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(* Title: HOL/Auth/Kerberos_BAN
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ID: $Id$
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Author: Giampaolo Bella, Cambridge University Computer Laboratory
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Copyright 1998 University of Cambridge
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The Kerberos protocol, BAN version.
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From page 251 of
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Burrows, Abadi and Needham. A Logic of Authentication.
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Proc. Royal Soc. 426 (1989)
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Confidentiality (secrecy) and authentication properties rely on
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temporal checks: strong guarantees in a little abstracted - but
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very realistic - model (see .thy).
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Tidied and converted to Isar by lcp.
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*)
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theory Kerberos_BAN = Shared:
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(* Temporal modelization: session keys can be leaked
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ONLY when they have expired *)
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syntax
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CT :: "event list=>nat"
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Expired :: "[nat, event list] => bool"
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RecentAuth :: "[nat, event list] => bool"
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consts
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(*Duration of the session key*)
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SesKeyLife :: nat
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(*Duration of the authenticator*)
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AutLife :: nat
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axioms
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(*The ticket should remain fresh for two journeys on the network at least*)
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SesKeyLife_LB: "2 <= SesKeyLife"
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(*The authenticator only for one journey*)
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AutLife_LB: "Suc 0 <= AutLife"
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translations
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"CT" == "length"
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"Expired T evs" == "SesKeyLife + T < CT evs"
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"RecentAuth T evs" == "CT evs <= AutLife + T"
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consts kerberos_ban :: "event list set"
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inductive "kerberos_ban"
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intros
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Nil: "[] \<in> kerberos_ban"
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Fake: "[| evsf \<in> kerberos_ban; X \<in> synth (analz (spies evsf)) |]
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==> Says Spy B X # evsf \<in> kerberos_ban"
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Kb1: "[| evs1 \<in> kerberos_ban |]
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==> Says A Server {|Agent A, Agent B|} # evs1
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\<in> kerberos_ban"
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Kb2: "[| evs2 \<in> kerberos_ban; Key KAB \<notin> used evs2;
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Says A' Server {|Agent A, Agent B|} \<in> set evs2 |]
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==> Says Server A
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(Crypt (shrK A)
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{|Number (CT evs2), Agent B, Key KAB,
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(Crypt (shrK B) {|Number (CT evs2), Agent A, Key KAB|})|})
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# evs2 \<in> kerberos_ban"
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Kb3: "[| evs3 \<in> kerberos_ban;
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Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
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\<in> set evs3;
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Says A Server {|Agent A, Agent B|} \<in> set evs3;
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~ Expired Ts evs3 |]
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==> Says A B {|X, Crypt K {|Agent A, Number (CT evs3)|} |}
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# evs3 \<in> kerberos_ban"
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Kb4: "[| evs4 \<in> kerberos_ban;
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Says A' B {|(Crypt (shrK B) {|Number Ts, Agent A, Key K|}),
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(Crypt K {|Agent A, Number Ta|}) |}: set evs4;
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~ Expired Ts evs4; RecentAuth Ta evs4 |]
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==> Says B A (Crypt K (Number Ta)) # evs4
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\<in> kerberos_ban"
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(*Old session keys may become compromised*)
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Oops: "[| evso \<in> kerberos_ban;
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Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
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\<in> set evso;
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Expired Ts evso |]
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==> Notes Spy {|Number Ts, Key K|} # evso \<in> kerberos_ban"
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declare Says_imp_knows_Spy [THEN parts.Inj, dest] parts.Body [dest]
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declare analz_subset_parts [THEN subsetD, dest]
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declare Fake_parts_insert [THEN subsetD, dest]
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declare SesKeyLife_LB [iff] AutLife_LB [iff]
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(*A "possibility property": there are traces that reach the end.*)
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lemma "\<exists>Timestamp K. \<exists>evs \<in> kerberos_ban.
