Cosmetic changes: margins, indentation, ...
(* Title: HOL/Auth/Yahalom
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "yahalom" for the Yahalom protocol.
From page 257 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open Yahalom;
proof_timing:=true;
HOL_quantifiers := false;
Pretty.setdepth 25;
(*A "possibility property": there are traces that reach the end*)
goal thy
"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \
\ ==> EX X NB K. EX evs: yahalom. \
\ Says A B {|X, Crypt K (Nonce NB)|} : set evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS
yahalom.YM4) 2);
by possibility_tac;
result();
(**** Inductive proofs about yahalom ****)
(*Nobody sends themselves messages*)
goal thy "!!evs. evs: yahalom ==> ALL A X. Says A A X ~: set evs";
by (etac yahalom.induct 1);
by (Auto_tac());
qed_spec_mp "not_Says_to_self";
Addsimps [not_Says_to_self];
AddSEs [not_Says_to_self RSN (2, rev_notE)];
(** For reasoning about the encrypted portion of messages **)
(*Lets us treat YM4 using a similar argument as for the Fake case.*)
goal thy "!!evs. Says S A {|Crypt (shrK A) Y, X|} : set evs ==> \
\ X : analz (sees Spy evs)";
by (blast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "YM4_analz_sees_Spy";
bind_thm ("YM4_parts_sees_Spy",
YM4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
(*Relates to both YM4 and Oops*)
goal thy "!!evs. Says S A {|Crypt (shrK A) {|B,K,NA,NB|}, X|} : set evs ==> \
\ K : parts (sees Spy evs)";
by (blast_tac (!claset addSEs partsEs
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
qed "YM4_Key_parts_sees_Spy";
(*For proving the easier theorems about X ~: parts (sees Spy evs).*)
fun parts_sees_tac i =
forward_tac [YM4_Key_parts_sees_Spy] (i+6) THEN
forward_tac [YM4_parts_sees_Spy] (i+5) THEN
prove_simple_subgoals_tac i;
(*Induction for regularity theorems. If induction formula has the form
X ~: analz (sees Spy evs) --> ... then it shortens the proof by discarding
needless information about analz (insert X (sees Spy evs)) *)
fun parts_induct_tac i =
etac yahalom.induct i
THEN
REPEAT (FIRSTGOAL analz_mono_contra_tac)
THEN parts_sees_tac i;
(** Theorems of the form X ~: parts (sees Spy evs) imply that NOBODY
sends messages containing X! **)
(*Spy never sees another agent's shared key! (unless it's lost at start)*)
goal thy
"!!evs. evs : yahalom ==> (Key (shrK A) : parts (sees Spy evs)) = (A : lost)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (Blast_tac 1);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : yahalom ==> (Key (shrK A) : analz (sees Spy evs)) = (A : lost)";
by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset));
qed "Spy_analz_shrK";
Addsimps [Spy_analz_shrK];
goal thy "!!A. [| Key (shrK A) : parts (sees Spy evs); \
\ evs : yahalom |] ==> A:lost";
by (blast_tac (!claset addDs [Spy_see_shrK]) 1);
qed "Spy_see_shrK_D";
bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
(*Nobody can have used non-existent keys! Needed to apply analz_insert_Key*)
goal thy "!!evs. evs : yahalom ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (sees Spy evs))";
by (parts_induct_tac 1);
(*YM4: Key K is not fresh!*)
by (blast_tac (!claset addSEs sees_Spy_partsEs) 3);
(*YM3*)
by (Blast_tac 2);
(*Fake*)
by (best_tac
(!claset addIs [impOfSubs analz_subset_parts]
addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
addss (!simpset)) 1);
qed_spec_mp "new_keys_not_used";
bind_thm ("new_keys_not_analzd",
[analz_subset_parts RS keysFor_mono,
new_keys_not_used] MRS contra_subsetD);
Addsimps [new_keys_not_used, new_keys_not_analzd];
(*Describes the form of K when the Server sends this message. Useful for
Oops as well as main secrecy property.*)
goal thy
"!!evs. [| Says Server A {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} \
\ : set evs; \
\ evs : yahalom |] \
\ ==> K ~: range shrK";
by (etac rev_mp 1);
by (etac yahalom.induct 1);
by (ALLGOALS Asm_simp_tac);
by (Blast_tac 1);
qed "Says_Server_message_form";
(*For proofs involving analz.*)
val analz_sees_tac =
forward_tac [YM4_analz_sees_Spy] 6 THEN
forward_tac [Says_Server_message_form] 7 THEN
assume_tac 7 THEN REPEAT ((etac exE ORELSE' hyp_subst_tac) 7);
(****
The following is to prove theorems of the form
Key K : analz (insert (Key KAB) (sees Spy evs)) ==>
Key K : analz (sees Spy evs)
A more general formula must be proved inductively.
