--- a/src/HOL/Auth/Yahalom.ML Fri Oct 18 11:42:17 1996 +0200
+++ b/src/HOL/Auth/Yahalom.ML Fri Oct 18 11:42:41 1996 +0200
@@ -8,8 +8,6 @@
From page 257 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
-
-DEFINE parts_induct_tac AS IN OtwayRees
*)
open Yahalom;
@@ -33,6 +31,7 @@
(**** Inductive proofs about yahalom ****)
+(*Monotonicity*)
goal thy "!!evs. lost' <= lost ==> yahalom lost' <= yahalom lost";
by (rtac subsetI 1);
by (etac yahalom.induct 1);
@@ -59,30 +58,44 @@
by (fast_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 Revl*)
goal thy "!!evs. Says S A {|Crypt {|B, K, NA, NB|} (shrK A), X|} \
\ : set_of_list evs ==> \
\ K : parts (sees lost Spy evs)";
by (fast_tac (!claset addSEs partsEs
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
-qed "YM4_parts_sees_Spy";
+qed "YM4_Key_parts_sees_Spy";
+
+(*We instantiate the variable to "lost". Leaving it as a Var makes proofs
+ harder: the simplifier does less.*)
+val parts_Fake_tac =
+ forw_inst_tac [("lost","lost")] YM4_parts_sees_Spy 6 THEN
+ forw_inst_tac [("lost","lost")] YM4_Key_parts_sees_Spy 7;
+(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
+fun parts_induct_tac i = SELECT_GOAL
+ (DETERM (etac yahalom.induct 1 THEN parts_Fake_tac THEN
+ (*Fake message*)
+ TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
+ impOfSubs Fake_parts_insert]
+ addss (!simpset)) 2)) THEN
+ (*Base case*)
+ fast_tac (!claset addss (!simpset)) 1 THEN
+ ALLGOALS Asm_simp_tac) i;
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
sends messages containing X! **)
-(*Spy never sees lost another agent's shared key! (unless it is leaked at start)*)
+(*Spy never sees another agent's shared key! (unless it is leaked at start)*)
goal thy
"!!evs. [| evs : yahalom lost; A ~: lost |] \
\ ==> Key (shrK A) ~: parts (sees lost Spy evs)";
-by (etac yahalom.induct 1);
-by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6);
-by (ALLGOALS Asm_simp_tac);
-by (stac insert_commute 3);
+by (parts_induct_tac 1);
by (Auto_tac());
-(*Fake and YM4 are similar*)
-by (ALLGOALS (best_tac (!claset addSDs [impOfSubs analz_subset_parts,
- impOfSubs Fake_parts_insert])));
qed "Spy_not_see_shrK";
bind_thm ("Spy_not_analz_shrK",
@@ -115,13 +128,11 @@
This has to be proved anew for each protocol description,
but should go by similar reasoning every time. Hardest case is the
standard Fake rule.
