Theory Yahalom

theory Yahalom
imports Public
(*  Title:      HOL/Auth/Yahalom.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)


header{*The Yahalom Protocol*}

theory Yahalom imports Public begin

text{*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.
*}


inductive_set yahalom :: "event list set"
where
(*Initial trace is empty*)
Nil: "[] ∈ 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: "[| evsf ∈ yahalom; X ∈ synth (analz (knows Spy evsf)) |]
==> Says Spy B X # evsf ∈ yahalom"


(*A message that has been sent can be received by the
intended recipient.*)

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


(*Alice initiates a protocol run*)
| YM1: "[| evs1 ∈ yahalom; Nonce NA ∉ used evs1 |]
==> Says A B {|Agent A, Nonce NA|} # evs1 ∈ yahalom"


(*Bob's response to Alice's message.*)
| YM2: "[| evs2 ∈ yahalom; Nonce NB ∉ used evs2;
Gets B {|Agent A, Nonce NA|} ∈ set evs2 |]
==> Says B Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
# evs2 ∈ 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: "[| evs3 ∈ yahalom; Key KAB ∉ used evs3; KAB ∈ symKeys;
Gets Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
∈ set evs3 |]
==> Says Server A
{|Crypt (shrK A) {|Agent B, Key KAB, Nonce NA, Nonce NB|},
Crypt (shrK B) {|Agent A, Key KAB|}|}
# evs3 ∈ yahalom"


| YM4:
--{*Alice receives the Server's (?) message, checks her Nonce, and
uses the new session key to send Bob his Nonce. The premise
@{term "A ≠ Server"} is needed for @{text Says_Server_not_range}.
Alice can check that K is symmetric by its length.*}

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


(*This message models possible leaks of session keys. The Nonces
identify the protocol run. Quoting Server here ensures they are
correct.*)

| Oops: "[| evso ∈ yahalom;
Says Server A {|Crypt (shrK A)
{|Agent B, Key K, Nonce NA, Nonce NB|},
X|} ∈ set evso |]
==> Notes Spy {|Nonce NA, Nonce NB, Key K|} # evso ∈ yahalom"



definition KeyWithNonce :: "[key, nat, event list] => bool" where
"KeyWithNonce K NB evs ==
∃A B na X.
Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}
∈ 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{*A "possibility property": there are traces that reach the end*}
lemma "[| A ≠ Server; K ∈ symKeys; Key K ∉ used [] |]
==> ∃X NB. ∃evs ∈ yahalom.
Says A B {|X, Crypt K (Nonce NB)|} ∈ 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{*Regularity Lemmas for Yahalom*}

lemma Gets_imp_Says:
"[| Gets B X ∈ set evs; evs ∈ yahalom |] ==> ∃A. Says A B X ∈ set evs"
by (erule rev_mp, erule yahalom.induct, auto)

text{*Must be proved separately for each protocol*}
lemma Gets_imp_knows_Spy:
"[| Gets B X ∈ set evs; evs ∈ yahalom |] ==> X ∈ 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{*Lets us treat YM4 using a similar argument as for the Fake case.*}
lemma YM4_analz_knows_Spy:
"[| Gets A {|Crypt (shrK A) Y, X|} ∈ set evs; evs ∈ yahalom |]
==> X ∈ analz (knows Spy evs)"

by blast

lemmas YM4_parts_knows_Spy =
YM4_analz_knows_Spy [THEN analz_into_parts]

text{*For Oops*}
lemma YM4_Key_parts_knows_Spy:
"Says Server A {|Crypt (shrK A) {|B,K,NA,NB|}, X|} ∈ set evs
==> K ∈ parts (knows Spy evs)"

by (metis parts.Body parts.Fst parts.Snd Says_imp_knows_Spy parts.Inj)

text{*Theorems of the form @{term "X ∉ parts (knows Spy evs)"} imply
that NOBODY sends messages containing X! *}


text{*Spy never sees a good agent's shared key!*}
lemma Spy_see_shrK [simp]:
"evs ∈ yahalom ==> (Key (shrK A) ∈ parts (knows Spy evs)) = (A ∈ bad)"
by (erule yahalom.induct, force,
drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+)

lemma Spy_analz_shrK [simp]:
"evs ∈ yahalom ==> (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 ∈ yahalom|] ==> A ∈ bad"
by (blast dest: Spy_see_shrK)

text{*Nobody can have used non-existent keys!
Needed to apply @{text analz_insert_Key}*}

lemma new_keys_not_used [simp]:
"[|Key K ∉ used evs; K ∈ symKeys; evs ∈ yahalom|]
==> K ∉ keysFor (parts (spies evs))"

apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt{*Fake*}
apply (force dest!: keysFor_parts_insert, auto)
done


