--- a/src/HOL/Auth/Kerberos_BAN.thy Sat Apr 26 12:38:17 2003 +0200
+++ b/src/HOL/Auth/Kerberos_BAN.thy Sat Apr 26 12:38:42 2003 +0200
@@ -8,17 +8,23 @@
From page 251 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
+
+ Confidentiality (secrecy) and authentication properties rely on
+ temporal checks: strong guarantees in a little abstracted - but
+ very realistic - model (see .thy).
+
+Tidied and converted to Isar by lcp.
*)
-Kerberos_BAN = Shared +
+theory Kerberos_BAN = Shared:
(* Temporal modelization: session keys can be leaked
ONLY when they have expired *)
syntax
- CT :: event list=>nat
- Expired :: [nat, event list] => bool
- RecentAuth :: [nat, event list] => bool
+ CT :: "event list=>nat"
+ Expired :: "[nat, event list] => bool"
+ RecentAuth :: "[nat, event list] => bool"
consts
@@ -28,12 +34,12 @@
(*Duration of the authenticator*)
AutLife :: nat
-rules
+axioms
(*The ticket should remain fresh for two journeys on the network at least*)
- SesKeyLife_LB "2 <= SesKeyLife"
+ SesKeyLife_LB: "2 <= SesKeyLife"
(*The authenticator only for one journey*)
- AutLife_LB "Suc 0 <= AutLife"
+ AutLife_LB: "Suc 0 <= AutLife"
translations
"CT" == "length"
@@ -42,51 +48,382 @@
"RecentAuth T evs" == "CT evs <= AutLife + T"
-consts kerberos_ban :: event list set
+consts kerberos_ban :: "event list set"
inductive "kerberos_ban"
- intrs
+ intros
+
+ Nil: "[] \<in> kerberos_ban"
+
+ Fake: "[| evsf \<in> kerberos_ban; X \<in> synth (analz (spies evsf)) |]
+ ==> Says Spy B X # evsf \<in> kerberos_ban"
+
+
+ Kb1: "[| evs1 \<in> kerberos_ban |]
+ ==> Says A Server {|Agent A, Agent B|} # evs1
+ \<in> kerberos_ban"
+
+
+ Kb2: "[| evs2 \<in> kerberos_ban; Key KAB \<notin> used evs2;
+ Says A' Server {|Agent A, Agent B|} \<in> set evs2 |]
+ ==> Says Server A
+ (Crypt (shrK A)
+ {|Number (CT evs2), Agent B, Key KAB,
+ (Crypt (shrK B) {|Number (CT evs2), Agent A, Key KAB|})|})
+ # evs2 \<in> kerberos_ban"
+
+
+ Kb3: "[| evs3 \<in> kerberos_ban;
+ Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs3;
+ Says A Server {|Agent A, Agent B|} \<in> set evs3;
+ ~ Expired Ts evs3 |]
+ ==> Says A B {|X, Crypt K {|Agent A, Number (CT evs3)|} |}
+ # evs3 \<in> kerberos_ban"
+
+
+ Kb4: "[| evs4 \<in> kerberos_ban;
+ Says A' B {|(Crypt (shrK B) {|Number Ts, Agent A, Key K|}),
+ (Crypt K {|Agent A, Number Ta|}) |}: set evs4;
+ ~ Expired Ts evs4; RecentAuth Ta evs4 |]
+ ==> Says B A (Crypt K (Number Ta)) # evs4
+ \<in> kerberos_ban"
- Nil "[] \\<in> kerberos_ban"
+ (*Old session keys may become compromised*)
+ Oops: "[| evso \<in> kerberos_ban;
+ Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evso;
+ Expired Ts evso |]
+ ==> Notes Spy {|Number Ts, Key K|} # evso \<in> kerberos_ban"
+
+
+declare Says_imp_knows_Spy [THEN parts.Inj, dest] parts.Body [dest]
+declare analz_subset_parts [THEN subsetD, dest]
+declare Fake_parts_insert [THEN subsetD, dest]
+
+declare SesKeyLife_LB [iff] AutLife_LB [iff]
+
- Fake "[| evsf \\<in> kerberos_ban; X \\<in> synth (analz (spies evsf)) |]
- ==> Says Spy B X # evsf \\<in> kerberos_ban"
+(*A "possibility property": there are traces that reach the end.*)
+lemma "\<exists>Timestamp K. \<exists>evs \<in> kerberos_ban.
