(* Title: HOL/Auth/OtwayRees_Bad
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
header{*The Otway-Rees Protocol: The Faulty BAN Version*}
theory OtwayRees_Bad imports Public begin
text{*The FAULTY version omitting encryption of Nonce NB, as suggested on
page 247 of
Burrows, Abadi and Needham (1988). A Logic of Authentication.
Proc. Royal Soc. 426
This file illustrates the consequences of such errors. We can still prove
impressive-looking properties such as @{text Spy_not_see_encrypted_key}, yet
the protocol is open to a middleperson attack. Attempting to prove some key
lemmas indicates the possibility of this attack.*}
consts otway :: "event list set"
inductive "otway"
intros
Nil: --{*The empty trace*}
"[] \<in> otway"
Fake: --{*The Spy may say anything he can say. The sender field is correct,
but agents don't use that information.*}
"[| evsf \<in> otway; X \<in> synth (analz (knows Spy evsf)) |]
==> Says Spy B X # evsf \<in> otway"
Reception: --{*A message that has been sent can be received by the
intended recipient.*}
"[| evsr \<in> otway; Says A B X \<in>set evsr |]
==> Gets B X # evsr \<in> otway"
OR1: --{*Alice initiates a protocol run*}
"[| evs1 \<in> otway; Nonce NA \<notin> used evs1 |]
==> Says A B {|Nonce NA, Agent A, Agent B,
Crypt (shrK A) {|Nonce NA, Agent A, Agent B|} |}
# evs1 \<in> otway"
OR2: --{*Bob's response to Alice's message.
This variant of the protocol does NOT encrypt NB.*}
"[| evs2 \<in> otway; Nonce NB \<notin> used evs2;
Gets B {|Nonce NA, Agent A, Agent B, X|} \<in> set evs2 |]
==> Says B Server
{|Nonce NA, Agent A, Agent B, X, Nonce NB,
Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
# evs2 \<in> otway"
OR3: --{*The Server receives Bob's message and checks that the three NAs
match. Then he sends a new session key to Bob with a packet for
forwarding to Alice.*}
"[| evs3 \<in> otway; Key KAB \<notin> used evs3;
Gets Server
{|Nonce NA, Agent A, Agent B,
Crypt (shrK A) {|Nonce NA, Agent A, Agent B|},
Nonce NB,
Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
\<in> set evs3 |]
==> Says Server B
{|Nonce NA,
Crypt (shrK A) {|Nonce NA, Key KAB|},
Crypt (shrK B) {|Nonce NB, Key KAB|}|}
# evs3 \<in> otway"
OR4: --{*Bob receives the Server's (?) message and compares the Nonces with
those in the message he previously sent the Server.
Need @{term "B \<noteq> Server"} because we allow messages to self.*}
"[| evs4 \<in> otway; B \<noteq> Server;
Says B Server {|Nonce NA, Agent A, Agent B, X', Nonce NB,
Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
\<in> set evs4;
Gets B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
\<in> set evs4 |]
==> Says B A {|Nonce NA, X|} # evs4 \<in> otway"
Oops: --{*This message models possible leaks of session keys. The nonces
identify the protocol run.*}
"[| evso \<in> otway;
Says Server B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
\<in> set evso |]
==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso \<in> otway"
declare Says_imp_knows_Spy [THEN analz.Inj, dest]
declare parts.Body [dest]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]
text{*A "possibility property": there are traces that reach the end*}
lemma "[| B \<noteq> Server; Key K \<notin> used [] |]
==> \<exists>NA. \<exists>evs \<in> otway.
