(* Title: HOL/Auth/Yahalom.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
section\<open>The Yahalom Protocol\<close>
theory Yahalom imports Public begin
text\<open>From page 257 of
Burrows, Abadi and Needham (1989). A Logic of Authentication.
Proc. Royal Soc. 426
This theory has the prototypical example of a secrecy relation, KeyCryptNonce.
\<close>
inductive_set yahalom :: "event list set"
where
(*Initial trace is empty*)
Nil: "[] \<in> yahalom"
(*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: "\<lbrakk>evsf \<in> yahalom; X \<in> synth (analz (knows Spy evsf))\<rbrakk>
\<Longrightarrow> Says Spy B X # evsf \<in> yahalom"
(*A message that has been sent can be received by the
intended recipient.*)
| Reception: "\<lbrakk>evsr \<in> yahalom; Says A B X \<in> set evsr\<rbrakk>
\<Longrightarrow> Gets B X # evsr \<in> yahalom"
(*Alice initiates a protocol run*)
| YM1: "\<lbrakk>evs1 \<in> yahalom; Nonce NA \<notin> used evs1\<rbrakk>
\<Longrightarrow> Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1 \<in> yahalom"
(*Bob's response to Alice's message.*)
| YM2: "\<lbrakk>evs2 \<in> yahalom; Nonce NB \<notin> used evs2;
Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk>
\<Longrightarrow> Says B Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
# evs2 \<in> yahalom"
(*The Server receives Bob's message. He responds by sending a
new session key to Alice, with a packet for forwarding to Bob.*)
| YM3: "\<lbrakk>evs3 \<in> yahalom; Key KAB \<notin> used evs3; KAB \<in> symKeys;
Gets Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
\<in> set evs3\<rbrakk>
\<Longrightarrow> Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA, Nonce NB\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key KAB\<rbrace>\<rbrace>
# evs3 \<in> yahalom"
| YM4:
\<comment> \<open>Alice receives the Server's (?) message, checks her Nonce, and
uses the new session key to send Bob his Nonce. The premise
@{term "A \<noteq> Server"} is needed for \<open>Says_Server_not_range\<close>.
Alice can check that K is symmetric by its length.\<close>
"\<lbrakk>evs4 \<in> yahalom; A \<noteq> Server; K \<in> symKeys;
Gets A \<lbrace>Crypt(shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace>
\<in> set evs4;
Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk>
\<Longrightarrow> Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4 \<in> yahalom"
(*This message models possible leaks of session keys. The Nonces
identify the protocol run. Quoting Server here ensures they are
correct.*)
| Oops: "\<lbrakk>evso \<in> yahalom;
Says Server A \<lbrace>Crypt (shrK A)
\<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>,
X\<rbrace> \<in> set evso\<rbrakk>
\<Longrightarrow> Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> yahalom"
definition KeyWithNonce :: "[key, nat, event list] => bool" where
"KeyWithNonce K NB evs ==
\<exists>A B na X.
Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace>
\<in> set evs"
declare Says_imp_analz_Spy [dest]
declare parts.Body [dest]
declare Fake_parts_insert_in_Un [dest]
declare analz_into_parts [dest]
text\<open>A "possibility property": there are traces that reach the end\<close>
lemma "\<lbrakk>A \<noteq> Server; K \<in> symKeys; Key K \<notin> used []\<rbrakk>
\<Longrightarrow> \<exists>X NB. \<exists>evs \<in> yahalom.
Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
apply (intro exI bexI)
apply (rule_tac [2] yahalom.Nil
[THEN yahalom.YM1, THEN yahalom.Reception,
THEN yahalom.YM2, THEN yahalom.Reception,
THEN yahalom.YM3, THEN yahalom.Reception,
THEN yahalom.YM4])
apply (possibility, simp add: used_Cons)
done
subsection\<open>Regularity Lemmas for Yahalom\<close>
lemma Gets_imp_Says:
"\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> \<exists>A. Says A B X \<in> set evs"
by (erule rev_mp, erule yahalom.induct, auto)
text\<open>Must be proved separately for each protocol\<close>
lemma Gets_imp_knows_Spy:
"\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> X \<in> knows Spy evs"
by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)
lemmas Gets_imp_analz_Spy = Gets_imp_knows_Spy [THEN analz.Inj]
declare Gets_imp_analz_Spy [dest]
text\<open>Lets us treat YM4 using a similar argument as for the Fake case.\<close>
lemma YM4_analz_knows_Spy:
"\<lbrakk>Gets A \<lbrace>Crypt (shrK A) Y, X\<rbrace> \<in> set evs; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> X \<in> analz (knows Spy evs)"
by blast
lemmas YM4_parts_knows_Spy =
YM4_analz_knows_Spy [THEN analz_into_parts]
text\<open>For Oops\<close>
lemma YM4_Key_parts_knows_Spy:
"Says Server A \<lbrace>Crypt (shrK A) \<lbrace>B,K,NA,NB\<rbrace>, X\<rbrace> \<in> set evs
\<Longrightarrow> K \<in> parts (knows Spy evs)"
by (metis parts.Body parts.Fst parts.Snd Says_imp_knows_Spy parts.Inj)
text\<open>Theorems of the form @{term "X \<notin> parts (knows Spy evs)"} imply
that NOBODY sends messages containing X!\<close>
text\<open>Spy never sees a good agent's shared key!\<close>
lemma Spy_see_shrK [simp]:
"evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
by (erule yahalom.induct, force,
drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)
lemma Spy_analz_shrK [simp]:
"evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
by auto
lemma Spy_see_shrK_D [dest!]:
"\<lbrakk>Key (shrK A) \<in> parts (knows Spy evs); evs \<in> yahalom\<rbrakk> \<Longrightarrow> A \<in> bad"
by (blast dest: Spy_see_shrK)
text\<open>Nobody can have used non-existent keys!
Needed to apply \<open>analz_insert_Key\<close>\<close>
lemma new_keys_not_used [simp]:
"\<lbrakk>Key K \<notin> used evs; K \<in> symKeys; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> K \<notin> keysFor (parts (spies evs))"
apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt\<open>Fake\<close>
apply (force dest!: keysFor_parts_insert, auto)
done
text\<open>Earlier, all protocol proofs declared this theorem.
But only a few proofs need it, e.g. Yahalom and Kerberos IV.\<close>
lemma new_keys_not_analzd:
"\<lbrakk>K \<in> symKeys; evs \<in> yahalom; Key K \<notin> used evs\<rbrakk>
\<Longrightarrow> K \<notin> keysFor (analz (knows Spy evs))"
by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD])
text\<open>Describes the form of K when the Server sends this message. Useful for
Oops as well as main secrecy property.\<close>
lemma Says_Server_not_range [simp]:
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace>
\<in> set evs; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> K \<notin> range shrK"
by (erule rev_mp, erule yahalom.induct, simp_all)
subsection\<open>Secrecy Theorems\<close>
(****
The following is to prove theorems of the form
Key K \<in> analz (insert (Key KAB) (knows Spy evs)) \<Longrightarrow>
Key K \<in> analz (knows Spy evs)
A more general formula must be proved inductively.
****)
text\<open>Session keys are not used to encrypt other session keys\<close>
lemma analz_image_freshK [rule_format]:
"evs \<in> yahalom \<Longrightarrow>
\<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 yahalom.induct,
drule_tac [7] YM4_analz_knows_Spy, analz_freshK, spy_analz, blast)
apply (simp only: Says_Server_not_range analz_image_freshK_simps)
apply safe
done
lemma analz_insert_freshK:
"\<lbrakk>evs \<in> yahalom; KAB \<notin> range shrK\<rbrakk> \<Longrightarrow>
(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\<open>The Key K uniquely identifies the Server's message.\<close>
lemma unique_session_keys:
"\<lbrakk>Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs;
Says Server A'
\<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs;
evs \<in> yahalom\<rbrakk>
\<Longrightarrow> A=A' & B=B' & na=na' & nb=nb'"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, simp_all)
txt\<open>YM3, by freshness, and YM4\<close>
apply blast+
done
text\<open>Crucial secrecy property: Spy does not see the keys sent in msg YM3\<close>
lemma secrecy_lemma:
"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs -->
Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs -->
Key K \<notin> analz (knows Spy evs)"
apply (erule yahalom.induct, force,
drule_tac [6] YM4_analz_knows_Spy)
apply (simp_all add: pushes analz_insert_eq analz_insert_freshK)
subgoal \<comment> \<open>Fake\<close> by spy_analz
subgoal \<comment> \<open>YM3\<close> by blast
subgoal \<comment> \<open>Oops\<close> by (blast dest: unique_session_keys)