+ Says B A (Crypt K (Number Timestamp))
+ \<in> set evs"
+apply (cut_tac SesKeyLife_LB)
+apply (intro exI bexI)
+apply (rule_tac [2]
+ kerberos_ban.Nil [THEN kerberos_ban.Kb1, THEN kerberos_ban.Kb2,
+ THEN kerberos_ban.Kb3, THEN kerberos_ban.Kb4], possibility)
+apply (simp_all (no_asm_simp))
+done
+
+
+(**** Inductive proofs about kerberos_ban ****)
+
+(*Forwarding Lemma for reasoning about the encrypted portion of message Kb3*)
+lemma Kb3_msg_in_parts_spies:
+ "Says S A (Crypt KA {|Timestamp, B, K, X|}) \<in> set evs
+ ==> X \<in> parts (spies evs)"
+by blast
+
+lemma Oops_parts_spies:
+ "Says Server A (Crypt (shrK A) {|Timestamp, B, K, X|}) \<in> set evs
+ ==> K \<in> parts (spies evs)"
+by blast
+
+
+(*Spy never sees another agent's shared key! (unless it's bad at start)*)
+lemma Spy_see_shrK [simp]:
+ "evs \<in> kerberos_ban ==> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)"
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
+apply blast+
+done
- Kb1 "[| evs1 \\<in> kerberos_ban |]
- ==> Says A Server {|Agent A, Agent B|} # evs1
- \\<in> kerberos_ban"
+lemma Spy_analz_shrK [simp]:
+ "evs \<in> kerberos_ban ==> (Key (shrK A) \<in> analz (spies evs)) = (A \<in> bad)"
+apply auto
+done
+
+
+lemma Spy_see_shrK_D [dest!]:
+ "[| Key (shrK A) \<in> parts (spies evs);
+ evs \<in> kerberos_ban |] ==> A:bad"
+apply (blast dest: Spy_see_shrK)
+done
+
+lemmas Spy_analz_shrK_D = analz_subset_parts [THEN subsetD, THEN Spy_see_shrK_D, dest!]
+
+
+(*Nobody can have used non-existent keys!*)
+lemma new_keys_not_used [rule_format, simp]:
+ "evs \<in> kerberos_ban ==>
+ Key K \<notin> used evs --> K \<notin> keysFor (parts (spies evs))"
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
+(*Fake*)
+apply (force dest!: keysFor_parts_insert)
+(*Kb2, Kb3, Kb4*)
+apply blast+
+done
+
+(** Lemmas concerning the form of items passed in messages **)
+
+(*Describes the form of K, X and K' when the Server sends this message.*)
+lemma Says_Server_message_form:
+ "[| Says Server A (Crypt K' {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs; evs \<in> kerberos_ban |]
+ ==> K \<notin> range shrK &
+ X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}) &
+ K' = shrK A"
+apply (erule rev_mp)
+apply (erule kerberos_ban.induct, auto)
+done
- Kb2 "[| evs2 \\<in> kerberos_ban; Key KAB \\<notin> used evs2;
- Says A' Server {|Agent A, Agent B|} \\<in> set evs2 |]
- ==> Says Server A
- (Crypt (shrK A)
- {|Number (CT evs2), Agent B, Key KAB,
- (Crypt (shrK B) {|Number (CT evs2), Agent A, Key KAB|})|})
- # evs2 \\<in> kerberos_ban"
+(*If the encrypted message appears then it originated with the Server
+ PROVIDED that A is NOT compromised!
+
+ This shows implicitly the FRESHNESS OF THE SESSION KEY to A
+*)
+lemma A_trusts_K_by_Kb2:
+ "[| Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
+ \<in> parts (spies evs);
+ A \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs"
+apply (erule rev_mp)
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
+apply blast
+done
+
+
+(*If the TICKET appears then it originated with the Server*)
+(*FRESHNESS OF THE SESSION KEY to B*)
+lemma B_trusts_K_by_Kb3:
+ "[| Crypt (shrK B) {|Number Ts, Agent A, Key K|} \<in> parts (spies evs);
+ B \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Says Server A
+ (Crypt (shrK A) {|Number Ts, Agent B, Key K,
+ Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
+ \<in> set evs"
+apply (erule rev_mp)
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
+apply blast
+done
+
+
+(*EITHER describes the form of X when the following message is sent,
+ OR reduces it to the Fake case.