Says B A {|Nonce NA, Crypt (shrK A) {|Nonce NA, Key K|}|}
\<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2] otway.Nil
[THEN otway.OR1, THEN otway.Reception,
THEN otway.OR2, THEN otway.Reception,
THEN otway.OR3, THEN otway.Reception, THEN otway.OR4])
apply (possibility, simp add: used_Cons)
done
lemma Gets_imp_Says [dest!]:
"[| Gets B X \<in> set evs; evs \<in> otway |] ==> \<exists>A. Says A B X \<in> set evs"
apply (erule rev_mp)
apply (erule otway.induct, auto)
done
subsection{*For reasoning about the encrypted portion of messages *}
lemma OR2_analz_knows_Spy:
"[| Gets B {|N, Agent A, Agent B, X|} \<in> set evs; evs \<in> otway |]
==> X \<in> analz (knows Spy evs)"
by blast
lemma OR4_analz_knows_Spy:
"[| Gets B {|N, X, Crypt (shrK B) X'|} \<in> set evs; evs \<in> otway |]
==> X \<in> analz (knows Spy evs)"
by blast
lemma Oops_parts_knows_Spy:
"Says Server B {|NA, X, Crypt K' {|NB,K|}|} \<in> set evs
==> K \<in> parts (knows Spy evs)"
by blast
text{*Forwarding lemma: see comments in OtwayRees.thy*}
lemmas OR2_parts_knows_Spy =
OR2_analz_knows_Spy [THEN analz_into_parts, standard]
text{*Theorems of the form @{term "X \<notin> parts (spies evs)"} imply that
NOBODY sends messages containing X! *}
text{*Spy never sees a good agent's shared key!*}
lemma Spy_see_shrK [simp]:
"evs \<in> otway ==> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
by (erule otway.induct, force,
drule_tac [4] OR2_parts_knows_Spy, simp_all, blast+)
lemma Spy_analz_shrK [simp]:
"evs \<in> otway ==> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
by auto
lemma Spy_see_shrK_D [dest!]:
"[|Key (shrK A) \<in> parts (knows Spy evs); evs \<in> otway|] ==> A \<in> bad"
by (blast dest: Spy_see_shrK)
subsection{*Proofs involving analz *}
text{*Describes the form of K and NA when the Server sends this message. Also
for Oops case.*}
lemma Says_Server_message_form:
"[| Says Server B {|NA, X, Crypt (shrK B) {|NB, Key K|}|} \<in> set evs;
evs \<in> otway |]
==> K \<notin> range shrK & (\<exists>i. NA = Nonce i) & (\<exists>j. NB = Nonce j)"
apply (erule rev_mp)
apply (erule otway.induct, simp_all, blast)
done
(****
The following is to prove theorems of the form
Key K \<in> analz (insert (Key KAB) (knows Spy evs)) ==>
Key K \<in> analz (knows Spy evs)
A more general formula must be proved inductively.
****)
text{*Session keys are not used to encrypt other session keys*}
text{*The equality makes the induction hypothesis easier to apply*}
lemma analz_image_freshK [rule_format]:
"evs \<in> otway ==>
\<forall>K KK. KK <= -(range shrK) -->
(Key K \<in> analz (Key`KK Un (knows Spy evs))) =
(K \<in> KK | Key K \<in> analz (knows Spy evs))"
apply (erule otway.induct)
apply (frule_tac [8] Says_Server_message_form)
apply (drule_tac [7] OR4_analz_knows_Spy)
apply (drule_tac [5] OR2_analz_knows_Spy, analz_freshK, spy_analz, auto)
done
lemma analz_insert_freshK:
"[| evs \<in> otway; KAB \<notin> range shrK |] ==>
(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) =
(K = KAB | Key K \<in> analz (knows Spy evs))"
by (simp only: analz_image_freshK analz_image_freshK_simps)
text{*The Key K uniquely identifies the Server's message. *}
lemma unique_session_keys:
"[| Says Server B {|NA, X, Crypt (shrK B) {|NB, K|}|} \<in> set evs;
Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} \<in> set evs;
