Separation of theory Event into two parts:
Shared for general shared-key material
NS_Shared for the Needham-Schroeder shared-key protocol
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Auth/NS_Shared.ML Wed Aug 21 13:25:27 1996 +0200
@@ -0,0 +1,497 @@
+(* 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;
+
+(**** 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)];
+
+goal thy "!!evs. evs : ns_shared ==> Notes A X ~: set_of_list evs";
+be ns_shared.induct 1;
+by (Auto_tac());
+qed "not_Notes";
+Addsimps [not_Notes];
+AddSEs [not_Notes RSN (2, rev_notE)];
+
+
+(*For reasoning about 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";
+
+
+(*** Server keys are not betrayed ***)
+
+(*Enemy never sees another agent's server key!*)
+goal thy
+ "!!evs. [| evs : ns_shared; A ~= Enemy |] ==> \
+\ Key (serverKey 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 synth_analz_parts_insert_subset_Un]) 1);
+qed "Enemy_not_see_serverKey";
+
+bind_thm ("Enemy_not_analz_serverKey",
+ [analz_subset_parts, Enemy_not_see_serverKey] MRS contra_subsetD);
+
+Addsimps [Enemy_not_see_serverKey,
+ not_sym RSN (2, Enemy_not_see_serverKey),
+ Enemy_not_analz_serverKey,
+ not_sym RSN (2, Enemy_not_analz_serverKey)];
+
+(*We go to some trouble to preserve R in the 3rd subgoal*)
+val major::prems =
+goal thy "[| Key (serverKey A) : parts (sees Enemy evs); \
+\ evs : ns_shared; \
+\ A=Enemy ==> R \
+\ |] ==> R";
+br ccontr 1;
+br ([major, Enemy_not_see_serverKey] MRS rev_notE) 1;
+by (swap_res_tac prems 2);
+by (ALLGOALS (fast_tac (!claset addIs prems)));
+qed "Enemy_see_serverKey_E";
+
+bind_thm ("Enemy_analz_serverKey_E",
+ analz_subset_parts RS subsetD RS Enemy_see_serverKey_E);
+
+(*Classical reasoner doesn't need the not_sym versions (with swapped ~=) *)
+AddSEs [Enemy_see_serverKey_E, Enemy_analz_serverKey_E];
+
+
+(*No Friend will ever see another agent's server key
+ (excluding the Enemy, who might transmit his).
+ The Server, of course, knows all server keys.*)
+goal thy
+ "!!evs. [| evs : ns_shared; A ~= Enemy; A ~= Friend j |] ==> \
+\ Key (serverKey A) ~: parts (sees (Friend j) evs)";
+br (sees_agent_subset_sees_Enemy RS parts_mono RS contra_subsetD) 1;
+by (ALLGOALS Asm_simp_tac);
+qed "Friend_not_see_serverKey";
+
+
+(*Not for Addsimps -- it can cause goals to blow up!*)
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ (Key (serverKey A) \
+\ : analz (insert (Key (serverKey B)) (sees Enemy evs))) = \
+\ (A=B | A=Enemy)";
+by (best_tac (!claset addDs [impOfSubs analz_subset_parts]
+ addIs [impOfSubs (subset_insertI RS analz_mono)]
+ addss (!simpset)) 1);
+qed "serverKey_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 (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 : 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;
+by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Enemy]
+ addIs [impOfSubs parts_mono, leI]) 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 addSDs [newK_invKey]
+ 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 [newK_invKey, 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 *and age* of K, and the form of X,
+ when the following message is sent*)
+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|} (serverKey B)) & \
+\ K' = serverKey A & \
+\ length evt < length evs)";
+be rev_mp 1;
+be ns_shared.induct 1;
+by (ALLGOALS (fast_tac (!claset addIs [less_SucI] addss (!simpset))));
+qed "Says_Server_message_form";
+
+
+(*Describes the form of X when the following message is sent*)
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ ALL A NA B K X. \
+\ (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
+\ : parts (sees Enemy evs) & A ~= Enemy --> \
+\ (EX evt:ns_shared. K = newK evt & \
+\ X = (Crypt {|Key K, Agent A|} (serverKey B)))";
+be ns_shared.induct 1;
+bd NS3_msg_in_parts_sees_Enemy 5;
+by (Step_tac 1);
+by (ALLGOALS Asm_full_simp_tac);