+ Use Says_Server_message_form if applicable.*)
+lemma Says_S_message_form:
+ "[| Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs;
+ evs \<in> kerberos_ban |]
+ ==> (K \<notin> range shrK & X = (Crypt (shrK B) {|Number Ts, Agent A, Key K|}))
+ | X \<in> analz (spies evs)"
+apply (case_tac "A \<in> bad")
+apply (force dest!: Says_imp_spies [THEN analz.Inj])
+apply (frule Says_imp_spies [THEN parts.Inj])
+apply (blast dest!: A_trusts_K_by_Kb2 Says_Server_message_form)
+done
+
- Kb3 "[| evs3 \\<in> kerberos_ban;
- Says S A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
- \\<in> set evs3;
- Says A Server {|Agent A, Agent B|} \\<in> set evs3;
- ~ Expired Ts evs3 |]
- ==> Says A B {|X, Crypt K {|Agent A, Number (CT evs3)|} |}
- # evs3 \\<in> kerberos_ban"
+(****
+ The following is to prove theorems of the form
+
+ Key K \<in> analz (insert (Key KAB) (spies evs)) ==>
+ Key K \<in> analz (spies evs)
+
+ A more general formula must be proved inductively.
+
+****)
+
+
+(** Session keys are not used to encrypt other session keys **)
+
+lemma analz_image_freshK [rule_format (no_asm)]:
+ "evs \<in> kerberos_ban ==>
+ \<forall>K KK. KK <= - (range shrK) -->
+ (Key K \<in> analz (Key`KK Un (spies evs))) =
+ (K \<in> KK | Key K \<in> analz (spies evs))"
+apply (erule kerberos_ban.induct)
+apply (drule_tac [7] Says_Server_message_form)
+apply (erule_tac [5] Says_S_message_form [THEN disjE], analz_freshK, spy_analz)
+done
+
+
+
+lemma analz_insert_freshK:
+ "[| evs \<in> kerberos_ban; KAB \<notin> range shrK |] ==>
+ (Key K \<in> analz (insert (Key KAB) (spies evs))) =
+ (K = KAB | Key K \<in> analz (spies evs))"
+by (simp only: analz_image_freshK analz_image_freshK_simps)
+
+
+(** The session key K uniquely identifies the message **)
+
+lemma unique_session_keys:
+ "[| Says Server A
+ (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}) \<in> set evs;
+ Says Server A'
+ (Crypt (shrK A') {|Number Ts', Agent B', Key K, X'|}) \<in> set evs;
+ evs \<in> kerberos_ban |] ==> A=A' & Ts=Ts' & B=B' & X = X'"
+apply (erule rev_mp)
+apply (erule rev_mp)
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Kb3_msg_in_parts_spies, simp_all)
+(*Kb2: it can't be a new key*)
+apply blast
+done
+
+
+(** Lemma: the session key sent in msg Kb2 would be EXPIRED
+ if the spy could see it!
+**)
+
+lemma lemma2 [rule_format (no_asm)]:
+ "[| A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Says Server A
+ (Crypt (shrK A) {|Number Ts, Agent B, Key K,
+ Crypt (shrK B) {|Number Ts, Agent A, Key K|}|})
+ \<in> set evs -->
+ Key K \<in> analz (spies evs) --> Expired Ts evs"
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Says_Server_message_form)
+apply (frule_tac [5] Says_S_message_form [THEN disjE])
+apply (simp_all (no_asm_simp) add: less_SucI analz_insert_eq analz_insert_freshK pushes)
+txt{*Fake*}
+apply spy_analz
+txt{*Kb2*}
+apply (blast intro: parts_insertI less_SucI)
+txt{*Kb3*}
+apply (case_tac "Aa \<in> bad")
+ prefer 2 apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
+apply (blast dest: Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad elim!: MPair_analz intro: less_SucI)
+txt{*Oops: PROOF FAILED if addIs below*}
+apply (blast dest: unique_session_keys intro!: less_SucI)
+done
- Kb4 "[| evs4 \\<in> kerberos_ban;
- Says A' B {|(Crypt (shrK B) {|Number Ts, Agent A, Key K|}),
- (Crypt K {|Agent A, Number Ta|}) |}: set evs4;
- ~ Expired Ts evs4; RecentAuth Ta evs4 |]
- ==> Says B A (Crypt K (Number Ta)) # evs4
- \\<in> kerberos_ban"
+(** CONFIDENTIALITY for the SERVER:
+ Spy does not see the keys sent in msg Kb2
+ as long as they have NOT EXPIRED
+**)
+lemma Confidentiality_S:
+ "[| Says Server A
+ (Crypt K' {|Number T, Agent B, Key K, X|}) \<in> set evs;
