Up to index of Isabelle/HOL/Auth
theory Guard_Shared(****************************************************************************** date: march 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{*lemmas on guarded messages for protocols with symmetric keys*} theory Guard_Shared imports Guard GuardK Shared begin subsection{*Extensions to Theory @{text Shared}*} declare initState.simps [simp del] subsubsection{*a little abbreviation*} abbreviation Ciph :: "agent => msg => msg" where "Ciph A X == Crypt (shrK A) X" subsubsection{*agent associated to a key*} constdefs agt :: "key => agent" "agt K == @A. K = shrK A" lemma agt_shrK [simp]: "agt (shrK A) = A" by (simp add: agt_def) subsubsection{*basic facts about @{term initState}*} lemma no_Crypt_in_parts_init [simp]: "Crypt K X ~:parts (initState A)" by (cases A, auto simp: initState.simps) lemma no_Crypt_in_analz_init [simp]: "Crypt K X ~:analz (initState A)" by auto lemma no_shrK_in_analz_init [simp]: "A ~:bad ==> Key (shrK A) ~:analz (initState Spy)" by (auto simp: initState.simps) lemma shrK_notin_initState_Friend [simp]: "A ~= Friend C ==> Key (shrK A) ~: parts (initState (Friend C))" by (auto simp: initState.simps) lemma keyset_init [iff]: "keyset (initState A)" by (cases A, auto simp: keyset_def initState.simps) subsubsection{*sets of symmetric keys*} constdefs shrK_set :: "key set => bool" "shrK_set Ks == ALL K. K:Ks --> (EX A. K = shrK A)" lemma in_shrK_set: "[| shrK_set Ks; K:Ks |] ==> EX A. K = shrK A" by (simp add: shrK_set_def) lemma shrK_set1 [iff]: "shrK_set {shrK A}" by (simp add: shrK_set_def) lemma shrK_set2 [iff]: "shrK_set {shrK A, shrK B}" by (simp add: shrK_set_def) subsubsection{*sets of good keys*} constdefs good :: "key set => bool" "good Ks == ALL K. K:Ks --> agt K ~:bad" lemma in_good: "[| good Ks; K:Ks |] ==> agt K ~:bad" by (simp add: good_def) lemma good1 [simp]: "A ~:bad ==> good {shrK A}" by (simp add: good_def) lemma good2 [simp]: "[| A ~:bad; B ~:bad |] ==> good {shrK A, shrK B}" by (simp add: good_def) subsection{*Proofs About Guarded Messages*} subsubsection{*small hack*} lemma shrK_is_invKey_shrK: "shrK A = invKey (shrK A)" by simp lemmas shrK_is_invKey_shrK_substI = shrK_is_invKey_shrK [THEN ssubst] lemmas invKey_invKey_substI = invKey [THEN ssubst] lemma "Nonce n:parts {X} ==> Crypt (shrK A) X:guard n {shrK A}" apply (rule shrK_is_invKey_shrK_substI, rule invKey_invKey_substI) by (rule Guard_Nonce, simp+) subsubsection{*guardedness results on nonces*} lemma guard_ciph [simp]: "shrK A:Ks ==> Ciph A X:guard n Ks" by (rule Guard_Nonce, simp) lemma guardK_ciph [simp]: "shrK A:Ks ==> Ciph A X:guardK n Ks" by (rule Guard_Key, simp) lemma Guard_init [iff]: "Guard n Ks (initState B)" by (induct B, auto simp: Guard_def initState.simps) lemma Guard_knows_max': "Guard n Ks (knows_max' C evs) ==> Guard n Ks (knows_max C evs)" by (simp add: knows_max_def) lemma Nonce_not_used_Guard_spies [dest]: "Nonce n ~:used evs ==> Guard n Ks (spies evs)" by (auto simp: Guard_def dest: not_used_not_known parts_sub) lemma Nonce_not_used_Guard [dest]: "[| evs:p; Nonce n ~:used evs; Gets_correct p; one_step p |] ==> Guard n Ks (knows (Friend C) evs)" by (auto simp: Guard_def dest: known_used parts_trans) lemma Nonce_not_used_Guard_max [dest]: "[| evs:p; Nonce n ~:used evs; Gets_correct p; one_step p |] ==> Guard n Ks (knows_max (Friend C) evs)" by (auto simp: Guard_def dest: known_max_used parts_trans) lemma Nonce_not_used_Guard_max' [dest]: "[| evs:p; Nonce n ~:used evs; Gets_correct p; one_step p |] ==> Guard n Ks (knows_max' (Friend C) evs)" apply (rule_tac H="knows_max (Friend C) evs" in Guard_mono) by (auto simp: knows_max_def) subsubsection{*guardedness results on keys*} lemma GuardK_init [simp]: "n ~:range shrK ==> GuardK n Ks (initState B)" by (induct B, auto simp: GuardK_def initState.simps) lemma GuardK_knows_max': "[| GuardK n A (knows_max' C evs); n ~:range shrK |] ==> GuardK n A (knows_max C evs)" by (simp add: knows_max_def) lemma Key_not_used_GuardK_spies [dest]: "Key n ~:used evs ==> GuardK n A (spies evs)" by (auto simp: GuardK_def dest: not_used_not_known parts_sub) lemma Key_not_used_GuardK [dest]: "[| evs:p; Key n ~:used evs; Gets_correct p; one_step p |] ==> GuardK n A (knows (Friend C) evs)" by (auto simp: GuardK_def dest: known_used parts_trans) lemma Key_not_used_GuardK_max [dest]: "[| evs:p; Key n ~:used evs; Gets_correct p; one_step p |] ==> GuardK n A (knows_max (Friend C) evs)" by (auto simp: GuardK_def dest: known_max_used parts_trans) lemma Key_not_used_GuardK_max' [dest]: "[| evs:p; Key n ~:used evs; Gets_correct p; one_step p |] ==> GuardK n A (knows_max' (Friend C) evs)" apply (rule_tac H="knows_max (Friend C) evs" in GuardK_mono) by (auto simp: knows_max_def) subsubsection{*regular protocols*} constdefs regular :: "event list set => bool" "regular p == ALL evs A. evs:p --> (Key (shrK A):parts (spies evs)) = (A:bad)" lemma shrK_parts_iff_bad [simp]: "[| evs:p; regular p |] ==> (Key (shrK A):parts (spies evs)) = (A:bad)" by (auto simp: regular_def) lemma shrK_analz_iff_bad [simp]: "[| evs:p; regular p |] ==> (Key (shrK A):analz (spies evs)) = (A:bad)" by auto lemma Guard_Nonce_analz: "[| Guard n Ks (spies evs); evs:p; shrK_set Ks; good Ks; regular p |] ==> Nonce n ~:analz (spies evs)" apply (clarify, simp only: knows_decomp) apply (drule Guard_invKey_keyset, simp+, safe) apply (drule in_good, simp) apply (drule in_shrK_set, simp+, clarify) apply (frule_tac A=A in shrK_analz_iff_bad) by (simp add: knows_decomp)+ lemma GuardK_Key_analz: "[| GuardK n Ks (spies evs); evs:p; shrK_set Ks; good Ks; regular p; n ~:range shrK |] ==> Key n ~:analz (spies evs)" apply (clarify, simp only: knows_decomp) apply (drule GuardK_invKey_keyset, clarify, simp+, simp add: initState.simps) apply clarify apply (drule in_good, simp) apply (drule in_shrK_set, simp+, clarify) apply (frule_tac A=A in shrK_analz_iff_bad) by (simp add: knows_decomp)+ end
