Theory OtwayRees_Bad

theory OtwayRees_Bad
imports Public
(*  Title:      HOL/Auth/OtwayRees_Bad.thy
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.*}


inductive_set otway :: "event list set"
where
Nil: --{*The empty trace*}
"[] ∈ otway"

| Fake: --{*The Spy may say anything he can say. The sender field is correct,
but agents don't use that information.*}

"[| evsf ∈ otway; X ∈ synth (analz (knows Spy evsf)) |]
==> Says Spy B X # evsf ∈ otway"



| Reception: --{*A message that has been sent can be received by the
intended recipient.*}

"[| evsr ∈ otway; Says A B X ∈set evsr |]
==> Gets B X # evsr ∈ otway"


| OR1: --{*Alice initiates a protocol run*}
"[| evs1 ∈ otway; Nonce NA ∉ used evs1 |]
==> Says A B {|Nonce NA, Agent A, Agent B,
Crypt (shrK A) {|Nonce NA, Agent A, Agent B|} |}
# evs1 ∈ otway"


| OR2: --{*Bob's response to Alice's message.
This variant of the protocol does NOT encrypt NB.*}

"[| evs2 ∈ otway; Nonce NB ∉ used evs2;
Gets B {|Nonce NA, Agent A, Agent B, X|} ∈ set evs2 |]
==> Says B Server
{|Nonce NA, Agent A, Agent B, X, Nonce NB,
Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
# evs2 ∈ 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 ∈ otway; Key KAB ∉ 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|}|}
∈ set evs3 |]
==> Says Server B
{|Nonce NA,
Crypt (shrK A) {|Nonce NA, Key KAB|},
Crypt (shrK B) {|Nonce NB, Key KAB|}|}
# evs3 ∈ 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 ≠ Server"} because we allow messages to self.*}

"[| evs4 ∈ otway; B ≠ Server;
Says B Server {|Nonce NA, Agent A, Agent B, X', Nonce NB,
Crypt (shrK B) {|Nonce NA, Agent A, Agent B|}|}
∈ set evs4;
Gets B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
∈ set evs4 |]
==> Says B A {|Nonce NA, X|} # evs4 ∈ otway"


| Oops: --{*This message models possible leaks of session keys. The nonces
identify the protocol run.*}

"[| evso ∈ otway;
Says Server B {|Nonce NA, X, Crypt (shrK B) {|Nonce NB, Key K|}|}
∈ set evso |]
==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso ∈ 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 ≠ Server; Key K ∉ used [] |]
==> ∃NA. ∃evs ∈ otway.
Says B A {|Nonce NA, Crypt (shrK A) {|Nonce NA, Key K|}|}
∈ 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 ∈ set evs; evs ∈ otway |] ==> ∃A. Says A B X ∈ 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|} ∈ set evs; evs ∈ otway |]
==> X ∈ analz (knows Spy evs)"

by blast

lemma OR4_analz_knows_Spy:
"[| Gets B {|N, X, Crypt (shrK B) X'|} ∈ set evs; evs ∈ otway |]
==> X ∈ analz (knows Spy evs)"

by blast

lemma Oops_parts_knows_Spy:
"Says Server B {|NA, X, Crypt K' {|NB,K|}|} ∈ set evs
==> K ∈ 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]


text{*Theorems of the form @{term "X ∉ 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 ∈ otway ==> (Key (shrK A) ∈ parts (knows Spy evs)) = (A ∈ bad)"
by (erule otway.induct, force,
drule_tac [4] OR2_parts_knows_Spy, simp_all, blast+)


lemma Spy_analz_shrK [simp]:
"evs ∈ otway ==> (Key (shrK A) ∈ analz (knows Spy evs)) = (A ∈ bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
"[|Key (shrK A) ∈ parts (knows Spy evs); evs ∈ otway|] ==> A ∈ 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|}|} ∈ set evs;
evs ∈ otway |]
==> K ∉ range shrK & (∃i. NA = Nonce i) & (∃j. NB = Nonce j)"

apply (erule rev_mp)
apply (erule otway.induct, simp_all)
done


(****
The following is to prove theorems of the form

Key K ∈ analz (insert (Key KAB) (knows Spy evs)) ==>
Key K ∈ 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 ∈ otway ==>
∀K KK. KK <= -(range shrK) -->
(Key K ∈ analz (Key`KK Un (knows Spy evs))) =
(K ∈ KK | Key K ∈ 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 ∈ otway; KAB ∉ range shrK |] ==>
(Key K ∈ analz (insert (Key KAB) (knows Spy evs))) =
(K = KAB | Key K ∈ 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|}|} ∈ set evs;
Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} ∈ set evs;
evs ∈ 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 ∉ bad; B ∉ bad; evs ∈ otway |]
==> Says Server B
{|NA, Crypt (shrK A) {|NA, Key K|},
Crypt (shrK B) {|NB, Key K|}|} ∈ set evs -->
Notes Spy {|NA, NB, Key K|} ∉ set evs -->
Key K ∉ 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|}|} ∈ set evs;
Notes Spy {|NA, NB, Key K|} ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ otway |]
==> Key K ∉ 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 ≠ 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 ∉ bad; A ≠ B; evs ∈ otway |]
==> Crypt (shrK A) {|NA, Agent A, Agent B|} ∈ parts (knows Spy evs) -->
Says A B {|NA, Agent A, Agent B,
Crypt (shrK A) {|NA, Agent A, Agent B|}|} ∈ 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 ≠ 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 ∉ bad; A ≠ B; evs ∈ otway |]
==> Crypt (shrK A) {|NA, Key K|} ∈ parts (knows Spy evs) -->
Says A B {|NA, Agent A, Agent B,
Crypt (shrK A) {|NA, Agent A, Agent B|}|}
∈ set evs -->
(∃B NB. Says Server B
{|NA,
Crypt (shrK A) {|NA, Key K|},
Crypt (shrK B) {|NB, Key K|}|} ∈ 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|}|}
∈ set evs3
Says A B
{|Nonce NB, Agent A, Agent B,
Crypt (shrK A) {|Nonce NB, Agent A, Agent B|}|}
∈ set evs3;
*)



(*Thus the key property A_can_trust probably fails too.*)
oops

end