****)
(** Session keys are not used to encrypt other session keys **)
goal thy
"!!evs. evs : yahalom ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (sees Spy evs))) = \
\ (K : KK | Key K : analz (sees Spy evs))";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (REPEAT_FIRST (resolve_tac [allI, impI]));
by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
(*Fake*)
by (spy_analz_tac 2);
(*Base*)
by (Blast_tac 1);
qed_spec_mp "analz_image_freshK";
goal thy
"!!evs. [| evs : yahalom; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (sees Spy evs)) = \
\ (K = KAB | Key K : analz (sees Spy evs))";
by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
qed "analz_insert_freshK";
(*** The Key K uniquely identifies the Server's message. **)
goal thy
"!!evs. evs : yahalom ==> \
\ EX A' B' na' nb' X'. ALL A B na nb X. \
\ Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, X|} \
\ : set evs --> A=A' & B=B' & na=na' & nb=nb' & X=X'";
by (etac yahalom.induct 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
by (Step_tac 1);
by (ex_strip_tac 2);
by (Blast_tac 2);
(*Remaining case: YM3*)
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
(*...we assume X is a recent message and handle this case by contradiction*)
by (blast_tac (!claset addSEs sees_Spy_partsEs
delrules [conjI] (*no split-up to 4 subgoals*)) 1);
val lemma = result();
goal thy
"!!evs. [| 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'";
by (prove_unique_tac lemma 1);
qed "unique_session_keys";
(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set evs --> \
\ Says A Spy {|na, nb, Key K|} ~: set evs --> \
\ Key K ~: analz (sees Spy evs)";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [analz_insert_eq, not_parts_not_analz,
analz_insert_freshK]
setloop split_tac [expand_if])));
(*Oops*)
by (blast_tac (!claset addDs [unique_session_keys]) 3);
(*YM3*)
by (blast_tac (!claset delrules [impCE]
addSEs sees_Spy_partsEs
addIs [impOfSubs analz_subset_parts]) 2);
(*Fake*)
by (spy_analz_tac 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);
(*Final version*)
goal thy
"!!evs. [| Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set evs; \
\ Says A Spy {|na, nb, Key K|} ~: set evs; \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> Key K ~: analz (sees Spy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (blast_tac (!claset addSEs [lemma]) 1);
qed "Spy_not_see_encrypted_key";
(** Security Guarantee for A upon receiving YM3 **)
(*If the encrypted message appears then it originated with the Server*)
goal thy
"!!evs. [| Crypt (shrK A) {|Agent B, Key K, na, nb|} \
\ : parts (sees Spy evs); \
\ A ~: lost; evs : yahalom |] \
\ ==> Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, na, nb|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
qed "A_trusts_YM3";
(** Security Guarantees for B upon receiving YM4 **)
(*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.*)
goal thy
"!!evs. [| Crypt (shrK B) {|Agent A, Key K|} : parts (sees Spy evs); \
\ B ~: lost; evs : yahalom |] \
\ ==> EX 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";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
(*YM3*)
by (Blast_tac 1);
qed "B_trusts_YM4_shrK";
(*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.*)
goal thy
"!!evs. evs : yahalom \
\ ==> Nonce NB ~: analz (sees Spy evs) --> \
\ Crypt K (Nonce NB) : parts (sees Spy evs) --> \
\ (EX 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)";
by (parts_induct_tac 1);
(*YM3 & Fake*)
by (Blast_tac 2);
by (Fake_parts_insert_tac 1);
(*YM4*)
by (Step_tac 1);
(*A is uncompromised because NB is secure*)
by (not_lost_tac "A" 1);
(*A's certificate guarantees the existence of the Server message*)
by (blast_tac (!claset addDs [Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS
A_trusts_YM3]) 1);
bind_thm ("B_trusts_YM4_newK", result() RS mp RSN (2, rev_mp));
(**** Towards proving secrecy of Nonce NB ****)