- The length comparison, and Union over C, are essential for the
- induction! *)
+ The Union over C is essential for the induction! *)
goal thy "!!evs. evs : yahalom lost ==> \
\ length evs <= length evs' --> \
\ Key (newK evs') ~: (UN C. parts (sees lost C evs))";
-by (etac yahalom.induct 1);
-by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6);
+by (parts_induct_tac 1);
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
@@ -154,10 +165,9 @@
goal thy "!!evs. evs : yahalom lost ==> \
\ length evs <= length evs' --> \
\ newK evs' ~: keysFor (UN C. parts (sees lost C evs))";
-by (etac yahalom.induct 1);
-by (forward_tac [YM4_parts_sees_Spy] 6);
-by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6);
-by (ALLGOALS Asm_full_simp_tac);
+by (parts_induct_tac 1);
+by (dresolve_tac [YM4_Key_parts_sees_Spy] 5);
+
(*YM1, YM2 and YM3*)
by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,3,2]));
(*Fake and YM4: these messages send unknown (X) components*)
@@ -165,14 +175,14 @@
by (Simp_tac 2);
(*YM4: the only way K could have been used is if it had been seen,
contradicting new_keys_not_seen*)
-by (ALLGOALS
+by (REPEAT
(best_tac
(!claset addDs [impOfSubs analz_subset_parts,
impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono),
Suc_leD]
addEs [new_keys_not_seen RSN(2,rev_notE)]
- addss (!simpset))));
+ addss (!simpset)) 1));
val lemma = result();
goal thy
@@ -188,7 +198,40 @@
Addsimps [new_keys_not_used, new_keys_not_analzd];
-(** Lemmas concerning the form of items passed in messages **)
+(*Describes the form of Key K when the following message is sent. The use of
+ "parts" strengthens the induction hyp for proving the Fake case. The
+ assumption A ~: lost prevents its being a Faked message. (Based
+ on NS_Shared/Says_S_message_form) *)
+goal thy
+ "!!evs. evs: yahalom lost ==> \
+\ Crypt {|B, Key K, NA, NB|} (shrK A) : parts (sees lost Spy evs) \
+\ --> A ~: lost --> (EX evt: yahalom lost. K = newK evt)";
+by (parts_induct_tac 1);
+by (Auto_tac());
+qed_spec_mp "Reveal_message_lemma";
+
+(*EITHER describes the form of Key K when the following message is sent,
+ OR reduces it to the Fake case.*)
+
+goal thy
+ "!!evs. [| Says S A {|Crypt {|B, Key K, NA, NB|} (shrK A), X|} \
+\ : set_of_list evs; \
+\ evs : yahalom lost |] \
+\ ==> (EX evt: yahalom lost. K = newK evt) \
+\ | Key K : analz (sees lost Spy evs)";
+br (Reveal_message_lemma RS disjCI) 1;
+ba 1;
+by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
+ addDs [impOfSubs analz_subset_parts]) 1);
+by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]
+ addss (!simpset)) 1);
+qed "Reveal_message_form";
+
+
+(*For proofs involving analz. We again instantiate the variable to "lost".*)
+val analz_Fake_tac =
+ dres_inst_tac [("lost","lost")] YM4_analz_sees_Spy 6 THEN
+ forw_inst_tac [("lost","lost")] Reveal_message_form 7;
(****
@@ -210,14 +253,12 @@
\ (Crypt X (newK evt)) : parts (sees lost Spy evs) & \
\ Key K : parts {X} --> Key K : parts (sees lost Spy evs)";
by (etac yahalom.induct 1);
-by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6);
+by parts_Fake_tac;
by (ALLGOALS Asm_simp_tac);
(*Deals with Faked messages*)