evs \<in> otway |] ==> X=X' & B=B' & NA=NA' & NB=NB'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule otway.induct, simp_all)
apply blast+ --{*OR3 and OR4*}
done
text{*Crucial secrecy property: Spy does not see the keys sent in msg OR3
Does not in itself guarantee security: an attack could violate
the premises, e.g. by having @{term "A=Spy"} *}
lemma secrecy_lemma:
"[| A \<notin> bad; B \<notin> bad; evs \<in> otway |]
==> Says Server B
{|NA, Crypt (shrK A) {|NA, Key K|},
Crypt (shrK B) {|NB, Key K|}|} \<in> set evs -->
Notes Spy {|NA, NB, Key K|} \<notin> set evs -->
Key K \<notin> analz (knows Spy evs)"
apply (erule otway.induct, force)
apply (frule_tac [7] Says_Server_message_form)
apply (drule_tac [6] OR4_analz_knows_Spy)
apply (drule_tac [4] OR2_analz_knows_Spy)
apply (simp_all add: analz_insert_eq analz_insert_freshK pushes)
apply spy_analz --{*Fake*}
apply (blast dest: unique_session_keys)+ --{*OR3, OR4, Oops*}
done
lemma Spy_not_see_encrypted_key:
"[| Says Server B
{|NA, Crypt (shrK A) {|NA, Key K|},
Crypt (shrK B) {|NB, Key K|}|} \<in> set evs;
Notes Spy {|NA, NB, Key K|} \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> otway |]
==> Key K \<notin> analz (knows Spy evs)"
by (blast dest: Says_Server_message_form secrecy_lemma)
subsection{*Attempting to prove stronger properties *}
text{*Only OR1 can have caused such a part of a message to appear. The premise
@{term "A \<noteq> B"} prevents OR2's similar-looking cryptogram from being picked
up. Original Otway-Rees doesn't need it.*}
lemma Crypt_imp_OR1 [rule_format]:
"[| A \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Crypt (shrK A) {|NA, Agent A, Agent B|} \<in> parts (knows Spy evs) -->
Says A B {|NA, Agent A, Agent B,
Crypt (shrK A) {|NA, Agent A, Agent B|}|} \<in> set evs"
by (erule otway.induct, force,
drule_tac [4] OR2_parts_knows_Spy, simp_all, blast+)
text{*Crucial property: If the encrypted message appears, and A has used NA
to start a run, then it originated with the Server!
The premise @{term "A \<noteq> B"} allows use of @{text Crypt_imp_OR1}*}
text{*Only it is FALSE. Somebody could make a fake message to Server
substituting some other nonce NA' for NB.*}
lemma "[| A \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Crypt (shrK A) {|NA, Key K|} \<in> parts (knows Spy evs) -->
Says A B {|NA, Agent A, Agent B,
Crypt (shrK A) {|NA, Agent A, Agent B|}|}
\<in> set evs -->
(\<exists>B NB. Says Server B
{|NA,
Crypt (shrK A) {|NA, Key K|},
Crypt (shrK B) {|NB, Key K|}|} \<in> set evs)"
apply (erule otway.induct, force,
drule_tac [4] OR2_parts_knows_Spy, simp_all)
apply blast --{*Fake*}
apply blast --{*OR1: it cannot be a new Nonce, contradiction.*}
txt{*OR3 and OR4*}
apply (simp_all add: ex_disj_distrib)
prefer 2 apply (blast intro!: Crypt_imp_OR1) --{*OR4*}
txt{*OR3*}
apply clarify
(*The hypotheses at this point suggest an attack in which nonce NB is used
in two different roles:
Gets Server
{|Nonce NA, Agent Aa, Agent A,
Crypt (shrK Aa) {|Nonce NA, Agent Aa, Agent A|}, Nonce NB,
Crypt (shrK A) {|Nonce NA, Agent Aa, Agent A|}|}
\<in> set evs3
Says A B
{|Nonce NB, Agent A, Agent B,
Crypt (shrK A) {|Nonce NB, Agent A, Agent B|}|}
\<in> set evs3;
*)
(*Thus the key property A_can_trust probably fails too.*)
oops
end