+(*Remaining cases are Fake and NS2*)
+by (fast_tac (!claset addSDs [spec]) 2);
+(*Now for the Fake case*)
+by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
+ impOfSubs synth_analz_parts_insert_subset_Un]
+ addss (!simpset)) 1);
+qed_spec_mp "encrypted_form";
+
+
+(*For eliminating the A ~= Enemy condition from the previous result*)
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ ALL S A NA B K X. \
+\ Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)) \
+\ : set_of_list evs --> \
+\ S = Server | S = Enemy";
+be ns_shared.induct 1;
+by (ALLGOALS Asm_simp_tac);
+(*We are left with NS3*)
+by (subgoal_tac "S = Server | S = Enemy" 1);
+(*First justify this assumption!*)
+by (fast_tac (!claset addSEs [allE, mp] addss (!simpset)) 2);
+by (Step_tac 1);
+bd Says_Server_message_form 1;
+by (ALLGOALS Full_simp_tac);
+(*Final case. Clear out needless quantifiers to speed the following step*)
+by (eres_inst_tac [("V","ALL x. ?P(x)")] thin_rl 1);
+bd encrypted_form 1;
+br (parts.Inj RS conjI) 1;
+auto();
+qed_spec_mp "Server_or_Enemy";
+
+
+(*Describes the form of X when the following message is sent;
+ use Says_Server_message_form if applicable*)
+goal thy
+ "!!evs. [| Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey 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|} (serverKey B)))";
+by (forward_tac [Server_or_Enemy] 1);
+ba 1;
+by (Step_tac 1);
+by (fast_tac (!claset addSDs [Says_Server_message_form] addss (!simpset)) 1);
+by (forward_tac [encrypted_form] 1);
+br (parts.Inj RS conjI) 1;
+by (auto_tac (!claset addIs [Says_imp_old_keys], !simpset));
+qed "Says_S_message_form";
+
+
+
+(****
+ The following is to prove theorems of the form
+
+ Key K : analz (insert (Key (newK evt))
+ (insert (Key (serverKey C)) (sees Enemy evs))) ==>
+ Key K : analz (insert (Key (serverKey C)) (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 analz_subset_parts,
+ impOfSubs parts_insert_subset_Un]
+ addss (!simpset)) 1);
+(*NS4 and NS5*)
+by (ALLGOALS (fast_tac (!claset addss (!simpset))));
+result();
+
+
+(** Specialized rewriting for this proof **)
+
+Delsimps [image_insert];
+Addsimps [image_insert 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 **)
+
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ ALL K E. (Key K : analz (insert (Key (serverKey C)) \
+\ (Key``(newK``E) Un (sees Enemy evs)))) = \
+\ (K : newK``E | \
+\ Key K : analz (insert (Key (serverKey C)) \
+\ (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] ORELSE' hyp_subst_tac) 5));
+by (ALLGOALS
+ (asm_simp_tac
+ (!simpset addsimps ([insert_Key_singleton, insert_Key_image, pushKey_newK]
+ @ pushes)
+ setloop split_tac [expand_if])));
+(*Cases NS2 and NS3!! Simple, thanks to auto case splits*)
+by (REPEAT (Fast_tac 3));
+(*Base case*)
+by (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1);
+(** LEVEL 7 **)
+(*Fake case*)
+by (REPEAT (Safe_step_tac 1));
+by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 2);
+by (subgoal_tac
+ "Key K : analz \
+\ (synth \
+\ (analz (insert (Key (serverKey C)) \
+\ (Key``(newK``E) Un (sees Enemy evsa)))))" 1);
+(*First, justify this subgoal*)
+(*Discard formulae for better speed*)
+by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
+by (eres_inst_tac [("V","?Q ~: ?QQ")] thin_rl 2);
+by (best_tac (!claset addIs [impOfSubs (analz_mono RS synth_mono)]
+ addSEs [impOfSubs analz_mono]) 2);
+by (Asm_full_simp_tac 1);
+by (deepen_tac (!claset addIs [impOfSubs analz_mono]) 0 1);
+qed_spec_mp "analz_image_newK";
+
+
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ Key K : analz (insert (Key (newK evt)) \
+\ (insert (Key (serverKey C)) \
+\ (sees Enemy evs))) = \
+\ (K = newK evt | \
+\ Key K : analz (insert (Key (serverKey C)) \
+\ (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";
+
+
+
+(*This says that the Key, K, uniquely identifies the message.
