(* Title: HOL/Auth/OtwayRees
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "otway" for the Otway-Rees protocol.
Abadi-Needham simplified version: minimal encryption, explicit messages
Note that the formalization does not even assume that nonces are fresh.
This is because the protocol does not rely on uniqueness of nonces for
security, only for freshness, and the proof script does not prove freshness
properties.
From page 11 of
Abadi and Needham. Prudent Engineering Practice for Cryptographic Protocols.
IEEE Trans. SE 22 (1), 1996
*)
theory OtwayRees_AN = Shared:
consts otway :: "event list set"
inductive "otway"
intros
(*Initial trace is empty*)
Nil: "[] \<in> otway"
(*The spy MAY say anything he CAN say. We do not expect him to
invent new nonces here, but he can also use NS1. Common to
all similar protocols.*)
Fake: "[| evsf \<in> otway; X \<in> synth (analz (knows Spy evsf)) |]
==> Says Spy B X # evsf \<in> otway"
(*A message that has been sent can be received by the
intended recipient.*)
Reception: "[| evsr \<in> otway; Says A B X \<in>set evsr |]
==> Gets B X # evsr \<in> otway"
(*Alice initiates a protocol run*)
OR1: "evs1 \<in> otway
==> Says A B {|Agent A, Agent B, Nonce NA|} # evs1 \<in> otway"
(*Bob's response to Alice's message.*)
OR2: "[| evs2 \<in> otway;
Gets B {|Agent A, Agent B, Nonce NA|} \<in>set evs2 |]
==> Says B Server {|Agent A, Agent B, Nonce NA, Nonce NB|}
# evs2 \<in> otway"
(*The Server receives Bob's message. Then he sends a new
session key to Bob with a packet for forwarding to Alice.*)
OR3: "[| evs3 \<in> otway; Key KAB \<notin> used evs3;
Gets Server {|Agent A, Agent B, Nonce NA, Nonce NB|}
\<in>set evs3 |]
==> Says Server B
{|Crypt (shrK A) {|Nonce NA, Agent A, Agent B, Key KAB|},
Crypt (shrK B) {|Nonce NB, Agent A, Agent B, Key KAB|}|}
# evs3 \<in> otway"
(*Bob receives the Server's (?) message and compares the Nonces with
those in the message he previously sent the Server.
Need B \<noteq> Server because we allow messages to self.*)
OR4: "[| evs4 \<in> otway; B \<noteq> Server;
Says B Server {|Agent A, Agent B, Nonce NA, Nonce NB|} \<in>set evs4;
Gets B {|X, Crypt(shrK B){|Nonce NB,Agent A,Agent B,Key K|}|}
\<in>set evs4 |]
==> Says B A X # evs4 \<in> otway"
(*This message models possible leaks of session keys. The nonces
identify the protocol run. B is not assumed to know shrK A.*)
Oops: "[| evso \<in> otway;
Says Server B
{|Crypt (shrK A) {|Nonce NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|Nonce NB, Agent A, Agent B, 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]
(*A "possibility property": there are traces that reach the end*)
lemma "B \<noteq> Server
==> \<exists>K. \<exists>evs \<in> otway.
Says B A (Crypt (shrK A) {|Nonce NA, Agent A, Agent B, 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], possibility)
done
lemma Gets_imp_Says [dest!]:
"[| Gets B X \<in> set evs; evs \<in> otway |] ==> \<exists>A. Says A B X \<in> set evs"
by (erule rev_mp, erule otway.induct, auto)
(**** Inductive proofs about otway ****)
(** For reasoning about the encrypted portion of messages **)
lemma OR4_analz_knows_Spy:
"[| Gets B {|X, Crypt(shrK B) X'|} \<in> set evs; evs \<in> otway |]
==> X \<in> analz (knows Spy evs)"
by blast
(** Theorems of the form X \<notin> parts (knows Spy evs) imply that NOBODY
sends messages containing X! **)
(*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)"
apply (erule otway.induct, simp_all, blast+)
done
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)
(*** Proofs involving analz ***)
(*Describes the form of K and NA when the Server sends this message.*)
lemma Says_Server_message_form:
"[| Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, 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.