(** Lemmas about the predicate KeyWithNonce **)
goalw thy [KeyWithNonce_def]
"!!evs. Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, na, Nonce NB|}, X|} \
\ : set evs ==> KeyWithNonce K NB evs";
by (Blast_tac 1);
qed "KeyWithNonceI";
goalw thy [KeyWithNonce_def]
"KeyWithNonce K NB (Says S A X # evs) = \
\ (Server = S & \
\ (EX B n X'. X = {|Crypt (shrK A) {|Agent B, Key K, n, Nonce NB|}, X'|}) \
\ | KeyWithNonce K NB evs)";
by (Simp_tac 1);
by (Blast_tac 1);
qed "KeyWithNonce_Says";
Addsimps [KeyWithNonce_Says];
(*A fresh key cannot be associated with any nonce
(with respect to a given trace). *)
goalw thy [KeyWithNonce_def]
"!!evs. Key K ~: used evs ==> ~ KeyWithNonce K NB evs";
by (blast_tac (!claset addSEs sees_Spy_partsEs) 1);
qed "fresh_not_KeyWithNonce";
(*The Server message associates K with NB' and therefore not with any
other nonce NB.*)
goalw thy [KeyWithNonce_def]
"!!evs. [| 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 (blast_tac (!claset addDs [unique_session_keys]) 1);
qed "Says_Server_KeyWithNonce";
(*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 analz_image_freshK, but it is much more complicated.*)
(*As with analz_image_freshK, we take some pains to express the property
as a logical equivalence so that the simplifier can apply it.*)
goal thy
"!!evs. P --> (X : analz (G Un H)) --> (X : analz H) ==> \
\ P --> (X : analz (G Un H)) = (X : analz H)";
by (blast_tac (!claset addIs [impOfSubs analz_mono]) 1);
val lemma = result();
goal thy
"!!evs. evs : yahalom ==> \
\ (ALL KK. KK <= Compl (range shrK) --> \
\ (ALL K: KK. ~ KeyWithNonce K NB evs) --> \
\ (Nonce NB : analz (Key``KK Un (sees Spy evs))) = \
\ (Nonce NB : analz (sees Spy evs)))";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (REPEAT_FIRST (resolve_tac [impI RS allI]));
by (REPEAT_FIRST (rtac lemma));
(*For Oops, simplification proves NBa~=NB. By Says_Server_KeyWithNonce,
we get (~ KeyWithNonce K NB evsa); then simplification can apply the
induction hypothesis with KK = {K}.*)
by (ALLGOALS (*22 seconds*)
(asm_simp_tac
(analz_image_freshK_ss addsimps
([all_conj_distrib, not_parts_not_analz, analz_image_freshK,
KeyWithNonce_Says, fresh_not_KeyWithNonce,
imp_disj_not1, (*Moves NBa~=NB to the front*)
Says_Server_KeyWithNonce]
@ pushes))));
(*Base*)
by (Blast_tac 1);
(*Fake*)
by (spy_analz_tac 1);
(*YM4*) (** LEVEL 7 **)
by (not_lost_tac "A" 1);
by (dtac (Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1
THEN REPEAT (assume_tac 1));
by (blast_tac (!claset addIs [KeyWithNonceI]) 1);
qed_spec_mp "Nonce_secrecy";
(*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.*)
goal thy
"!!evs. [| 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) (sees Spy evs))) = \
\ (Nonce NB : analz (sees Spy evs))";
by (asm_simp_tac (analz_image_freshK_ss addsimps
[Nonce_secrecy, Says_Server_KeyWithNonce]) 1);
qed "single_Nonce_secrecy";
(*** The Nonce NB uniquely identifies B's message. ***)
goal thy
"!!evs. evs : yahalom ==> \
\ EX NA' A' B'. ALL NA A B. \
\ Crypt (shrK B) {|Agent A, Nonce NA, nb|} : parts(sees Spy evs) \
\ --> B ~: lost --> NA = NA' & A = A' & B = B'";
by (parts_induct_tac 1);
(*Fake*)
by (REPEAT (etac (exI RSN (2,exE)) 1) (*stripping EXs makes proof faster*)
THEN Fake_parts_insert_tac 1);
by (asm_simp_tac (!simpset addsimps [all_conj_distrib]) 1);
(*YM2: creation of new Nonce. Move assertion into global context*)
by (expand_case_tac "nb = ?y" 1);
by (REPEAT (resolve_tac [exI, conjI, impI, refl] 1));
by (blast_tac (!claset addSEs sees_Spy_partsEs) 1);
val lemma = result();
goal thy
"!!evs.[| Crypt (shrK B) {|Agent A, Nonce NA, nb|} \
\ : parts (sees Spy evs); \
\ Crypt (shrK B') {|Agent A', Nonce NA', nb|} \
\ : parts (sees Spy evs); \
\ evs : yahalom; B ~: lost; B' ~: lost |] \
\ ==> NA' = NA & A' = A & B' = B";
by (prove_unique_tac lemma 1);
qed "unique_NB";