-by (EVERY
- (map (best_tac (!claset addSEs partsEs
- addDs [impOfSubs parts_insert_subset_Un]
- addss (!simpset)))
- [3,2]));
+by (best_tac (!claset addSEs partsEs
+ addDs [impOfSubs parts_insert_subset_Un]
+ addss (!simpset)) 2);
(*Base case*)
by (Auto_tac());
result();
@@ -230,21 +271,19 @@
\ ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
\ (K : newK``E | Key K : analz (sees lost Spy evs))";
by (etac yahalom.induct 1);
-by (dtac YM4_analz_sees_Spy 6);
+by analz_Fake_tac;
by (REPEAT_FIRST (resolve_tac [allI, analz_image_newK_lemma]));
-by (ALLGOALS
+by (REPEAT ((eresolve_tac [bexE, disjE] ORELSE' hyp_subst_tac) 8));
+by (ALLGOALS (*Takes 26 secs*)
(asm_simp_tac
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
@ pushes)
setloop split_tac [expand_if])));
-(*YM4*)
-by (spy_analz_tac 4);
-(*YM3*)
-by (Fast_tac 3);
-(*Fake case*)
-by (spy_analz_tac 2);
-(*Base case*)
-by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
+(** LEVEL 5 **)
+(*Reveal case 2, YM4, Fake*)
+by (EVERY (map spy_analz_tac [6, 4, 2]));
+(*Reveal case 1, YM3, Base*)
+by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
qed_spec_mp "analz_image_newK";
goal thy
@@ -257,12 +296,66 @@
qed "analz_insert_Key_newK";
+(*** The Key K uniquely identifies the Server's message. **)
+
+goal thy
+ "!!evs. evs : yahalom lost ==> \
+\ EX A' B' NA' NB'. ALL A B NA NB. \
+\ Says Server A \
+\ {|Crypt {|Agent B, Key K, NA, NB|} (shrK A), \
+\ Crypt {|Agent A, Key K|} (shrK B)|} \
+\ : set_of_list evs --> A=A' & B=B' & NA=NA' & NB=NB'";
+by (etac yahalom.induct 1);
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
+by (Step_tac 1);
+(*Remaining case: YM3*)
+by (expand_case_tac "K = ?y" 1);
+by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
+(*...we assume X is a very new message, and handle this case by contradiction*)
+by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
+ delrules [conjI] (*prevent split-up into 4 subgoals*)
+ addss (!simpset addsimps [parts_insertI])) 1);
+val lemma = result();
+
+goal thy
+"!!evs. [| Says Server A \
+\ {|Crypt {|Agent B, Key K, NA, NB|} (shrK A), \
+\ Crypt {|Agent A, Key K|} (shrK B)|} \
+\ : set_of_list evs; \
+\ Says Server A' \
+\ {|Crypt {|Agent B', Key K, NA', NB'|} (shrK A'), \
+\ Crypt {|Agent A', Key K|} (shrK B')|} \
+\ : set_of_list evs; \
+\ evs : yahalom lost |] \
+\ ==> A=A' & B=B' & NA=NA' & NB=NB'";
+by (dtac lemma 1);
+by (REPEAT (etac exE 1));
+(*Duplicate the assumption*)
+by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
+by (fast_tac (!claset addSDs [spec]) 1);
+qed "unique_session_keys";
+
+
+(*If the encrypted message appears then it originated with the Server*)
+goal thy
+ "!!evs. [| Crypt {|Agent B, Key K, NA, NB|} (shrK A) \
+\ : parts (sees lost Spy evs); \
+\ A ~: lost; evs : yahalom lost |] \
+\ ==> Says Server A \
+\ {|Crypt {|Agent B, Key K, NA, NB|} (shrK A), \
+\ Crypt {|Agent A, Key K|} (shrK B)|} \
+\ : set_of_list evs";
+by (etac rev_mp 1);
+by (parts_induct_tac 1);
+qed "A_trust_YM3";
+
+
(*Describes the form of K when the Server sends this message.*)
goal thy
"!!evs. [| Says Server A \
\ {|Crypt {|Agent B, K, NA, NB|} (shrK A), \
