(* Title: HOL/Auth/NS_Shared
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "ns_shared" for Needham-Schroeder Shared-Key protocol.
From page 247 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open NS_Shared;
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 N K. EX evs: ns_shared. \
\ Says A B (Crypt {|Nonce N, Nonce N|} K) : set_of_list evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
br (ns_shared.Nil RS ns_shared.NS1 RS ns_shared.NS2 RS ns_shared.NS3 RS ns_shared.NS4 RS ns_shared.NS5) 2;
by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
by (REPEAT_FIRST (resolve_tac [refl, conjI]));
by (ALLGOALS (fast_tac (!claset addss (!simpset setsolver safe_solver))));
result();
(**** Inductive proofs about ns_shared ****)
(*The Enemy can see more than anybody else, except for their initial state*)
goal thy
"!!evs. evs : ns_shared ==> \
\ sees A evs <= initState A Un sees Enemy evs";
be ns_shared.induct 1;
by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD]
addss (!simpset))));
qed "sees_agent_subset_sees_Enemy";
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : ns_shared ==> ALL A X. Says A A X ~: set_of_list evs";
be ns_shared.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 message NS3*)
goal thy "!!evs. (Says S A (Crypt {|N, B, K, X|} KA)) : set_of_list evs ==> \
\ X : parts (sees Enemy evs)";
by (fast_tac (!claset addSEs partsEs
addSDs [Says_imp_sees_Enemy RS parts.Inj]) 1);
qed "NS3_msg_in_parts_sees_Enemy";
(** Theorems of the form X ~: parts (sees Enemy evs) imply that NOBODY
sends messages containing X! **)
(*Enemy never sees another agent's shared key!*)
goal thy
"!!evs. [| evs : ns_shared; A ~: bad |] \
\ ==> Key (shrK A) ~: parts (sees Enemy evs)";
be ns_shared.induct 1;
bd NS3_msg_in_parts_sees_Enemy 5;
by (Auto_tac());
(*Deals with Fake message*)
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]) 1);
qed "Enemy_not_see_shrK";
bind_thm ("Enemy_not_analz_shrK",
[analz_subset_parts, Enemy_not_see_shrK] MRS contra_subsetD);
Addsimps [Enemy_not_see_shrK, Enemy_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 Enemy evs); \
\ evs : ns_shared; \
\ A:bad ==> R \
\ |] ==> R";
br ccontr 1;
br ([major, Enemy_not_see_shrK] MRS rev_notE) 1;
by (swap_res_tac prems 2);
by (ALLGOALS (fast_tac (!claset addIs prems)));
qed "Enemy_see_shrK_E";
bind_thm ("Enemy_analz_shrK_E",
analz_subset_parts RS subsetD RS Enemy_see_shrK_E);
AddSEs [Enemy_see_shrK_E, Enemy_analz_shrK_E];
goal thy
"!!evs. evs : ns_shared ==> \
\ (Key (shrK A) : analz (sees Enemy evs)) = (A : bad)";
by (best_tac (!claset addIs [impOfSubs (subset_insertI RS analz_mono)]
addss (!simpset)) 1);
qed "shrK_mem_analz";
Addsimps [shrK_mem_analz];
(*** 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 length comparison, and Union over C, are essential for the
induction! *)
goal thy "!!evs. evs : ns_shared ==> \
\ length evs <= length evs' --> \
\ Key (newK evs') ~: (UN C. parts (sees C evs))";
be ns_shared.induct 1;
bd NS3_msg_in_parts_sees_Enemy 5;
(*auto_tac does not work here, as it performs safe_tac first*)
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS (fast_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 : ns_shared; length evs <= length evs' |] \
\ ==> Key (newK evs') ~: parts (sees 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 : ns_shared \
\ |] ==> length evt < length evs";
br ccontr 1;
bd leI 1;
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Enemy]
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 : ns_shared ==> \
\ length evs <= length evs' --> \
\ newK evs' ~: keysFor (UN C. parts (sees C evs))";
be ns_shared.induct 1;
bd NS3_msg_in_parts_sees_Enemy 5;
by (ALLGOALS Asm_simp_tac);
(*NS1 and NS2*)
map (by o fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [3,2];
(*Fake and NS3*)
map (by o best_tac
(!claset addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono),
Suc_leD]
addEs [new_keys_not_seen RS not_parts_not_analz RSN (2,rev_notE)]
addss (!simpset)))
[2,1];
(*NS4 and NS5: nonce exchange*)