(*Variant useful for proving secrecy of NB: the Says... form allows
not_lost_tac to remove the assumption B' ~: lost.*)
goal thy
"!!evs.[| Says C D {|X, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set evs; B ~: lost; \
\ Says C' D' {|X', Crypt (shrK B') {|Agent A', Nonce NA', nb|}|} \
\ : set evs; \
\ nb ~: analz (sees Spy evs); evs : yahalom |] \
\ ==> NA' = NA & A' = A & B' = B";
by (not_lost_tac "B'" 1);
by (blast_tac (!claset addSDs [Says_imp_sees_Spy RS parts.Inj]
addSEs [MPair_parts]
addDs [unique_NB]) 1);
qed "Says_unique_NB";
(** A nonce value is never used both as NA and as NB **)
goal thy
"!!evs. [| B ~: lost; evs : yahalom |] \
\ ==> Nonce NB ~: analz (sees Spy evs) --> \
\ Crypt (shrK B') {|Agent A', Nonce NB, nb'|} \
\ : parts(sees Spy evs) \
\ --> Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|} \
\ ~: parts(sees Spy evs)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (blast_tac (!claset addDs [Says_imp_sees_Spy RS analz.Inj]
addSIs [parts_insertI]
addSEs partsEs) 1);
bind_thm ("no_nonce_YM1_YM2", result() RS mp RSN (2,rev_mp) RSN (2,rev_notE));
(*The Server sends YM3 only in response to YM2.*)
goal thy
"!!evs. [| Says Server A \
\ {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} : set evs; \
\ evs : yahalom |] \
\ ==> EX B'. Says B' Server \
\ {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |} \
\ : set evs";
by (etac rev_mp 1);
by (etac yahalom.induct 1);
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS Blast_tac);
qed "Says_Server_imp_YM2";
(*A vital theorem for B, that nonce NB remains secure from the Spy.*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> Says B Server \
\ {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
\ : set evs --> \
\ (ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs) --> \
\ Nonce NB ~: analz (sees Spy evs)";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_insert_eq, not_parts_not_analz,
analz_insert_freshK] @ pushes)
setloop split_tac [expand_if])));
(*Prove YM3 by showing that no NB can also be an NA*)
by (blast_tac (!claset addDs [Says_imp_sees_Spy RS parts.Inj]
addSEs [MPair_parts]
addDs [no_nonce_YM1_YM2, Says_unique_NB]) 4
THEN flexflex_tac);
(*YM2: similar freshness reasoning*)
by (blast_tac (!claset addSEs partsEs
addDs [Says_imp_sees_Spy RS analz.Inj,
impOfSubs analz_subset_parts]) 3);
(*YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!*)
by (blast_tac (!claset addSIs [parts_insertI]
addSEs sees_Spy_partsEs) 2);
(*Fake*)
by (spy_analz_tac 1);
(** LEVEL 7: YM4 and Oops remain **)
(*YM4: key K is visible to Spy, contradicting session key secrecy theorem*)
by (REPEAT (Safe_step_tac 1));
by (not_lost_tac "Aa" 1);
by (dtac (Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS A_trusts_YM3) 1);
by (forward_tac [Says_Server_message_form] 3);
by (forward_tac [Says_Server_imp_YM2] 4);
by (REPEAT_FIRST (eresolve_tac [asm_rl, bexE, exE, disjE]));
(* use Says_unique_NB to identify message components: Aa=A, Ba=B, NAa=NA *)
by (blast_tac (!claset addDs [Says_unique_NB, Spy_not_see_encrypted_key,
impOfSubs Fake_analz_insert]) 1);
(** LEVEL 14 **)
(*Oops case: if the nonce is betrayed now, show that the Oops event is
covered by the quantified Oops assumption.*)
by (full_simp_tac (!simpset addsimps [all_conj_distrib]) 1);
by (step_tac (!claset delrules [disjE, conjI]) 1);
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1 THEN etac exE 1);
by (expand_case_tac "NB = NBa" 1);
(*If NB=NBa then all other components of the Oops message agree*)
by (blast_tac (!claset addDs [Says_unique_NB]) 1 THEN flexflex_tac);
(*case NB ~= NBa*)
by (asm_simp_tac (!simpset addsimps [single_Nonce_secrecy]) 1);
by (blast_tac (!claset addSEs [MPair_parts]
addDs [Says_imp_sees_Spy RS parts.Inj,
no_nonce_YM1_YM2 (*to prove NB~=NAa*) ]) 1);
bind_thm ("Spy_not_see_NB", result() RSN(2,rev_mp) RSN(2,rev_mp));
(*B's session key guarantee from YM4. The two certificates contribute to a