+ But if C=Enemy then he could send all sorts of nonsense.*)
+goal thy
+ "!!evs. evs : ns_shared ==> \
+\ EX X'. ALL C S A Y N B X. \
+\ C ~= Enemy --> \
+\ Says S A Y : set_of_list evs --> \
+\ ((Crypt {|N, Agent B, Key K, X|} (serverKey C)) : parts{Y} --> \
+\ (X = X'))";
+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, imp_conj_distrib])));
+(*NS2: Case split propagates some context to other subgoal...*)
+by (excluded_middle_tac "K = newK evsa" 2);
+by (Asm_simp_tac 2);
+(*...we assume X is a very new message, and handle this case by contradiction*)
+by (fast_tac (!claset addIs [impOfSubs (subset_insertI RS parts_mono)]
+ addSEs partsEs
+ addEs [Says_imp_old_keys RS less_irrefl]
+ addss (!simpset)) 2);
+(*NS3: No relevant messages*)
+by (fast_tac (!claset addSEs [exI] addss (!simpset)) 2);
+(*Fake*)
+by (Step_tac 1);
+br exI 1;
+br conjI 1;
+ba 2;
+by (Step_tac 1);
+(** LEVEL 12 **)
+by (subgoal_tac "Crypt {|N, Agent Ba, Key K, Xa|} (serverKey C) \
+\ : parts (sees Enemy evsa)" 1);
+by (eres_inst_tac [("V","ALL S.?P(S)")] thin_rl 2);
+by (best_tac (!claset addSEs [impOfSubs analz_subset_parts]
+ addDs [impOfSubs parts_insert_subset_Un]
+ addss (!simpset)) 2);
+by (eres_inst_tac [("V","?aa : parts {X}")] thin_rl 1);
+bd parts_singleton 1;
+by (Step_tac 1);
+bd seesD 1;
+by (Step_tac 1);
+by (Full_simp_tac 2);
+by (fast_tac (!claset addSDs [spec]) 1);
+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|} (serverKey C)) \
+\ : set_of_list evs; \
+ \ Says S' A' \
+\ (Crypt {|N', Agent B', Key K, X'|} (serverKey C')) \
+\ : set_of_list evs; \
+\ evs : ns_shared; C ~= Enemy; C' ~= Enemy |] ==> X = X'";
+bd lemma 1;
+be exE 1;
+by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
+by (Fast_tac 1);
+qed "unique_session_keys";
+
+
+
+(*Crucial secrecy property: Enemy does not see the keys sent in msg NS2
+ -- even if another key is compromised*)
+goal thy
+ "!!evs. [| Says Server (Friend i) \
+\ (Crypt {|N, Agent(Friend j), K, X|} K') : set_of_list evs; \
+\ evs : ns_shared; Friend i ~= C; Friend j ~= C \
+\ |] ==> \
+\ K ~: analz (insert (Key (serverKey C)) (sees Enemy evs))";
+be rev_mp 1;
+be ns_shared.induct 1;
+by (ALLGOALS (asm_simp_tac (!simpset addsimps pushes)));
+(*Next 3 steps infer that K has the form "Key (newK evs'" ... *)
+by (REPEAT_FIRST (resolve_tac [conjI, impI]));
+by (TRYALL (forward_tac [Says_Server_message_form] THEN' assume_tac));