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Says B A (Crypt K (Number Timestamp))
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\<in> set evs"
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apply (cut_tac SesKeyLife_LB)
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apply (intro exI bexI)
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apply (rule_tac [2]
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kerberos_ban.Nil [THEN kerberos_ban.Kb1, THEN kerberos_ban.Kb2,
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THEN kerberos_ban.Kb3, THEN kerberos_ban.Kb4], possibility)
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apply (simp_all (no_asm_simp))
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done
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(**** Inductive proofs about kerberos_ban ****)
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(*Forwarding Lemma for reasoning about the encrypted portion of message Kb3*)
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lemma Kb3_msg_in_parts_spies:
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"Says S A (Crypt KA {|Timestamp, B, K, X|}) \<in> set evs
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==> X \<in> parts (spies evs)"
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by blast
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lemma Oops_parts_spies:
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"Says Server A (Crypt (shrK A) {|Timestamp, B, K, X|}) \<in> set evs
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==> K \<in> parts (spies evs)"
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by blast
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(*Spy never sees another agent's shared key! (unless it's bad at start)*)
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lemma Spy_see_shrK [simp]:
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"evs \<in> kerberos_ban ==> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)"
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Oops_parts_spies)
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apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
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apply blast+
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done
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lemma Spy_analz_shrK [simp]:
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"evs \<in> kerberos_ban ==> (Key (shrK A) \<in> analz (spies evs)) = (A \<in> bad)"
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apply auto
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done
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lemma Spy_see_shrK_D [dest!]:
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"[| Key (shrK A) \<in> parts (spies evs);
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evs \<in> kerberos_ban |] ==> A:bad"
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apply (blast dest: Spy_see_shrK)
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done
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lemmas Spy_analz_shrK_D = analz_subset_parts [THEN subsetD, THEN Spy_see_shrK_D, dest!]
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(*Nobody can have used non-existent keys!*)
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lemma new_keys_not_used [rule_format, simp]:
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"evs \<in> kerberos_ban ==>
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Key K \<notin> used evs --> K \<notin> keysFor (parts (spies evs))"
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Oops_parts_spies)
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apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
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(*Fake*)
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apply (force dest!: keysFor_parts_insert)
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(*Kb2, Kb3, Kb4*)
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apply blast+
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done
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(** Lemmas concerning the form of items passed in messages **)
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(*Describes the form of K, X and K' when the Server sends this message.*)
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lemma Says_Server_message_form:
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"[| Says Server A (Crypt K' {|Number Ts, Agent B, Key K, X|})
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\<in> set evs; evs \<in> kerberos_ban |]
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==> K \<notin> range shrK &
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X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}) &
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K' = shrK A"
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apply (erule rev_mp)
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apply (erule kerberos_ban.induct, auto)
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done
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(*If the encrypted message appears then it originated with the Server
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PROVIDED that A is NOT compromised!
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This shows implicitly the FRESHNESS OF THE SESSION KEY to A
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*)
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lemma A_trusts_K_by_Kb2:
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"[| Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
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\<in> parts (spies evs);
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A \<notin> bad; evs \<in> kerberos_ban |]
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==> Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
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\<in> set evs"
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apply (erule rev_mp)
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Oops_parts_spies)
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apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
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apply blast
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done
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(*If the TICKET appears then it originated with the Server*)
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(*FRESHNESS OF THE SESSION KEY to B*)
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lemma B_trusts_K_by_Kb3:
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"[| Crypt (shrK B) {|Number Ts, Agent A, Key K|} \<in> parts (spies evs);
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B \<notin> bad; evs \<in> kerberos_ban |]
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==> Says Server A
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(Crypt (shrK A) {|Number Ts, Agent B, Key K,
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Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
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\<in> set evs"
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apply (erule rev_mp)
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Oops_parts_spies)
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apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
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apply blast
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done
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(*EITHER describes the form of X when the following message is sent,
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OR reduces it to the Fake case.