single conclusion about the Server's message. Note that the "Says A Spy"
assumption must quantify over ALL 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.*)
goal thy
"!!evs. [| Says B Server \
\ {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
\ : set evs; \
\ Says A' B {|Crypt (shrK B) {|Agent A, Key K|}, \
\ Crypt K (Nonce NB)|} : set evs; \
\ ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs; \
\ A ~: lost; B ~: lost; 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 (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1));
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1 THEN
dtac B_trusts_YM4_shrK 1);
by (dtac B_trusts_YM4_newK 3);
by (REPEAT_FIRST (eresolve_tac [asm_rl, exE]));
by (forward_tac [Says_Server_imp_YM2] 1 THEN assume_tac 1);
by (dtac unique_session_keys 1 THEN REPEAT (assume_tac 1));
by (blast_tac (!claset addDs [Says_unique_NB]) 1);
qed "B_trusts_YM4";
(*** Authenticating B to A ***)
(*The encryption in message YM2 tells us it cannot be faked.*)
goal thy
"!!evs. evs : yahalom \
\ ==> Crypt (shrK B) {|Agent A, Nonce NA, nb|} \
\ : parts (sees Spy evs) --> \
\ B ~: lost --> \
\ Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set evs";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
bind_thm ("B_Said_YM2", result() RSN (2, rev_mp) RS mp);
(*If the server sends YM3 then B sent YM2*)
goal thy
"!!evs. evs : yahalom \
\ ==> Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|} \
\ : set evs --> \
\ B ~: lost --> \
\ Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set evs";
by (etac yahalom.induct 1);
by (ALLGOALS Asm_simp_tac);
(*YM4*)
by (Blast_tac 2);
(*YM3*)
by (best_tac (!claset addSDs [B_Said_YM2, Says_imp_sees_Spy RS parts.Inj]
addSEs [MPair_parts]) 1);
val lemma = result() RSN (2, rev_mp) RS mp |> standard;
(*If A receives YM3 then B has used nonce NA (and therefore is alive)*)
goal thy
"!!evs. [| Says S A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|} \
\ : set evs; \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set evs";
by (blast_tac (!claset addSDs [A_trusts_YM3, lemma]
addEs sees_Spy_partsEs) 1);
qed "YM3_auth_B_to_A";
(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)
(*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.*)
goal thy
"!!evs. evs : yahalom \
\ ==> Key K ~: analz (sees Spy evs) --> \
\ Crypt K (Nonce NB) : parts (sees Spy evs) --> \
\ Crypt (shrK B) {|Agent A, Key K|} \
\ : parts (sees Spy evs) --> \
\ B ~: lost --> \
\ (EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs)";
by (parts_induct_tac 1);
(*Fake*)
by (Fake_parts_insert_tac 1);
(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
by (fast_tac (!claset addSDs [Crypt_imp_invKey_keysFor] addss (!simpset)) 1);
(*YM4: was Crypt K (Nonce NB) the very last message? If not, use ind. hyp.*)
by (asm_simp_tac (!simpset addsimps [ex_disj_distrib]) 1);
(*yes: apply unicity of session keys*)
by (not_lost_tac "Aa" 1);
by (blast_tac (!claset addSEs [MPair_parts]
addSDs [A_trusts_YM3, B_trusts_YM4_shrK]
addDs [Says_imp_sees_Spy RS parts.Inj,
unique_session_keys]) 1);
val lemma = normalize_thm [RSspec, RSmp] (result()) |> standard;
(*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.*)
goal thy
"!!evs. [| Says B Server \
\ {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
\ : set evs; \
\ Says A' B {|Crypt (shrK B) {|Agent A, Key K|}, \
\ Crypt K (Nonce NB)|} : set evs; \
\ (ALL NA k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set evs); \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs";
by (dtac B_trusts_YM4 1);
by (REPEAT_FIRST (eresolve_tac [asm_rl, spec]));
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1);
by (rtac lemma 1);
by (rtac Spy_not_see_encrypted_key 2);
by (REPEAT_FIRST assume_tac);
by (blast_tac (!claset addSEs [MPair_parts]
addDs [Says_imp_sees_Spy RS parts.Inj]) 1);
qed_spec_mp "YM4_imp_A_Said_YM3";