by (ALLGOALS (deepen_tac (!claset addSDs [Says_imp_old_keys]
addIs [less_SucI, impOfSubs keysFor_mono]
addss (!simpset addsimps [le_def])) 0));
val lemma = result();
goal thy
"!!evs. [| evs : ns_shared; length evs <= length evs' |] \
\ ==> newK evs' ~: keysFor (parts (sees 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];
(** Lemmas concerning the form of items passed in messages **)
(*Describes the form of K, X and K' when the Server sends this message.*)
goal thy
"!!evs. [| Says Server A (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
\ evs : ns_shared |] \
\ ==> (EX evt:ns_shared. \
\ K = Key(newK evt) & \
\ X = (Crypt {|K, Agent A|} (shrK B)) & \
\ K' = shrK A)";
be rev_mp 1;
be ns_shared.induct 1;
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "Says_Server_message_form";
(*Describes the form of X when the following message is sent. The use of
"parts" strengthens the induction hyp for proving the Fake case. The
assumptions on A are needed to prevent its being a Faked message.*)
goal thy
"!!evs. evs : ns_shared ==> \
\ Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A) \
\ : parts (sees Enemy evs) & \
\ A ~: bad --> \
\ (EX evt:ns_shared. K = newK evt & \
\ X = (Crypt {|Key K, Agent A|} (shrK B)))";
be ns_shared.induct 1;
bd NS3_msg_in_parts_sees_Enemy 5;
(*Fake case*)
by (best_tac (!claset addSDs [impOfSubs Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 2);
by (Auto_tac());
val lemma = result() RS mp;
(*The following theorem is proved by cases. If the message was sent with a
bad key then the Enemy reads it -- even if he didn't send it in the first
place.*)
(*EITHER describes the form of X when the following message is sent,
OR reduces it to the Fake case.
Use Says_Server_message_form if applicable.*)
goal thy
"!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (shrK A)) \
\ : set_of_list evs; evs : ns_shared |] \
\ ==> (EX evt:ns_shared. K = newK evt & length evt < length evs & \
\ X = (Crypt {|Key K, Agent A|} (shrK B))) | \
\ X : analz (sees Enemy evs)";
by (excluded_middle_tac "A : bad" 1);
by (fast_tac (!claset addSDs [Says_imp_sees_Enemy RS analz.Inj]
addss (!simpset)) 2);
by (forward_tac [lemma] 1);
by (fast_tac (!claset addEs partsEs
addSDs [Says_imp_sees_Enemy RS parts.Inj]) 1);
by (fast_tac (!claset addIs [Says_imp_old_keys] addss (!simpset)) 1);
qed "Says_S_message_form";
(****
The following is to prove theorems of the form
Key K : analz (insert (Key (newK evt)) (sees Enemy evs)) ==>
Key K : analz (sees Enemy 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 : ns_shared ==> \
\ (Crypt X (newK evt)) : parts (sees Enemy evs) & \
\ Key K : parts {X} --> Key K : parts (sees Enemy evs)";
be ns_shared.induct 1;
bd NS3_msg_in_parts_sees_Enemy 5;
by (ALLGOALS (asm_simp_tac (!simpset addsimps pushes)));
(*Deals with Faked messages*)
by (best_tac (!claset addSEs partsEs
addDs [impOfSubs parts_insert_subset_Un]
addss (!simpset)) 2);
(*Base, NS4 and NS5*)
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
result();
(** Specialized rewriting for this proof **)
Delsimps [image_insert];
Addsimps [image_insert RS sym];
Delsimps [image_Un];
Addsimps [image_Un RS sym];
goal thy "insert (Key (newK x)) (sees A evs) = \
\ Key `` (newK``{x}) Un (sees A evs)";
by (Fast_tac 1);
val insert_Key_singleton = result();
goal thy "insert (Key (f x)) (Key``(f``E) Un C) = \
\ Key `` (f `` (insert x E)) Un C";
by (Fast_tac 1);
val insert_Key_image = result();
(** Session keys are not used to encrypt other session keys **)
(*Lemma for the trivial direction of the if-and-only-if*)
goal thy
"!!evs. (Key K : analz (Key``nE Un sEe)) --> \
\ (K : nE | Key K : analz sEe) ==> \
\ (Key K : analz (Key``nE Un sEe)) = (K : nE | Key K : analz sEe)";
by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 1);
val lemma = result();
(*The equality makes the induction hypothesis easier to apply*)
goal thy
"!!evs. evs : ns_shared ==> \
\ ALL K E. (Key K : analz (Key``(newK``E) Un (sees Enemy evs))) = \
\ (K : newK``E | Key K : analz (sees Enemy evs))";
be ns_shared.induct 1;