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Use Says_Server_message_form if applicable.*)
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lemma Says_S_message_form:
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"[| Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
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\<in> set evs;
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evs \<in> kerberos_ban |]
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==> (K \<notin> range shrK & X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}))
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| X \<in> analz (spies evs)"
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apply (case_tac "A \<in> bad")
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apply (force dest!: Says_imp_spies [THEN analz.Inj])
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apply (frule Says_imp_spies [THEN parts.Inj])
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apply (blast dest!: A_trusts_K_by_Kb2 Says_Server_message_form)
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done
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(****
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The following is to prove theorems of the form
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Key K \<in> analz (insert (Key KAB) (spies evs)) ==>
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Key K \<in> analz (spies evs)
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A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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lemma analz_image_freshK [rule_format (no_asm)]:
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"evs \<in> kerberos_ban ==>
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\<forall>K KK. KK <= - (range shrK) -->
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(Key K \<in> analz (Key`KK Un (spies evs))) =
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(K \<in> KK | Key K \<in> analz (spies evs))"
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apply (erule kerberos_ban.induct)
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apply (drule_tac [7] Says_Server_message_form)
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apply (erule_tac [5] Says_S_message_form [THEN disjE], analz_freshK, spy_analz)
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done
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lemma analz_insert_freshK:
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"[| evs \<in> kerberos_ban; KAB \<notin> range shrK |] ==>
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(Key K \<in> analz (insert (Key KAB) (spies evs))) =
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(K = KAB | Key K \<in> analz (spies evs))"
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by (simp only: analz_image_freshK analz_image_freshK_simps)
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(** The session key K uniquely identifies the message **)
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lemma unique_session_keys:
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"[| Says Server A
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(Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) \<in> set evs;
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Says Server A'
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(Crypt (shrK A') {|Number Ts', Agent B', Key K, X'|}) \<in> set evs;
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evs \<in> kerberos_ban |] ==> A=A' & Ts=Ts' & B=B' & X = X'"
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apply (erule rev_mp)
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apply (erule rev_mp)
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Oops_parts_spies)
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apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
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(*Kb2: it can't be a new key*)
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apply blast
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done
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(** Lemma: the session key sent in msg Kb2 would be EXPIRED
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if the spy could see it!
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**)
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lemma lemma2 [rule_format (no_asm)]:
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"[| A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
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==> Says Server A
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(Crypt (shrK A) {|Number Ts, Agent B, Key K,
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Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
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\<in> set evs -->
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Key K \<in> analz (spies evs) --> Expired Ts evs"
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apply (erule kerberos_ban.induct)
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apply (frule_tac [7] Says_Server_message_form)
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apply (frule_tac [5] Says_S_message_form [THEN disjE])
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apply (simp_all (no_asm_simp) add: less_SucI analz_insert_eq analz_insert_freshK pushes)
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txt{*Fake*}
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apply spy_analz
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txt{*Kb2*}
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apply (blast intro: parts_insertI less_SucI)
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txt{*Kb3*}
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apply (case_tac "Aa \<in> bad")
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prefer 2 apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
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apply (blast dest: Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad elim!: MPair_analz intro: less_SucI)
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txt{*Oops: PROOF FAILED if addIs below*}
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apply (blast dest: unique_session_keys intro!: less_SucI)
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done
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(** CONFIDENTIALITY for the SERVER:
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Spy does not see the keys sent in msg Kb2
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as long as they have NOT EXPIRED
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**)
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lemma Confidentiality_S:
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"[| Says Server A
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(Crypt K' {|Number T, Agent B, Key K, X|}) \<in> set evs;
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~ Expired T evs;
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A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
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|] ==> Key K \<notin> analz (spies evs)"