done
text\<open>Final version\<close>
lemma Spy_not_see_encrypted_key:
"\<lbrakk>Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs;
Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
by (blast dest: secrecy_lemma)
subsubsection\<open>Security Guarantee for A upon receiving YM3\<close>
text\<open>If the encrypted message appears then it originated with the Server\<close>
lemma A_trusts_YM3:
"\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
A \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs"
apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt\<open>Fake, YM3\<close>
apply blast+
done
text\<open>The obvious combination of \<open>A_trusts_YM3\<close> with
\<open>Spy_not_see_encrypted_key\<close>\<close>
lemma A_gets_good_key:
"\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs);
Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
by (metis A_trusts_YM3 secrecy_lemma)
subsubsection\<open>Security Guarantees for B upon receiving YM4\<close>
text\<open>B knows, by the first part of A's message, that the Server distributed
the key for A and B. But this part says nothing about nonces.\<close>
lemma B_trusts_YM4_shrK:
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs);
B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> \<exists>NA NB. Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K,
Nonce NA, Nonce NB\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs"
apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt\<open>Fake, YM3\<close>
apply blast+
done
text\<open>B knows, by the second part of A's message, that the Server
distributed the key quoting nonce NB. This part says nothing about
agent names. Secrecy of NB is crucial. Note that @{term "Nonce NB
\<notin> analz(knows Spy evs)"} must be the FIRST antecedent of the
induction formula.\<close>
lemma B_trusts_YM4_newK [rule_format]:
"\<lbrakk>Crypt K (Nonce NB) \<in> parts (knows Spy evs);
Nonce NB \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk>
\<Longrightarrow> \<exists>A B NA. Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
subgoal \<comment> \<open>Fake\<close> by blast
subgoal \<comment> \<open>YM3\<close> by blast
txt\<open>YM4. A is uncompromised because NB is secure
A's certificate guarantees the existence of the Server message\<close>
apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
dest: Says_imp_spies
parts.Inj [THEN parts.Fst, THEN A_trusts_YM3])
done
subsubsection\<open>Towards proving secrecy of Nonce NB\<close>
text\<open>Lemmas about the predicate KeyWithNonce\<close>
lemma KeyWithNonceI:
"Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace>
\<in> set evs \<Longrightarrow> KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast)
lemma KeyWithNonce_Says [simp]:
"KeyWithNonce K NB (Says S A X # evs) =
(Server = S &
(\<exists>B n X'. X = \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, n, Nonce NB\<rbrace>, X'\<rbrace>)
| KeyWithNonce K NB evs)"
by (simp add: KeyWithNonce_def, blast)
lemma KeyWithNonce_Notes [simp]:
"KeyWithNonce K NB (Notes A X # evs) = KeyWithNonce K NB evs"
by (simp add: KeyWithNonce_def)
lemma KeyWithNonce_Gets [simp]:
"KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs"
by (simp add: KeyWithNonce_def)
text\<open>A fresh key cannot be associated with any nonce
(with respect to a given trace).\<close>
lemma fresh_not_KeyWithNonce:
"Key K \<notin> used evs \<Longrightarrow> ~ KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast)
text\<open>The Server message associates K with NB' and therefore not with any
other nonce NB.\<close>
lemma Says_Server_KeyWithNonce:
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB'\<rbrace>, X\<rbrace>
\<in> set evs;
NB \<noteq> NB'; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> ~ KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast dest: unique_session_keys)
text\<open>The only nonces that can be found with the help of session keys are
those distributed as nonce NB by the Server. The form of the theorem
recalls \<open>analz_image_freshK\<close>, but it is much more complicated.\<close>
text\<open>As with \<open>analz_image_freshK\<close>, we take some pains to express the
property as a logical equivalence so that the simplifier can apply it.\<close>
lemma Nonce_secrecy_lemma:
"P --> (X \<in> analz (G Un H)) --> (X \<in> analz H) \<Longrightarrow>
P --> (X \<in> analz (G Un H)) = (X \<in> analz H)"
by (blast intro: analz_mono [THEN subsetD])
lemma Nonce_secrecy:
"evs \<in> yahalom \<Longrightarrow>
(\<forall>KK. KK <= - (range shrK) -->
(\<forall>K \<in> KK. K \<in> symKeys --> ~ KeyWithNonce K NB evs) -->
(Nonce NB \<in> analz (Key`KK Un (knows Spy evs))) =
(Nonce NB \<in> analz (knows Spy evs)))"
apply (erule yahalom.induct,
frule_tac [7] YM4_analz_knows_Spy)
apply (safe del: allI impI intro!: Nonce_secrecy_lemma [THEN impI, THEN allI])
apply (simp_all del: image_insert image_Un
add: analz_image_freshK_simps split_ifs
all_conj_distrib ball_conj_distrib
analz_image_freshK fresh_not_KeyWithNonce