lemma agt_shrK:
agt (shrK A) = A
lemma no_Crypt_in_parts_init:
Crypt K X ∉ parts (initState A)
lemma no_Crypt_in_analz_init:
Crypt K X ∉ analz (initState A)
lemma no_shrK_in_analz_init:
A ∉ bad ==> Key (shrK A) ∉ analz (initState Spy)
lemma shrK_notin_initState_Friend:
A ≠ Friend C ==> Key (shrK A) ∉ parts (initState (Friend C))
lemma keyset_init:
keyset (initState A)
lemma in_shrK_set:
[| shrK_set Ks; K ∈ Ks |] ==> ∃A. K = shrK A
lemma shrK_set1:
shrK_set {shrK A}
lemma shrK_set2:
shrK_set {shrK A, shrK B}
lemma in_good:
[| good Ks; K ∈ Ks |] ==> agt K ∉ bad
lemma good1:
A ∉ bad ==> good {shrK A}
lemma good2:
[| A ∉ bad; B ∉ bad |] ==> good {shrK A, shrK B}
lemma shrK_is_invKey_shrK:
shrK A = invKey (shrK A)
lemma shrK_is_invKey_shrK_substI:
P (invKey (shrK A1)) ==> P (shrK A1)
lemma invKey_invKey_substI:
P s ==> P (invKey (invKey s))
lemma
Nonce n ∈ parts {X} ==> Ciph A X ∈ guard n {shrK A}
lemma guard_ciph:
shrK A ∈ Ks ==> Ciph A X ∈ guard n Ks
lemma guardK_ciph:
shrK A ∈ Ks ==> Ciph A X ∈ guardK n Ks
lemma Guard_init:
Guard n Ks (initState B)
lemma Guard_knows_max':
Guard n Ks (knows_max' C evs) ==> Guard n Ks (knows_max C evs)
lemma Nonce_not_used_Guard_spies:
Nonce n ∉ used evs ==> Guard n Ks (knows Spy evs)
lemma Nonce_not_used_Guard:
[| evs ∈ p; Nonce n ∉ used evs; Gets_correct p; one_step p |]
==> Guard n Ks (knows (Friend C) evs)
lemma Nonce_not_used_Guard_max:
[| evs ∈ p; Nonce n ∉ used evs; Gets_correct p; one_step p |]
==> Guard n Ks (knows_max (Friend C) evs)
lemma Nonce_not_used_Guard_max':
[| evs ∈ p; Nonce n ∉ used evs; Gets_correct p; one_step p |]
==> Guard n Ks (knows_max' (Friend C) evs)
lemma GuardK_init:
n ∉ range shrK ==> GuardK n Ks (initState B)
lemma GuardK_knows_max':
[| GuardK n A (knows_max' C evs); n ∉ range shrK |]
==> GuardK n A (knows_max C evs)
lemma Key_not_used_GuardK_spies:
Key n ∉ used evs ==> GuardK n A (knows Spy evs)
lemma Key_not_used_GuardK:
[| evs ∈ p; Key n ∉ used evs; Gets_correct p; one_step p |]
==> GuardK n A (knows (Friend C) evs)
lemma Key_not_used_GuardK_max:
[| evs ∈ p; Key n ∉ used evs; Gets_correct p; one_step p |]
==> GuardK n A (knows_max (Friend C) evs)
lemma Key_not_used_GuardK_max':
[| evs ∈ p; Key n ∉ used evs; Gets_correct p; one_step p |]
==> GuardK n A (knows_max' (Friend C) evs)
lemma shrK_parts_iff_bad:
[| evs ∈ p; regular p |] ==> (Key (shrK A) ∈ parts (knows Spy evs)) = (A ∈ bad)
lemma shrK_analz_iff_bad:
[| evs ∈ p; regular p |] ==> (Key (shrK A) ∈ analz (knows Spy evs)) = (A ∈ bad)
lemma Guard_Nonce_analz:
[| Guard n Ks (knows Spy evs); evs ∈ p; shrK_set Ks; good Ks; regular p |]
==> Nonce n ∉ analz (knows Spy evs)
lemma GuardK_Key_analz:
[| GuardK n Ks (knows Spy evs); evs ∈ p; shrK_set Ks; good Ks; regular p;
n ∉ range shrK |]
==> Key n ∉ analz (knows Spy evs)