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apply (frule Says_Server_message_form, assumption)
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apply (blast intro: lemma2)
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done
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(**** THE COUNTERPART OF CONFIDENTIALITY
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[|...; Expired Ts evs; ...|] ==> Key K \<in> analz (spies evs)
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WOULD HOLD ONLY IF AN OOPS OCCURRED! ---> Nothing to prove! ****)
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(** CONFIDENTIALITY for ALICE: **)
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(** Also A_trusts_K_by_Kb2 RS Confidentiality_S **)
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lemma Confidentiality_A:
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"[| Crypt (shrK A) {|Number T, Agent B, Key K, X|} \<in> parts (spies evs);
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~ Expired T evs;
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A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
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|] ==> Key K \<notin> analz (spies evs)"
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apply (blast dest!: A_trusts_K_by_Kb2 Confidentiality_S)
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done
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(** CONFIDENTIALITY for BOB: **)
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(** Also B_trusts_K_by_Kb3 RS Confidentiality_S **)
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lemma Confidentiality_B:
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"[| Crypt (shrK B) {|Number Tk, Agent A, Key K|}
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\<in> parts (spies evs);
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~ Expired Tk evs;
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A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
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|] ==> Key K \<notin> analz (spies evs)"
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apply (blast dest!: B_trusts_K_by_Kb3 Confidentiality_S)
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done
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lemma lemma_B [rule_format]:
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"[| B \<notin> bad; evs \<in> kerberos_ban |]
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==> Key K \<notin> analz (spies evs) -->
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Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
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362 |
\<in> set evs -->
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363 |
Crypt K (Number Ta) \<in> parts (spies evs) -->
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364 |
Says B A (Crypt K (Number Ta)) \<in> set evs"
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365 |
apply (erule kerberos_ban.induct)
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366 |
apply (frule_tac [7] Oops_parts_spies)
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367 |
apply (frule_tac [5] Says_S_message_form)
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368 |
apply (drule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
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369 |
apply (simp_all (no_asm_simp) add: all_conj_distrib)
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370 |
txt{*Fake*}
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371 |
apply blast
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372 |
txt{*Kb2*}
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373 |
apply (force dest: Crypt_imp_invKey_keysFor)
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374 |
txt{*Kb4*}
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375 |
apply (blast dest: B_trusts_K_by_Kb3 unique_session_keys
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376 |
Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad)
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377 |
done
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378 |
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379 |
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|
380 |
(*AUTHENTICATION OF B TO A*)
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381 |
lemma Authentication_B:
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382 |
"[| Crypt K (Number Ta) \<in> parts (spies evs);
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383 |
Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
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|
384 |
\<in> parts (spies evs);
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|
385 |
~ Expired Ts evs;
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|
386 |
A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
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387 |
==> Says B A (Crypt K (Number Ta)) \<in> set evs"
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388 |
by (blast dest!: A_trusts_K_by_Kb2
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389 |
intro!: lemma_B elim!: Confidentiality_S [THEN [2] rev_notE])
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390 |
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|
391 |
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|
392 |
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|
393 |
lemma lemma_A [rule_format]:
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|
394 |
"[| A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
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|
395 |
==>
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|
396 |
Key K \<notin> analz (spies evs) -->
|
|
397 |
Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
|
|
398 |
\<in> set evs -->
|
|
399 |
Crypt K {|Agent A, Number Ta|} \<in> parts (spies evs) -->
|
|
400 |
Says A B {|X, Crypt K {|Agent A, Number Ta|}|}
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|
401 |
\<in> set evs"
|
|
402 |
apply (erule kerberos_ban.induct)
|
|
403 |
apply (frule_tac [7] Oops_parts_spies)
|
|
404 |
apply (frule_tac [5] Says_S_message_form)
|
|
405 |
apply (frule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
|
|
406 |
apply (simp_all (no_asm_simp) add: all_conj_distrib)
|
|
407 |
txt{*Fake*}
|
|
408 |
apply blast
|
|
409 |
txt{*Kb2*}
|
|
410 |
apply (force dest: Crypt_imp_invKey_keysFor)
|
|
411 |
txt{*Kb3*}
|
|
412 |
apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
|
|
413 |
done
|
|
414 |
|
|
415 |
|
|
416 |
(*AUTHENTICATION OF A TO B*)
|
|
417 |
lemma Authentication_A:
|
|
418 |
"[| Crypt K {|Agent A, Number Ta|} \<in> parts (spies evs);
|
|
419 |
Crypt (shrK B) {|Number Ts, Agent A, Key K|}
|
|
420 |
\<in> parts (spies evs);
|
|
421 |
~ Expired Ts evs;
|
|
422 |
A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
|
|
423 |
==> Says A B {|Crypt (shrK B) {|Number Ts, Agent A, Key K|},
|
|
424 |
Crypt K {|Agent A, Number Ta|}|} \<in> set evs"
|
|
425 |
by (blast dest!: B_trusts_K_by_Kb3
|
|
426 |
intro!: lemma_A
|
|
427 |
elim!: Confidentiality_S [THEN [2] rev_notE])
|
5053
|
428 |
|
|
429 |
end
|