(* 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)
*)
(*to HOL/simpdata.ML ????????????????*)
fun prove nm thm = qed_goal nm HOL.thy thm (fn _ => [blast_tac HOL_cs 1]);
prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)";
prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)";
open Yahalom;
proof_timing:=true;
HOL_quantifiers := false;
Pretty.setdepth 25;
(*Replacing the variable by a constant improves speed*)
val Says_imp_sees_Spy' = read_instantiate [("lost","lost")] Says_imp_sees_Spy;
(*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 lost. \
\ Says A B {|X, Crypt K (Nonce NB)|} : set_of_list 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 ****)
(*Monotonicity*)
goal thy "!!evs. lost' <= lost ==> yahalom lost' <= yahalom lost";
by (rtac subsetI 1);
by (etac yahalom.induct 1);
by (REPEAT_FIRST
(blast_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
:: yahalom.intrs))));
qed "yahalom_mono";
(*Nobody sends themselves messages*)
goal thy "!!evs. evs: yahalom lost ==> ALL A X. Says A A X ~: set_of_list 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_of_list evs ==> \
\ X : analz (sees lost 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_of_list evs ==> \
\ K : parts (sees lost 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 lost Spy evs).
We instantiate the variable to "lost" since leaving it as a Var would
interfere with simplification.*)
val parts_sees_tac =
forw_inst_tac [("lost","lost")] YM4_parts_sees_Spy 6 THEN
forw_inst_tac [("lost","lost")] YM4_Key_parts_sees_Spy 7 THEN
prove_simple_subgoals_tac 1;
val parts_induct_tac =
etac yahalom.induct 1 THEN parts_sees_tac;
(** Theorems of the form X ~: parts (sees lost 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 lost \
\ ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
by parts_induct_tac;
by (Fake_parts_insert_tac 1);
by (Blast_tac 1);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : yahalom lost \
\ ==> (Key (shrK A) : analz (sees lost 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 lost Spy evs); \
\ evs : yahalom lost |] ==> 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 lost ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (sees lost Spy evs))";
by parts_induct_tac;
(*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_of_list evs; \
\ evs : yahalom lost |] \
\ ==> 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. We again instantiate the variable to "lost".*)
val analz_sees_tac =
forw_inst_tac [("lost","lost")] YM4_analz_sees_Spy 6 THEN
forw_inst_tac [("lost","lost")] 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 lost Spy evs)) ==>
Key K : analz (sees lost 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 lost ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (sees lost Spy evs))) = \
\ (K : KK | Key K : analz (sees lost 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));
(*Base*)
by (Blast_tac 1);
(*YM4, Fake*)
by (REPEAT (spy_analz_tac 1));
qed_spec_mp "analz_image_freshK";
goal thy
"!!evs. [| evs : yahalom lost; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (sees lost Spy evs)) = \
\ (K = KAB | Key K : analz (sees lost 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 lost ==> \
\ 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_of_list 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_of_list evs; \
\ Says Server A' \
\ {|Crypt (shrK A') {|Agent B', Key K, NA', NB'|}, X'|} \
\ : set_of_list evs; \
\ evs : yahalom lost |] \
\ ==> 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 lost |] \
\ ==> Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, NA, NB|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set_of_list evs --> \
\ Says A Spy {|NA, NB, Key K|} ~: set_of_list evs --> \
\ Key K ~: analz (sees lost Spy evs)";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [not_parts_not_analz, analz_insert_freshK]
setloop split_tac [expand_if])));
(*YM3*)
by (blast_tac (!claset delrules [impCE]
addSEs sees_Spy_partsEs
addIs [impOfSubs analz_subset_parts]) 2);
(*OR4, Fake*)
by (REPEAT_FIRST spy_analz_tac);
(*Oops*)
by (blast_tac (!claset addDs [unique_session_keys]) 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_of_list evs; \
\ Says A Spy {|NA, NB, Key K|} ~: set_of_list evs; \
\ A ~: lost; B ~: lost; evs : yahalom lost |] \
\ ==> Key K ~: analz (sees lost 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";
(*And other agents don't see the key either.*)
goal thy
"!!evs. [| C ~: {A,B,Server}; \
\ Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, NA, NB|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set_of_list evs; \
\ Says A Spy {|NA, NB, Key K|} ~: set_of_list evs; \
\ A ~: lost; B ~: lost; evs : yahalom lost |] \
\ ==> Key K ~: analz (sees lost C evs)";
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
by (FIRSTGOAL (rtac Spy_not_see_encrypted_key));
by (REPEAT_FIRST (blast_tac (!claset addIs [yahalom_mono RS subsetD])));
qed "Agent_not_see_encrypted_key";
(*Induction for theorems of the form X ~: analz (sees lost Spy evs) --> ...