text{*Earlier, all protocol proofs declared this theorem.
But only a few proofs need it, e.g. Yahalom and Kerberos IV.*}

lemma new_keys_not_analzd:
"[|K ∈ symKeys; evs ∈ yahalom; Key K ∉ used evs|]
==> K ∉ keysFor (analz (knows Spy evs))"

by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD])


text{*Describes the form of K when the Server sends this message. Useful for
Oops as well as main secrecy property.*}

lemma Says_Server_not_range [simp]:
"[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|}
∈ set evs; evs ∈ yahalom |]
==> K ∉ range shrK"

by (erule rev_mp, erule yahalom.induct, simp_all)


subsection{*Secrecy Theorems*}

(****
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 *}

lemma analz_image_freshK [rule_format]:
"evs ∈ yahalom ==>
∀K KK. KK <= - (range shrK) -->
(Key K ∈ analz (Key`KK Un (knows Spy evs))) =
(K ∈ KK | Key K ∈ 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:
"[| evs ∈ yahalom; 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 A
{|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} ∈ set evs;
Says Server A'
{|Crypt (shrK A') {|Agent B', Key K, na', nb'|}, X'|} ∈ set evs;
evs ∈ yahalom |]
==> A=A' & B=B' & na=na' & nb=nb'"

apply (erule rev_mp, erule rev_mp)
apply (erule yahalom.induct, simp_all)
txt{*YM3, by freshness, and YM4*}
apply blast+
done


text{*Crucial secrecy property: Spy does not see the keys sent in msg YM3*}
lemma secrecy_lemma:
"[| A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Says Server A
{|Crypt (shrK A) {|Agent B, Key K, na, nb|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ set evs -->
Notes Spy {|na, nb, Key K|} ∉ set evs -->
Key K ∉ 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, spy_analz) --{*Fake*}
apply (blast dest: unique_session_keys)+ --{*YM3, Oops*}
done

text{*Final version*}
lemma Spy_not_see_encrypted_key:
"[| Says Server A
{|Crypt (shrK A) {|Agent B, Key K, na, nb|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ set evs;
Notes Spy {|na, nb, Key K|} ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Key K ∉ analz (knows Spy evs)"

by (blast dest: secrecy_lemma)


subsubsection{* Security Guarantee for A upon receiving YM3 *}

text{*If the encrypted message appears then it originated with the Server*}
lemma A_trusts_YM3:
"[| Crypt (shrK A) {|Agent B, Key K, na, nb|} ∈ parts (knows Spy evs);
A ∉ bad; evs ∈ yahalom |]
==> Says Server A
{|Crypt (shrK A) {|Agent B, Key K, na, nb|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ set evs"

apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt{*Fake, YM3*}
apply blast+
done

text{*The obvious combination of @{text A_trusts_YM3} with
@{text Spy_not_see_encrypted_key}*}

lemma A_gets_good_key:
"[| Crypt (shrK A) {|Agent B, Key K, na, nb|} ∈ parts (knows Spy evs);
Notes Spy {|na, nb, Key K|} ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Key K ∉ analz (knows Spy evs)"

by (metis A_trusts_YM3 secrecy_lemma)


subsubsection{* Security Guarantees for B upon receiving YM4 *}

text{*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.*}

lemma B_trusts_YM4_shrK:
"[| Crypt (shrK B) {|Agent A, Key K|} ∈ parts (knows Spy evs);
B ∉ bad; evs ∈ yahalom |]
==> ∃NA NB. Says Server A
{|Crypt (shrK A) {|Agent B, Key K,
Nonce NA, Nonce NB|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ set evs"

apply (erule rev_mp)
apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt{*Fake, YM3*}
apply blast+
done

text{*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
∉ analz(knows Spy evs)"} must be the FIRST antecedent of the
induction formula.*}


lemma B_trusts_YM4_newK [rule_format]:
"[|Crypt K (Nonce NB) ∈ parts (knows Spy evs);
Nonce NB ∉ analz (knows Spy evs); evs ∈ yahalom|]
==> ∃A B NA. Says Server A
{|Crypt (shrK A) {|Agent B, Key K, Nonce NA, Nonce NB|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ 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)
txt{*Fake, YM3*}
apply blast
apply blast
txt{*YM4. A is uncompromised because NB is secure
A's certificate guarantees the existence of the Server message*}