+ ~ Expired T evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
+ |] ==> Key K \<notin> analz (spies evs)"
+apply (frule Says_Server_message_form, assumption)
+apply (blast intro: lemma2)
+done
+
+(**** THE COUNTERPART OF CONFIDENTIALITY
+ [|...; Expired Ts evs; ...|] ==> Key K \<in> analz (spies evs)
+ WOULD HOLD ONLY IF AN OOPS OCCURRED! ---> Nothing to prove! ****)
+
+
+(** CONFIDENTIALITY for ALICE: **)
+(** Also A_trusts_K_by_Kb2 RS Confidentiality_S **)
+lemma Confidentiality_A:
+ "[| Crypt (shrK A) {|Number T, Agent B, Key K, X|} \<in> parts (spies evs);
+ ~ Expired T evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
+ |] ==> Key K \<notin> analz (spies evs)"
+apply (blast dest!: A_trusts_K_by_Kb2 Confidentiality_S)
+done
+
+
+(** CONFIDENTIALITY for BOB: **)
+(** Also B_trusts_K_by_Kb3 RS Confidentiality_S **)
+lemma Confidentiality_B:
+ "[| Crypt (shrK B) {|Number Tk, Agent A, Key K|}
+ \<in> parts (spies evs);
+ ~ Expired Tk evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban
+ |] ==> Key K \<notin> analz (spies evs)"
+apply (blast dest!: B_trusts_K_by_Kb3 Confidentiality_S)
+done
+
- (*Old session keys may become compromised*)
- Oops "[| evso \\<in> kerberos_ban;
- Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
- \\<in> set evso;
- Expired Ts evso |]
- ==> Notes Spy {|Number Ts, Key K|} # evso \\<in> kerberos_ban"
+lemma lemma_B [rule_format]:
+ "[| B \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Key K \<notin> analz (spies evs) -->
+ Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs -->
+ Crypt K (Number Ta) \<in> parts (spies evs) -->
+ Says B A (Crypt K (Number Ta)) \<in> set evs"
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Says_S_message_form)
+apply (drule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
+apply (simp_all (no_asm_simp) add: all_conj_distrib)
+txt{*Fake*}
+apply blast
+txt{*Kb2*}
+apply (force dest: Crypt_imp_invKey_keysFor)
+txt{*Kb4*}
+apply (blast dest: B_trusts_K_by_Kb3 unique_session_keys
+ Says_imp_spies [THEN analz.Inj] Crypt_Spy_analz_bad)
+done
+
+
+(*AUTHENTICATION OF B TO A*)
+lemma Authentication_B:
+ "[| Crypt K (Number Ta) \<in> parts (spies evs);
+ Crypt (shrK A) {|Number Ts, Agent B, Key K, X|}
+ \<in> parts (spies evs);
+ ~ Expired Ts evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Says B A (Crypt K (Number Ta)) \<in> set evs"
+by (blast dest!: A_trusts_K_by_Kb2
+ intro!: lemma_B elim!: Confidentiality_S [THEN [2] rev_notE])
+
+
+
+lemma lemma_A [rule_format]:
+ "[| A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
+ ==>
+ Key K \<notin> analz (spies evs) -->
+ Says Server A (Crypt (shrK A) {|Number Ts, Agent B, Key K, X|})
+ \<in> set evs -->
+ Crypt K {|Agent A, Number Ta|} \<in> parts (spies evs) -->
+ Says A B {|X, Crypt K {|Agent A, Number Ta|}|}
+ \<in> set evs"
+apply (erule kerberos_ban.induct)
+apply (frule_tac [7] Oops_parts_spies)
+apply (frule_tac [5] Says_S_message_form)
+apply (frule_tac [6] Kb3_msg_in_parts_spies, analz_mono_contra)
+apply (simp_all (no_asm_simp) add: all_conj_distrib)
+txt{*Fake*}
+apply blast
+txt{*Kb2*}
+apply (force dest: Crypt_imp_invKey_keysFor)
+txt{*Kb3*}
+apply (blast dest: A_trusts_K_by_Kb2 unique_session_keys)
+done
+
+
+(*AUTHENTICATION OF A TO B*)
+lemma Authentication_A:
+ "[| Crypt K {|Agent A, Number Ta|} \<in> parts (spies evs);
+ Crypt (shrK B) {|Number Ts, Agent A, Key K|}
+ \<in> parts (spies evs);
+ ~ Expired Ts evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> kerberos_ban |]
+ ==> Says A B {|Crypt (shrK B) {|Number Ts, Agent A, Key K|},
+ Crypt K {|Agent A, Number Ta|}|} \<in> set evs"
+by (blast dest!: B_trusts_K_by_Kb3
+ intro!: lemma_A
+ elim!: Confidentiality_S [THEN [2] rev_notE])
end