It simplifies the proof by discarding needless information about
analz (insert X (sees lost Spy evs))
*)
val analz_mono_parts_induct_tac =
etac yahalom.induct 1
THEN
REPEAT_FIRST
(rtac impI THEN'
dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD) THEN'
mp_tac)
THEN parts_sees_tac;
(** 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 lost Spy evs); \
\ A ~: lost; evs : yahalom lost |] \
\ ==> Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, NA, NB|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set_of_list evs";
by (etac rev_mp 1);
by parts_induct_tac;
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 lost Spy evs); \
\ B ~: lost; evs : yahalom lost |] \
\ ==> 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_of_list evs";
by (etac rev_mp 1);
by parts_induct_tac;
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 lost \
\ ==> Nonce NB ~: analz (sees lost Spy evs) --> \
\ Crypt K (Nonce NB) : parts (sees lost 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_of_list evs)";
by analz_mono_parts_induct_tac;
(*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);
val 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_of_list 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];
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_of_list evs; \
\ NB ~= NB'; evs : yahalom lost |] \
\ ==> ~ 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 lost ==> \
\ (ALL KK. KK <= Compl (range shrK) --> \
\ (ALL K: KK. ~ KeyWithNonce K NB evs) --> \
\ (Nonce NB : analz (Key``KK Un (sees lost Spy evs))) = \
\ (Nonce NB : analz (sees lost 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 (asm_simp_tac (*X contributes nothing to the result of analz*)
(analz_image_freshK_ss addsimps
([ball_conj_distrib, analz_image_freshK,
analz_insert_eq, impOfSubs (Un_upper2 RS analz_mono)])) 1);
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_of_list evs; \
\ NB ~= NB'; KAB ~: range shrK; evs : yahalom lost |] \
\ ==> (Nonce NB : analz (insert (Key KAB) (sees lost Spy evs))) = \
\ (Nonce NB : analz (sees lost 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 lost ==> \
\ EX NA' A' B'. ALL NA A B. \
\ Crypt (shrK B) {|Agent A, Nonce NA, NB|} : parts(sees lost Spy evs) \
\ --> B ~: lost --> NA = NA' & A = A' & B = B'";
by parts_induct_tac;
(*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 lost Spy evs); \
\ Crypt (shrK B') {|Agent A', Nonce NA', NB|} \
\ : parts (sees lost Spy evs); \
\ evs : yahalom lost; 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_of_list evs; B ~: lost; \
\ Says C' D' {|X', Crypt (shrK B') {|Agent A', Nonce NA', NB|}|} \
\ : set_of_list evs; \
\ NB ~: analz (sees lost Spy evs); evs : yahalom lost |] \
\ ==> 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";
val Says_unique_NB' = read_instantiate [("lost","lost")] Says_unique_NB;
(** A nonce value is never used both as NA and as NB **)
goal thy
"!!evs. [| B ~: lost; evs : yahalom lost |] \
\ ==> Nonce NB ~: analz (sees lost Spy evs) --> \
\ Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}: parts(sees lost Spy evs)\
\ --> Crypt (shrK B') {|Agent A', Nonce NB, NB'|} ~: parts(sees lost Spy evs)";
by analz_mono_parts_induct_tac;
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) RS notE);
(*YM3 can only be triggered by YM2*)
goal thy
"!!evs. [| Says Server A \
\ {|Crypt (shrK A) {|Agent B, k, na, nb|}, X|} : set_of_list evs; \
\ evs : yahalom lost |] \
\ ==> EX B'. Says B' Server \
\ {| Agent B, Crypt (shrK B) {|Agent A, na, nb|} |} \
\ : set_of_list 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";
(*Basic theorem for B: Nonce NB remains secure from the Spy.