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{* Towards proving secrecy of Nonce NB *}

text{*Lemmas about the predicate KeyWithNonce*}

lemma KeyWithNonceI:
"Says Server A
{|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|}
∈ set evs ==> KeyWithNonce K NB evs"

by (unfold KeyWithNonce_def, blast)

lemma KeyWithNonce_Says [simp]:
"KeyWithNonce K NB (Says S A X # evs) =
(Server = S &
(∃B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|})
| 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{*A fresh key cannot be associated with any nonce
(with respect to a given trace). *}

lemma fresh_not_KeyWithNonce:
"Key K ∉ used evs ==> ~ KeyWithNonce K NB evs"
by (unfold KeyWithNonce_def, blast)

text{*The Server message associates K with NB' and therefore not with any
other nonce NB.*}

lemma Says_Server_KeyWithNonce:
"[| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB'|}, X|}
∈ set evs;
NB ≠ NB'; evs ∈ yahalom |]
==> ~ KeyWithNonce K NB evs"

by (unfold KeyWithNonce_def, blast dest: unique_session_keys)


text{*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 @{text analz_image_freshK}, but it is much more complicated.*}



text{*As with @{text analz_image_freshK}, we take some pains to express the
property as a logical equivalence so that the simplifier can apply it.*}

lemma Nonce_secrecy_lemma:
"P --> (X ∈ analz (G Un H)) --> (X ∈ analz H) ==>
P --> (X ∈ analz (G Un H)) = (X ∈ analz H)"

by (blast intro: analz_mono [THEN subsetD])

lemma Nonce_secrecy:
"evs ∈ yahalom ==>
(∀KK. KK <= - (range shrK) -->
(∀K ∈ KK. K ∈ symKeys --> ~ KeyWithNonce K NB evs) -->
(Nonce NB ∈ analz (Key`KK Un (knows Spy evs))) =
(Nonce NB ∈ 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≠NB to the front*)
Says_Server_KeyWithNonce)
txt{*For Oops, simplification proves @{prop "NBa≠NB"}. By
@{term Says_Server_KeyWithNonce}, we get @{prop "~ KeyWithNonce K NB
evs"}; then simplification can apply the induction hypothesis with
@{term "KK = {K}"}.*}

txt{*Fake*}
apply spy_analz
txt{*YM2*}
apply blast
txt{*YM3*}
apply blast
txt{*YM4*}
apply (erule_tac V = "∀KK. ?P KK" in thin_rl, clarify)
txt{*If @{prop "A ∈ bad"} then @{term NBa} is known, therefore
@{prop "NBa ≠ NB"}. Previous two steps make the next step
faster.*}

apply (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{*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.*}

lemma single_Nonce_secrecy:
"[| Says Server A
{|Crypt (shrK A) {|Agent B, Key KAB, na, Nonce NB'|}, X|}
∈ set evs;
NB ≠ NB'; KAB ∉ range shrK; evs ∈ yahalom |]
==> (Nonce NB ∈ analz (insert (Key KAB) (knows Spy evs))) =
(Nonce NB ∈ 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{* The Nonce NB uniquely identifies B's message. *}

lemma unique_NB:
"[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} ∈ parts (knows Spy evs);
Crypt (shrK B') {|Agent A', Nonce NA', nb|} ∈ parts (knows Spy evs);
evs ∈ yahalom; B ∉ bad; B' ∉ bad |]
==> 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{*Fake, and YM2 by freshness*}
apply blast+
done


text{*Variant useful for proving secrecy of NB. Because nb is assumed to be
secret, we no longer must assume B, B' not bad.*}

lemma Says_unique_NB:
"[| Says C S {|X, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
∈ set evs;
Gets S' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|}
∈ set evs;
nb ∉ analz (knows Spy evs); evs ∈ yahalom |]
==> 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{* A nonce value is never used both as NA and as NB *}

lemma no_nonce_YM1_YM2:
"[|Crypt (shrK B') {|Agent A', Nonce NB, nb'|} ∈ parts(knows Spy evs);
Nonce NB ∉ analz (knows Spy evs); evs ∈ yahalom|]
==> Crypt (shrK B) {|Agent A, na, Nonce NB|} ∉ 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{*Fake, YM2*}
apply blast+
done

text{*The Server sends YM3 only in response to YM2.*}
lemma Says_Server_imp_YM2:
"[| Says Server A {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} ∈ set evs;
evs ∈ yahalom |]
==> Gets Server {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |}
∈ set evs"

by (erule rev_mp, erule yahalom.induct, auto)

text{*A vital theorem for B, that nonce NB remains secure from the Spy.*}
lemma Spy_not_see_NB :
"[| Says B Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
∈ set evs;
(∀k. Notes Spy {|Nonce NA, Nonce NB, k|} ∉ set evs);
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Nonce NB ∉ 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)
txt{*Fake*}
apply spy_analz
txt{*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*}
apply blast
txt{*YM2*}
apply blast
txt{*Prove YM3 by showing that no NB can also be an NA*}
apply (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB)
txt{*LEVEL 7: YM4 and Oops remain*}
apply (clarify, simp add: all_conj_distrib)
txt{*YM4: key K is visible to Spy, contradicting session key secrecy theorem*}
txt{*Case analysis on Aa:bad; PROOF FAILED problems
use @{text Says_unique_NB} to identify message components: @{term "Aa=A"}, @{term "Ba=B"}*}

