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