma Guard_invKey: "[| Nonce n:analz G; Guard n Ks G |] |
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==> EX K. K:Ks & Key K:analz G" |
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by (auto dest: analz_needs_only_finite Guard_invKey_finite) |
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subsection{*if the analyse of a finite guarded set and a (possibly infinite) set of keys |
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gives n then it must also gives Ks*} |
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lemma Guard_invKey_keyset: "[| Nonce n:analz (G Un H); Guard n Ks G; finite G; |
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keyset H |] ==> EX K. K:Ks & Key K:analz (G Un H)" |
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apply (frule_tac P="%G. Nonce n:G" and G2=G in analz_keyset_substD, simp_all) |
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apply (drule_tac G="G Un (H Int keysfor G)" in Guard_invKey_finite) |
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by (auto simp: Guard_def intro: analz_sub) |
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end |