+by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
+by (ALLGOALS
+ (asm_full_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 addSEs [less_irrefl]) 2);
+(** LEVEL 8 **)
+(*Now for the Fake case*)
+br notI 1;
+by (subgoal_tac
+ "Key (newK evt) : \
+\ analz (synth (analz (insert (Key (serverKey C)) \
+\ (sees Enemy evsa))))" 1);
+be (impOfSubs analz_mono) 2;
+by (deepen_tac (!claset addIs [analz_mono RS synth_mono RSN (2,rev_subsetD),
+ impOfSubs synth_increasing,
+ impOfSubs analz_increasing]) 0 2);
+(*Proves the Fake goal*)
+by (fast_tac (!claset addss (!simpset)) 1);
+
+(**LEVEL 13**)
+(*NS3: that message from the Server was sent earlier*)
+by (mp_tac 1);
+by (forward_tac [Says_S_message_form] 1 THEN assume_tac 1);
+by (REPEAT_FIRST (eresolve_tac [bexE, conjE] ORELSE' hyp_subst_tac));
+by (asm_full_simp_tac
+ (!simpset addsimps (mem_if::analz_insert_Key_newK::pushes)) 1);
+by (Step_tac 1);
+(**LEVEL 18 **)
+bd unique_session_keys 1;
+by (REPEAT_FIRST assume_tac);
+by (ALLGOALS Full_simp_tac);
+by (Step_tac 1);
+by (asm_full_simp_tac (!simpset addsimps [serverKey_mem_analz]) 1);
+qed "Enemy_not_see_encrypted_key";
+
+
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Auth/NS_Shared.thy Wed Aug 21 13:25:27 1996 +0200
@@ -0,0 +1,67 @@
+(* 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)
+*)
+
+NS_Shared = Shared +
+
+consts ns_shared :: "event list set"
+inductive ns_shared
+ intrs
+ (*Initial trace is empty*)
+ Nil "[]: ns_shared"
+
+ (*The enemy MAY say anything he CAN say. We do not expect him to
+ invent new nonces here, but he can also use NS1. Common to
+ all similar protocols.*)
+ Fake "[| evs: ns_shared; B ~= Enemy; X: synth (analz (sees Enemy evs))
+ |] ==> (Says Enemy B X) # evs : ns_shared"
+
+ (*Alice initiates a protocol run*)
+ NS1 "[| evs: ns_shared; A ~= Server
+ |] ==> (Says A Server {|Agent A, Agent B, Nonce (newN evs)|})
+ # evs : ns_shared"
+
+ (*Server's response to Alice's message.
+ !! It may respond more than once to A's request !!
+ Server doesn't know who the true sender is, hence the A' in
+ the sender field.*)
+ NS2 "[| evs: ns_shared; A ~= B; A ~= Server;
+ (Says A' Server {|Agent A, Agent B, Nonce NA|}) : set_of_list evs
+ |] ==> (Says Server A
+ (Crypt {|Nonce NA, Agent B, Key (newK evs),
+ (Crypt {|Key (newK evs), Agent A|} (serverKey B))|}
+ (serverKey A))) # evs : ns_shared"
+
+ (*We can't assume S=Server. Agent A "remembers" her nonce.