\ Crypt {|Agent A, K|} (shrK B)|} : set_of_list evs; \
-\ evs : yahalom lost |] \
+\ evs : yahalom lost |] \
\ ==> (EX evt: yahalom lost. K = Key(newK evt))";
by (etac rev_mp 1);
by (etac yahalom.induct 1);
@@ -270,8 +363,7 @@
qed "Says_Server_message_form";
-(** Crucial secrecy property: Spy does not see the keys sent in msg YM3
- As with Otway-Rees, proof does not need uniqueness of session keys. **)
+(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
goal thy
"!!evs. [| A ~: lost; B ~: lost; \
@@ -279,32 +371,44 @@
\ ==> Says Server A \
\ {|Crypt {|Agent B, Key K, NA, NB|} (shrK A), \
\ Crypt {|Agent A, Key K|} (shrK B)|} \
-\ : set_of_list evs --> \
+\ : 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 (dtac YM4_analz_sees_Spy 6);
+by analz_Fake_tac;
+by (REPEAT_FIRST (eresolve_tac [asm_rl, bexE, disjE] ORELSE' hyp_subst_tac));
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD,
analz_insert_Key_newK] @ pushes)
setloop split_tac [expand_if])));
-(*YM4*)
-by (spy_analz_tac 3);
(*YM3*)
by (fast_tac (!claset addIs [parts_insertI]
addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset)) 2);
-(*Fake*) (** LEVEL 10 **)
-by (spy_analz_tac 1);
-val lemma = result() RS mp RSN(2,rev_notE);
+(*Reveal case 2, OR4, Fake*)
+by (REPEAT_FIRST (resolve_tac [conjI, impI] ORELSE' spy_analz_tac));
+(*Reveal case 1*) (** LEVEL 6 **)
+by (case_tac "Aa : lost" 1);
+(*But this contradicts Key K ~: analz (sees lost Spy evsa) *)
+by (dtac (Says_imp_sees_Spy RS analz.Inj) 1);
+by (fast_tac (!claset addSDs [analz.Decrypt] addss (!simpset)) 1);
+(*So now we have Aa ~: lost *)
+bd (Says_imp_sees_Spy RS parts.Inj) 1;
+by (fast_tac (!claset delrules [disjE]
+ addSEs [MPair_parts]
+ addDs [A_trust_YM3, unique_session_keys]
+ addss (!simpset)) 1);
+val lemma = result() RS mp RS mp RSN(2,rev_notE);
(*Final version: Server's message in the most abstract form*)
goal thy
- "!!evs. [| Says Server A \
+ "!!evs. [| Says Server A \
\ {|Crypt {|Agent B, K, NA, NB|} (shrK A), \
\ Crypt {|Agent A, K|} (shrK B)|} : set_of_list evs; \
-\ A ~: lost; B ~: lost; evs : yahalom lost |] ==> \
+\ Says A Spy {|NA, NB, K|} ~: set_of_list evs; \
+\ A ~: lost; B ~: lost; evs : yahalom lost |] ==> \
\ K ~: analz (sees lost Spy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (fast_tac (!claset addSEs [lemma]) 1);
@@ -316,6 +420,7 @@
\ Says Server A \
\ {|Crypt {|Agent B, K, NA, NB|} (shrK A), \
\ Crypt {|Agent A, K|} (shrK B)|} : set_of_list evs; \
+\ Says A Spy {|NA, NB, K|} ~: set_of_list evs; \
\ A ~: lost; B ~: lost; evs : yahalom lost |] ==> \
\ K ~: analz (sees lost C evs)";
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
@@ -325,8 +430,10 @@
qed "Agent_not_see_encrypted_key";
-(** Towards proofs of stronger authenticity properties **)
+(*** Security Guarantee 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 {|Agent A, Key K|} (shrK B) : parts (sees lost Spy evs); \
\ B ~: lost; evs : yahalom lost |] \
@@ -336,46 +443,62 @@
\ Crypt {|Agent A, Key K|} (shrK B)|} \
\ : set_of_list evs";
by (etac rev_mp 1);
+by (parts_induct_tac 1);
+(*YM3*)
+by (Fast_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.*)
+goal thy