imp_disj_not1 (*Moves NBa\<noteq>NB to the front*)
Says_Server_KeyWithNonce)
txt\<open>For Oops, simplification proves @{prop "NBa\<noteq>NB"}. By
@{term Says_Server_KeyWithNonce}, we get @{prop "~ KeyWithNonce K NB
evs"}; then simplification can apply the induction hypothesis with
@{term "KK = {K}"}.\<close>
subgoal \<comment> \<open>Fake\<close> by spy_analz
subgoal \<comment> \<open>YM2\<close> by blast
subgoal \<comment> \<open>YM3\<close> by blast
subgoal \<comment> \<open>YM4: If @{prop "A \<in> bad"} then @{term NBa} is known, therefore @{prop "NBa \<noteq> NB"}.\<close>
by (metis A_trusts_YM3 Gets_imp_analz_Spy Gets_imp_knows_Spy KeyWithNonce_def
Spy_analz_shrK analz.Fst analz.Snd analz_shrK_Decrypt parts.Fst parts.Inj)
done
text\<open>Version required below: if NB can be decrypted using a session key then
it was distributed with that key. The more general form above is required
for the induction to carry through.\<close>
lemma single_Nonce_secrecy:
"\<lbrakk>Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, na, Nonce NB'\<rbrace>, X\<rbrace>
\<in> set evs;
NB \<noteq> NB'; KAB \<notin> range shrK; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) =
(Nonce NB \<in> analz (knows Spy evs))"
by (simp_all del: image_insert image_Un imp_disjL
add: analz_image_freshK_simps split_ifs
Nonce_secrecy Says_Server_KeyWithNonce)
subsubsection\<open>The Nonce NB uniquely identifies B's message.\<close>
lemma unique_NB:
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace> \<in> parts (knows Spy evs);
evs \<in> yahalom; B \<notin> bad; B' \<notin> bad\<rbrakk>
\<Longrightarrow> NA' = NA & A' = A & B' = B"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt\<open>Fake, and YM2 by freshness\<close>
apply blast+
done
text\<open>Variant useful for proving secrecy of NB. Because nb is assumed to be
secret, we no longer must assume B, B' not bad.\<close>
lemma Says_unique_NB:
"\<lbrakk>Says C S \<lbrace>X, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
\<in> set evs;
Gets S' \<lbrace>X', Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace>\<rbrace>
\<in> set evs;
nb \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk>
\<Longrightarrow> NA' = NA & A' = A & B' = B"
by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad
dest: Says_imp_spies unique_NB parts.Inj analz.Inj)
subsubsection\<open>A nonce value is never used both as NA and as NB\<close>
lemma no_nonce_YM1_YM2:
"\<lbrakk>Crypt (shrK B') \<lbrace>Agent A', Nonce NB, nb'\<rbrace> \<in> parts(knows Spy evs);
Nonce NB \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Crypt (shrK B) \<lbrace>Agent A, na, Nonce NB\<rbrace> \<notin> parts(knows Spy evs)"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
txt\<open>Fake, YM2\<close>
apply blast+
done
text\<open>The Server sends YM3 only in response to YM2.\<close>
lemma Says_Server_imp_YM2:
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, k, na, nb\<rbrace>, X\<rbrace> \<in> set evs;
evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, na, nb\<rbrace>\<rbrace>
\<in> set evs"
by (erule rev_mp, erule yahalom.induct, auto)
text\<open>A vital theorem for B, that nonce NB remains secure from the Spy.\<close>
theorem Spy_not_see_NB :
"\<lbrakk>Says B Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
\<in> set evs;
(\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs);
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Nonce NB \<notin> analz (knows Spy evs)"
apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_analz_knows_Spy)
apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq
analz_insert_freshK)
subgoal \<comment> \<open>Fake\<close> by spy_analz
subgoal \<comment> \<open>YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!\<close> by blast
subgoal \<comment> \<open>YM2\<close> by blast
subgoal \<comment> \<open>YM3: because no NB can also be an NA\<close>
by (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
subgoal \<comment> \<open>YM4: key K is visible to Spy, contradicting session key secrecy theorem\<close>
\<comment> \<open>Case analysis on whether Aa is bad;
use \<open>Says_unique_NB\<close> to identify message components: @{term "Aa=A"}, @{term "Ba=B"}\<close>
apply clarify
apply (blast dest!: Says_unique_NB analz_shrK_Decrypt
parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]
dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2
Spy_not_see_encrypted_key)
done
subgoal \<comment> \<open>Oops case: if the nonce is betrayed now, show that the Oops event is
covered by the quantified Oops assumption.\<close>
apply clarsimp
apply (metis Says_Server_imp_YM2 Gets_imp_Says Says_Server_not_range Says_unique_NB no_nonce_YM1_YM2 parts.Snd single_Nonce_secrecy spies_partsEs(1))
done
done
text\<open>B's session key guarantee from YM4. The two certificates contribute to a
single conclusion about the Server's message. Note that the "Notes Spy"
assumption must quantify over \<open>\<forall>\<close> POSSIBLE keys instead of our particular K.