Unusually, the Fake case requires Spy:lost.*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; Spy: lost; evs : yahalom lost |] \
\ ==> Says B Server \
\ {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, Nonce NB|}|} \
\ : set_of_list evs --> \
\ (ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set_of_list evs) --> \
\ Nonce NB ~: analz (sees lost 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])));
(*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);
(*YM2: similar freshness reasoning*)
by (blast_tac (!claset addSEs partsEs
addDs [Says_imp_sees_Spy' RS analz.Inj,
impOfSubs analz_subset_parts]) 2);
(*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']) 2
THEN flexflex_tac);
(*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));
(*What can B deduce from receipt of YM4? Note how the two components of
the message 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 has a certificate for 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_of_list evs; \
\ Says A' B {|Crypt (shrK B) {|Agent A, Key K|}, \
\ Crypt K (Nonce NB)|} : set_of_list evs; \
\ ALL k. Says A Spy {|Nonce NA, Nonce NB, k|} ~: set_of_list evs; \
\ A ~: lost; B ~: lost; Spy: lost; evs : yahalom lost |] \
\ ==> Says Server A \
\ {|Crypt (shrK A) {|Agent B, Key K, \
\ Nonce NA, Nonce NB|}, \
\ Crypt (shrK B) {|Agent A, Key K|}|} \
\ : set_of_list 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 lost \
\ ==> Crypt (shrK B) {|Agent A, Nonce NA, nb|} \
\ : parts (sees lost Spy evs) --> \
\ B ~: lost --> \
\ Says B Server {|Agent B, \
\ Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set_of_list evs";
by parts_induct_tac;
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 lost \
\ ==> Says Server A {|Crypt (shrK A) {|Agent B, Key K, Nonce NA, nb|}, X|} \
\ : set_of_list evs --> \
\ B ~: lost --> \
\ Says B Server {|Agent B, \
\ Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set_of_list 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_of_list evs; \
\ A ~: lost; B ~: lost; evs : yahalom lost |] \
\ ==> Says B Server {|Agent B, Crypt (shrK B) {|Agent A, Nonce NA, nb|}|} \
\ : set_of_list 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) ***)
(*Induction for theorems of the form X ~: analz (sees lost Spy evs) --> ...
It simplifies the proof by discarding needless information about
analz (insert X (sees lost Spy evs))
*)
val analz_mono_parts_induct_tac =
etac yahalom.induct 1
THEN
REPEAT_FIRST
(rtac impI THEN'
dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD) THEN'
mp_tac)
THEN parts_sees_tac;
(*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 lost \
\ ==> Key K ~: analz (sees lost Spy evs) --> \
\ Crypt K (Nonce NB) : parts (sees lost Spy evs) --> \
\ Crypt (shrK B) {|Agent A, Key K|} \
\ : parts (sees lost Spy evs) --> \
\ B ~: lost --> \
\ (EX X. Says A B {|X, Crypt K (Nonce NB)|} : set_of_list evs)";
by analz_mono_parts_induct_tac;
(*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_of_list evs; \
\ Says A' B {|Crypt (shrK B) {|Agent A, Key K|}, \
\ Crypt K (Nonce NB)|} : set_of_list evs; \
\ (ALL NA k. Says A Spy {|Nonce NA, Nonce NB, k|} \
\ ~: set_of_list evs); \
\ A ~: lost; B ~: lost; Spy: lost; evs : yahalom lost |] \
\ ==> EX X. Says A B {|X, Crypt K (Nonce NB)|} : set_of_list 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";