****)
(** Session keys are not used to encrypt other session keys **)
(*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, force)
apply (frule_tac [7] Says_Server_message_form)
apply (drule_tac [6] OR4_analz_knows_Spy, analz_freshK, spy_analz)
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)
(*** The Key K uniquely identifies the Server's message. **)
lemma unique_session_keys:
"[| Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, K|},
Crypt (shrK B) {|NB, Agent A, Agent B, K|}|}
\<in> set evs;
Says Server B'
{|Crypt (shrK A') {|NA', Agent A', Agent B', K|},
Crypt (shrK B') {|NB', Agent A', Agent B', K|}|}
\<in> set evs;
evs \<in> otway |]
==> A=A' & B=B' & NA=NA' & NB=NB'"
apply (erule rev_mp, erule rev_mp, erule otway.induct, simp_all)
(*Remaining cases: OR3 and OR4*)
apply blast+
done
(**** Authenticity properties relating to NA ****)
(*If the encrypted message appears then it originated with the Server!*)
lemma NA_Crypt_imp_Server_msg [rule_format]:
"[| A \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Crypt (shrK A) {|NA, Agent A, Agent B, Key K|} \<in> parts (knows Spy evs)
--> (\<exists>NB. Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs)"
apply (erule otway.induct, force)
apply (simp_all add: ex_disj_distrib)
(*Fake, OR3*)
apply blast+
done
(*Corollary: if A receives B's OR4 message then it originated with the Server.
Freshness may be inferred from nonce NA.*)
lemma A_trusts_OR4:
"[| Says B' A (Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}) \<in> set evs;
A \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> \<exists>NB. Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs"
by (blast intro!: NA_Crypt_imp_Server_msg)
(** 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 A=Spy **)
lemma secrecy_lemma:
"[| A \<notin> bad; B \<notin> bad; evs \<in> otway |]
==> Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, 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 (simp_all add: analz_insert_eq analz_insert_freshK pushes, spy_analz) (*Fake*)
(*OR3, OR4, Oops*)
apply (blast dest: unique_session_keys)+
done
lemma Spy_not_see_encrypted_key:
"[| Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, 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)
(*A's guarantee. The Oops premise quantifies over NB because A cannot know
what it is.*)
lemma A_gets_good_key:
"[| Says B' A (Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}) \<in> set evs;
\<forall>NB. Notes Spy {|NA, NB, Key K|} \<notin> set evs;
A \<notin> bad; B \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Key K \<notin> analz (knows Spy evs)"
by (blast dest!: A_trusts_OR4 Spy_not_see_encrypted_key)
(**** Authenticity properties relating to NB ****)
(*If the encrypted message appears then it originated with the Server!*)
lemma NB_Crypt_imp_Server_msg [rule_format]:
"[| B \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Crypt (shrK B) {|NB, Agent A, Agent B, Key K|} \<in> parts (knows Spy evs)
--> (\<exists>NA. Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs)"
apply (erule otway.induct, force, simp_all add: ex_disj_distrib)
(*Fake, OR3*)
apply blast+
done
(*Guarantee for B: if it gets a well-formed certificate then the Server
has sent the correct message in round 3.*)
lemma B_trusts_OR3:
"[| Says S B {|X, Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs;
B \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> \<exists>NA. Says Server B
{|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|},
Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs"
by (blast intro!: NB_Crypt_imp_Server_msg)
(*The obvious combination of B_trusts_OR3 with Spy_not_see_encrypted_key*)
lemma B_gets_good_key:
"[| Gets B {|X, Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|}
\<in> set evs;
\<forall>NA. Notes Spy {|NA, NB, Key K|} \<notin> set evs;
A \<notin> bad; B \<notin> bad; A \<noteq> B; evs \<in> otway |]
==> Key K \<notin> analz (knows Spy evs)"
by (blast dest: B_trusts_OR3 Spy_not_see_encrypted_key)
end