(* Title: HOL/Auth/Yahalom
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "otway" 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;
(*Weak liveness: 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 (Nonce NB) K|} : 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 (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
by (ALLGOALS Fast_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
(best_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 Y (shrK A), X|} : set_of_list evs ==> \
\ X : analz (sees lost Spy evs)";
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_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 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 (parts_induct_tac 1);
by (Auto_tac());
qed "Spy_not_see_shrK";
bind_thm ("Spy_not_analz_shrK",
[analz_subset_parts, Spy_not_see_shrK] MRS contra_subsetD);
Addsimps [Spy_not_see_shrK, Spy_not_analz_shrK];
(*We go to some trouble to preserve R in the 3rd and 4th subgoals
As usual fast_tac cannot be used because it uses the equalities too soon*)
val major::prems =
goal thy "[| Key (shrK A) : parts (sees lost Spy evs); \
\ evs : yahalom lost; \
\ A:lost ==> R \
\ |] ==> R";
by (rtac ccontr 1);
by (rtac ([major, Spy_not_see_shrK] MRS rev_notE) 1);
by (swap_res_tac prems 2);
by (ALLGOALS (fast_tac (!claset addIs prems)));
qed "Spy_see_shrK_E";
bind_thm ("Spy_analz_shrK_E",
analz_subset_parts RS subsetD RS Spy_see_shrK_E);
AddSEs [Spy_see_shrK_E, Spy_analz_shrK_E];
(*** Future keys can't be seen or used! ***)
(*Nobody can have SEEN keys that will be generated in the future.
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 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 (parts_induct_tac 1);
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
addss (!simpset))));
val lemma = result();
(*Variant needed for the main theorem below*)
goal thy
"!!evs. [| evs : yahalom lost; length evs <= length evs' |] \
\ ==> Key (newK evs') ~: parts (sees lost C evs)";
by (fast_tac (!claset addDs [lemma]) 1);
qed "new_keys_not_seen";
Addsimps [new_keys_not_seen];
(*Another variant: old messages must contain old keys!*)
goal thy
"!!evs. [| Says A B X : set_of_list evs; \
\ Key (newK evt) : parts {X}; \
\ evs : yahalom lost \
\ |] ==> length evt < length evs";
by (rtac ccontr 1);
by (dtac leI 1);
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
addIs [impOfSubs parts_mono]) 1);
qed "Says_imp_old_keys";
(*Nobody can have USED keys that will be generated in the future.
...very like new_keys_not_seen*)
goal thy "!!evs. evs : yahalom lost ==> \
\ length evs <= length evs' --> \
\ newK evs' ~: keysFor (UN C. parts (sees lost C evs))";
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*)
by (stac insert_commute 2);
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 (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)) 1));
val lemma = result();
goal thy
"!!evs. [| evs : yahalom lost; length evs <= length evs' |] \
\ ==> newK evs' ~: keysFor (parts (sees lost C evs))";
by (fast_tac (!claset addSDs [lemma] addss (!simpset)) 1);
qed "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 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;
(****
The following is to prove theorems of the form
Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==>
Key K : analz (sees lost Spy evs)
A more general formula must be proved inductively.
****)
(*NOT useful in this form, but it says that session keys are not used
to encrypt messages containing other keys, in the actual protocol.
We require that agents should behave like this subsequently also.*)
goal thy
"!!evs. evs : yahalom lost ==> \
\ (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 parts_Fake_tac;
by (ALLGOALS Asm_simp_tac);
(*Deals with Faked messages*)
by (best_tac (!claset addSEs partsEs
addDs [impOfSubs parts_insert_subset_Un]
addss (!simpset)) 2);
(*Base case*)
by (Auto_tac());
result();
(** Session keys are not used to encrypt other session keys **)
goal thy
"!!evs. evs : yahalom lost ==> \
\ 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 analz_Fake_tac;
by (REPEAT_FIRST (resolve_tac [allI, analz_image_newK_lemma]));
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])));
(** 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
"!!evs. evs : yahalom lost ==> \
\ Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \
\ (K = newK evt | Key K : analz (sees lost Spy evs))";
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK,
insert_Key_singleton]) 1);
by (Fast_tac 1);
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 |] \
\ ==> (EX evt: yahalom lost. K = Key(newK evt))";
by (etac rev_mp 1);
by (etac yahalom.induct 1);
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "Says_Server_message_form";
(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
goal thy
"!!evs. [| A ~: lost; B ~: lost; \
\ evs : yahalom lost; evt : 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 --> \
\ 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_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])));
(*YM3*)
by (fast_tac (!claset addIs [parts_insertI]
addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset)) 2);
(*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 \
\ {|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 Spy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (fast_tac (!claset addSEs [lemma]) 1);
qed "Spy_not_see_encrypted_key";
goal thy
"!!evs. [| C ~: {A,B,Server}; \
\ 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);
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 (fast_tac (!claset addIs [yahalom_mono RS subsetD])));
qed "Agent_not_see_encrypted_key";
(*** 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 |] \
\ ==> EX NA NB. 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 (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 parts_Fake_tac;
by (fast_tac (!claset addss (!simpset)) 1);
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? 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; \
\ 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";
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";