+ "!!evs. evs : yahalom lost \
+\ ==> Key K ~: analz (sees lost Spy evs) --> \
+\ Crypt (Nonce NB) K : parts (sees lost Spy evs) --> \
+\ (EX A B NA. Says Server A \
+\ {|Crypt {|Agent B, Key K, \
+\ Nonce NA, Nonce NB|} (shrK A), \
+\ Crypt {|Agent A, Key K|} (shrK B)|} \
+\ : set_of_list evs)";
by (etac yahalom.induct 1);
-by (dtac (YM4_analz_sees_Spy RS synth.Inj) 6);
-by (ALLGOALS Asm_simp_tac);
-(*YM3*)
-by (Fast_tac 3);
-(*Base case*)
+by parts_Fake_tac;
by (fast_tac (!claset addss (!simpset)) 1);
-(*Prepare YM4*)
-by (stac insert_commute 2 THEN Simp_tac 2);
-(*Fake and YM4 are similar*)
-by (ALLGOALS (best_tac (!claset addSDs [impOfSubs analz_subset_parts,
- impOfSubs Fake_parts_insert])));
-qed "Crypt_imp_Server_msg";
+by (TRYALL (rtac impI));
+by (REPEAT_FIRST
+ (dtac (sees_subset_sees_Says RS analz_mono RS contra_subsetD)));
+by (ALLGOALS Asm_simp_tac);
+(*Fake, YM3, YM4*)
+by (fast_tac (!claset addSDs [Crypt_Fake_parts_insert]
+ addDs [impOfSubs analz_subset_parts]) 1);
+by (Fast_tac 1);
+(*YM4*)
+by (Step_tac 1);
+by (case_tac "A : lost" 1);
+(*But this contradicts Key K ~: analz (sees lost Spy evsa) *)
+by (dtac (Says_imp_sees_Spy RS analz.Inj) 1);
+by (fast_tac (!claset addSDs [analz.Decrypt] addss (!simpset)) 1);
+by (fast_tac (!claset addDs [Says_imp_sees_Spy RS parts.Inj RS parts.Fst RS
+ A_trust_YM3]) 1);
+val B_trusts_YM4_newK = result() RS mp RSN (2, rev_mp);
-
-(*What can B deduce from receipt of YM4? NOT THAT THE NONCES AGREE. Cf the
- BAN paper page 259. "If A chose to replay an old key to B in message 4,
- B could not detect the fraud." *)
+(*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.
+ It's annoying that the "Says A Spy" assumption must quantify over
+ ALL POSSIBLE nonces instead of our particular NB. Perhaps a different
+ proof of B_trusts_YM4_newK could eliminate this problem.*)
goal thy
"!!evs. [| Says A' B {|Crypt {|Agent A, Key K|} (shrK B), \
\ Crypt (Nonce NB) K|} : set_of_list evs; \
-\ B ~: lost; evs : yahalom lost |] \
-\ ==> EX NA NB. Says Server A \
+\ ALL N N'. Says A Spy {|N,N', Key K|} ~: set_of_list evs; \
+\ A ~: lost; B ~: lost; evs : yahalom lost |] \
+\ ==> EX NA. Says Server A \
\ {|Crypt {|Agent B, Key K, \
\ Nonce NA, Nonce NB|} (shrK A), \
\ Crypt {|Agent A, Key K|} (shrK B)|} \
\ : set_of_list evs";
-by (etac rev_mp 1);
-by (etac yahalom.induct 1);
-by (dtac YM4_analz_sees_Spy 6);
-by (ALLGOALS Asm_simp_tac);
-by (ALLGOALS (fast_tac (!claset addSDs [impOfSubs analz_subset_parts RS
- Crypt_imp_Server_msg])));
-qed "YM4_imp_Says_Server_A";
-
-goal thy
- "!!evs. [| Says A' B {|Crypt {|Agent A, Key K|} (shrK B), \
-\ Crypt (Nonce NB) K|} : set_of_list evs; \
-\ A ~: lost; B ~: lost; evs : yahalom lost |] \
-\ ==> Key K ~: analz (sees lost Spy evs)";
-by (fast_tac (!claset addSDs [YM4_imp_Says_Server_A,
- Spy_not_see_encrypted_key]) 1);
-qed "B_gets_secure_key";
+be (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1;
+bd B_trusts_YM4_shrK 1;
+bd B_trusts_YM4_newK 3;
+by (REPEAT_FIRST (eresolve_tac [asm_rl, exE]));
+by (fast_tac (!claset addDs [unique_session_keys]) 2);
+by (fast_tac (!claset addDs [Spy_not_see_encrypted_key]) 1);
+qed "B_trust_YM4";