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)
txt{*Oops case: if the nonce is betrayed now, show that the Oops event is
covered by the quantified Oops assumption.*}

apply (clarify, simp add: all_conj_distrib)
apply (frule Says_Server_imp_YM2, assumption)
apply (metis Gets_imp_Says Says_Server_not_range Says_unique_NB no_nonce_YM1_YM2 parts.Snd single_Nonce_secrecy spies_partsEs(1))
done


text{*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 @{text ∀} 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.*}

lemma B_trusts_YM4:
"[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
Crypt K (Nonce NB)|} ∈ set evs;
Says B Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
∈ set evs;
∀k. Notes Spy {|Nonce NA, Nonce NB, k|} ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Says Server A
{|Crypt (shrK A) {|Agent B, Key K,
Nonce NA, Nonce NB|},
Crypt (shrK B) {|Agent A, Key K|}|}
∈ set evs"

by (blast dest: Spy_not_see_NB Says_unique_NB
Says_Server_imp_YM2 B_trusts_YM4_newK)



text{*The obvious combination of @{text B_trusts_YM4} with
@{text Spy_not_see_encrypted_key}*}

lemma B_gets_good_key:
"[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
Crypt K (Nonce NB)|} ∈ set evs;
Says B Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
∈ set evs;
∀k. Notes Spy {|Nonce NA, Nonce NB, k|} ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Key K ∉ analz (knows Spy evs)"

by (metis B_trusts_YM4 Spy_not_see_encrypted_key)


subsection{*Authenticating B to A*}

text{*The encryption in message YM2 tells us it cannot be faked.*}
lemma B_Said_YM2 [rule_format]:
"[|Crypt (shrK B) {|Agent A, Nonce NA, nb|} ∈ parts (knows Spy evs);
evs ∈ yahalom|]
==> B ∉ bad -->
Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
∈ set evs"

apply (erule rev_mp, erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy, simp_all)
txt{*Fake*}
apply blast
done

text{*If the server sends YM3 then B sent YM2*}
lemma YM3_auth_B_to_A_lemma:
"[|Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
∈ set evs; evs ∈ yahalom|]
==> B ∉ bad -->
Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
∈ set evs"

apply (erule rev_mp, erule yahalom.induct, simp_all)
txt{*YM3, YM4*}
apply (blast dest!: B_Said_YM2)+
done

text{*If A receives YM3 then B has used nonce NA (and therefore is alive)*}
lemma YM3_auth_B_to_A:
"[| Gets A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|}
∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|}
∈ set evs"

by (metis A_trusts_YM3 Gets_imp_analz_Spy YM3_auth_B_to_A_lemma analz.Fst
not_parts_not_analz)


subsection{*Authenticating A to B using the certificate
@{term "Crypt K (Nonce NB)"}*}


text{*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.*}

lemma A_Said_YM3_lemma [rule_format]:
"evs ∈ yahalom
==> Key K ∉ analz (knows Spy evs) -->
Crypt K (Nonce NB) ∈ parts (knows Spy evs) -->
Crypt (shrK B) {|Agent A, Key K|} ∈ parts (knows Spy evs) -->
B ∉ bad -->
(∃X. Says A B {|X, Crypt K (Nonce NB)|} ∈ set evs)"

apply (erule yahalom.induct, force,
frule_tac [6] YM4_parts_knows_Spy)
apply (analz_mono_contra, simp_all)
txt{*Fake*}
apply blast
txt{*YM3: by @{text new_keys_not_used}, the message
@{term "Crypt K (Nonce NB)"} could not exist*}

apply (force dest!: Crypt_imp_keysFor)
txt{*YM4: was @{term "Crypt K (Nonce NB)"} the very last message?
If not, use the induction hypothesis*}

apply (simp add: ex_disj_distrib)
txt{*yes: apply unicity of session keys*}
apply (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{*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.*}

lemma YM4_imp_A_Said_YM3 [rule_format]:
"[| Gets B {|Crypt (shrK B) {|Agent A, Key K|},
Crypt K (Nonce NB)|} ∈ set evs;
Says B Server
{|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|}
∈ set evs;
(∀NA k. Notes Spy {|Nonce NA, Nonce NB, k|} ∉ set evs);
A ∉ bad; B ∉ bad; evs ∈ yahalom |]
==> ∃X. Says A B {|X, Crypt K (Nonce NB)|} ∈ 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