(* Title: HOL/Auth/Kerberos_BAN
ID: $Id$
Author: Giampaolo Bella, Cambridge University Computer Laboratory
Copyright 1998 University of Cambridge
The Kerberos protocol, BAN version.
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.
*)
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"
consts
(*Duration of the session key*)
SesKeyLife :: nat
(*Duration of the authenticator*)
AutLife :: nat
axioms
(*The ticket should remain fresh for two journeys on the network at least*)
SesKeyLife_LB: "2 <= SesKeyLife"
(*The authenticator only for one journey*)
AutLife_LB: "Suc 0 <= AutLife"
translations
"CT" == "length"
"Expired T evs" == "SesKeyLife + T < CT evs"
"RecentAuth T evs" == "CT evs <= AutLife + T"
consts kerberos_ban :: "event list set"
inductive "kerberos_ban"
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"
(*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]
(*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
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
(*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
(****
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
(** 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
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