If this run is broken and the spy substitutes a certificate containing an
old key, B has no means of telling.\<close>
lemma B_trusts_YM4:
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
Crypt K (Nonce NB)\<rbrace> \<in> set evs;
Says B Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
\<in> set evs;
\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Says Server A
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K,
Nonce NA, Nonce NB\<rbrace>,
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace>
\<in> set evs"
by (blast dest: Spy_not_see_NB Says_unique_NB
Says_Server_imp_YM2 B_trusts_YM4_newK)
text\<open>The obvious combination of \<open>B_trusts_YM4\<close> with
\<open>Spy_not_see_encrypted_key\<close>\<close>
lemma B_gets_good_key:
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
Crypt K (Nonce NB)\<rbrace> \<in> set evs;
Says B Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
\<in> set evs;
\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
by (metis B_trusts_YM4 Spy_not_see_encrypted_key)
subsection\<open>Authenticating B to A\<close>
text\<open>The encryption in message YM2 tells us it cannot be faked.\<close>
lemma B_Said_YM2 [rule_format]:
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs);
evs \<in> yahalom\<rbrakk>
\<Longrightarrow> B \<notin> bad -->
Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
\<in> set evs"
apply (erule rev_mp, erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt\<open>Fake\<close>
apply blast
done
text\<open>If the server sends YM3 then B sent YM2\<close>
lemma YM3_auth_B_to_A_lemma:
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
\<in> set evs; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> B \<notin> bad -->
Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
\<in> set evs"
apply (erule rev_mp, erule yahalom.induct, simp_all)
txt\<open>YM3, YM4\<close>
apply (blast dest!: B_Said_YM2)+
done
text\<open>If A receives YM3 then B has used nonce NA (and therefore is alive)\<close>
theorem YM3_auth_B_to_A:
"\<lbrakk>Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace>
\<in> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace>
\<in> set evs"
by (metis A_trusts_YM3 Gets_imp_analz_Spy YM3_auth_B_to_A_lemma analz.Fst
not_parts_not_analz)
subsection\<open>Authenticating A to B using the certificate
@{term "Crypt K (Nonce NB)"}\<close>
text\<open>Assuming the session key is secure, if both certificates are present then
A has said NB. We can't be sure about the rest of A's message, but only
NB matters for freshness.\<close>
theorem A_Said_YM3_lemma [rule_format]:
"evs \<in> yahalom
\<Longrightarrow> Key K \<notin> analz (knows Spy evs) -->
Crypt K (Nonce NB) \<in> parts (knows Spy evs) -->
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs) -->
B \<notin> bad -->
(\<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs)"
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
subgoal \<comment> \<open>Fake\<close> by blast
subgoal \<comment> \<open>YM3 because the message @{term "Crypt K (Nonce NB)"} could not exist\<close>
by (force dest!: Crypt_imp_keysFor)
subgoal \<comment> \<open>YM4: was @{term "Crypt K (Nonce NB)"} the very last message? If not, use the induction hypothesis,
otherwise by unicity of session keys\<close>
by (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK Crypt_Spy_analz_bad
dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys)
done
text\<open>If B receives YM4 then A has used nonce NB (and therefore is alive).
Moreover, A associates K with NB (thus is talking about the same run).
Other premises guarantee secrecy of K.\<close>
theorem YM4_imp_A_Said_YM3 [rule_format]:
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>,
Crypt K (Nonce NB)\<rbrace> \<in> set evs;
Says B Server
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace>
\<in> set evs;
(\<forall>NA k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs);
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk>
\<Longrightarrow> \<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
by (metis A_Said_YM3_lemma B_gets_good_key Gets_imp_analz_Spy YM4_parts_knows_Spy analz.Fst not_parts_not_analz)
end