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
by (REPEAT ((eresolve_tac [bexE, conjE, disjE] ORELSE' hyp_subst_tac) 5));
by (REPEAT_FIRST (resolve_tac [allI, lemma]));
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
@ pushes)
setloop split_tac [expand_if])));
(** LEVEL 5 **)
(*NS3, Fake subcase*)
by (enemy_analz_tac 5);
(*Cases NS2 and NS3!! Simple, thanks to auto case splits*)
by (REPEAT (Fast_tac 3));
(*Fake case*) (** LEVEL 7 **)
by (enemy_analz_tac 2);
(*Base case*)
by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
qed_spec_mp "analz_image_newK";
goal thy
"!!evs. evs : ns_shared ==> \
\ Key K : analz (insert (Key (newK evt)) (sees Enemy evs)) = \
\ (K = newK evt | Key K : analz (sees Enemy 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 session key K uniquely identifies the message, if encrypted
with a secure key **)
fun ex_strip_tac i = REPEAT (ares_tac [exI, conjI] i) THEN assume_tac (i+1);
goal thy
"!!evs. evs : ns_shared ==> \
\ EX X'. ALL A X N B. \
\ A ~: bad --> \
\ Crypt {|N, Agent B, Key K, X|} (shrK A) : parts (sees Enemy evs) --> \
\ X=X'";
by (Simp_tac 1); (*Miniscoping*)
be ns_shared.induct 1;
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
by (ALLGOALS
(asm_simp_tac (!simpset addsimps [all_conj_distrib, ex_disj_distrib,
imp_conj_distrib, parts_insert_sees])));
by (REPEAT_FIRST (eresolve_tac [exE,disjE]));
(*NS2: Cextraction of K = newK evsa to global context...*)
(** LEVEL 5 **)
by (excluded_middle_tac "K = newK evsa" 3);
by (Asm_simp_tac 3);
be exI 3;
(*...we assume X is a very new message, and handle this case by contradiction*)
by (fast_tac (!claset addSEs partsEs
addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset)) 3);
(*Base, Fake, NS3*) (** LEVEL 9 **)
by (REPEAT_FIRST ex_strip_tac);
bd synth.Inj 4;
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs Fake_parts_insert]
addss (!simpset))));
val lemma = result();
(*In messages of this form, the session key uniquely identifies the rest*)
goal thy
"!!evs. [| Says S A \
\ (Crypt {|N, Agent B, Key K, X|} (shrK C)) \
\ : set_of_list evs; \
\ Says S' A' \
\ (Crypt {|N', Agent B', Key K, X'|} (shrK C')) \
\ : set_of_list evs; \
\ evs : ns_shared; C ~= Enemy; C ~: bad; C' ~: bad |] ==> X = X'";
bd lemma 1;
be exE 1;
(*Duplicate the assumption*)
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
by (fast_tac (!claset addSDs [Says_imp_sees_Enemy RS parts.Inj]) 1);
qed "unique_session_keys";
(** Crucial secrecy property: Enemy does not see the keys sent in msg NS2 **)
goal thy
"!!evs. [| A ~: bad; B ~: bad; evs : ns_shared; evt: ns_shared |] \
\ ==> Says Server A \
\ (Crypt {|N, Agent B, Key(newK evt), \
\ Crypt {|Key(newK evt), Agent A|} (shrK B)|} (shrK A)) \
\ : set_of_list evs --> \
\ Key(newK evt) ~: analz (sees Enemy evs)";
be ns_shared.induct 1;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD,
analz_insert_Key_newK] @ pushes)
setloop split_tac [expand_if])));
(*NS2*)
by (fast_tac (!claset addIs [parts_insertI]
addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset)) 2);
(*Fake case*)
by (enemy_analz_tac 1);
(*NS3: that message from the Server was sent earlier*)
by (forward_tac [Says_S_message_form] 1 THEN assume_tac 1);
by (Step_tac 1);
by (enemy_analz_tac 2); (*Prove the Fake subcase*)
by (asm_full_simp_tac
(!simpset addsimps (mem_if::analz_insert_Key_newK::pushes)) 1);
by (Step_tac 1);
(**LEVEL 9 **)
by (excluded_middle_tac "Aa : bad" 1);
(*But this contradicts Key(newK evt) ~: analz (sees Enemy evsa) *)
bd (Says_imp_sees_Enemy RS analz.Inj) 2;
by (fast_tac (!claset addSDs [analz.Decrypt]
addss (!simpset)) 2);
(*So now we have Aa ~: bad *)
bd unique_session_keys 1;
by (Auto_tac ());
val lemma = result() RS mp RSN(2,rev_notE);
(*Final version: Server's message in the most abstract form*)
goal thy
"!!evs. [| Says Server A \
\ (Crypt {|N, Agent B, K, X|} K') : set_of_list evs; \
\ A ~: bad; B ~: bad; evs : ns_shared \
\ |] ==> \
\ K ~: analz (sees Enemy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (fast_tac (!claset addSEs [lemma]) 1);
qed "Enemy_not_see_encrypted_key";