+ May assume WLOG that she is NOT the Enemy: the Fake rule
+ covers this case. Can inductively show A ~= Server*)
+ NS3 "[| evs: ns_shared; A ~= B;
+ (Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)))
+ : set_of_list evs;
+ A = Friend i;
+ (Says A Server {|Agent A, Agent B, Nonce NA|}) : set_of_list evs
+ |] ==> (Says A B X) # evs : ns_shared"
+
+ (*Bob's nonce exchange. He does not know who the message came
+ from, but responds to A because she is mentioned inside.*)
+ NS4 "[| evs: ns_shared; A ~= B;
+ (Says A' B (Crypt {|Key K, Agent A|} (serverKey B)))
+ : set_of_list evs
+ |] ==> (Says B A (Crypt (Nonce (newN evs)) K)) # evs : ns_shared"
+
+ (*Alice responds with (Suc N), if she has seen the key before.*)
+ NS5 "[| evs: ns_shared; A ~= B;
+ (Says B' A (Crypt (Nonce N) K)) : set_of_list evs;
+ (Says S A (Crypt {|Nonce NA, Agent B, Key K, X|} (serverKey A)))
+ : set_of_list evs
+ |] ==> (Says A B (Crypt (Nonce (Suc N)) K)) # evs : ns_shared"
+
+end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Auth/Shared.ML Wed Aug 21 13:25:27 1996 +0200
@@ -0,0 +1,216 @@
+(* Title: HOL/Auth/Message
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1996 University of Cambridge
+
+Theory of Shared Keys (common to all symmetric-key protocols)
+
+Server keys; initial states of agents; new nonces and keys; function "sees"
+
+
+*)
+
+Addsimps [parts_cut_eq];
+
+proof_timing:=true;
+
+(*IN SET.ML*)
+goal Set.thy "(a : (if Q then x else y)) = ((Q --> a:x) & (~Q --> a : y))";
+by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
+qed "mem_if";
+
+(*FUN.ML?? WE NEED A NOTION OF INVERSE IMAGE, OR GRAPH!!*)
+goal Set.thy "!!f. B <= range f = (B = f`` {x. f x: B})";
+by (fast_tac (!claset addEs [equalityE]) 1);
+val subset_range_iff = result();
+
+
+open Shared;
+
+Addsimps [Un_insert_left, Un_insert_right];
+
+(*By default only o_apply is built-in. But in the presence of eta-expansion
+ this means that some terms displayed as (f o g) will be rewritten, and others
+ will not!*)
+Addsimps [o_def];
+
+(*** Basic properties of serverKey and newK ***)
+
+(* invKey (serverKey A) = serverKey A *)
+bind_thm ("invKey_serverKey", rewrite_rule [isSymKey_def] isSym_serverKey);
+
+(* invKey (newK evs) = newK evs *)
+bind_thm ("invKey_newK", rewrite_rule [isSymKey_def] isSym_newK);
+Addsimps [invKey_serverKey, invKey_newK];
+
+
+(*New keys and nonces are fresh*)
+val serverKey_inject = inj_serverKey RS injD;
+val newN_inject = inj_newN RS injD
+and newK_inject = inj_newK RS injD;
+AddSEs [serverKey_inject, newN_inject, newK_inject,
+ fresh_newK RS notE, fresh_newN RS notE];
+Addsimps [inj_serverKey RS inj_eq, inj_newN RS inj_eq, inj_newK RS inj_eq];
+Addsimps [fresh_newN, fresh_newK];
+
+(** Rewrites should not refer to initState(Friend i)
+ -- not in normal form! **)
+
+goal thy "newK evs ~= serverKey B";
+by (subgoal_tac "newK evs = serverKey B --> \
+\ Key (newK evs) : parts (initState B)" 1);
+by (Fast_tac 1);
+by (agent.induct_tac "B" 1);
+by (auto_tac (!claset addIs [range_eqI], !simpset));
+qed "newK_neq_serverKey";
+
+Addsimps [newK_neq_serverKey, newK_neq_serverKey RS not_sym];
+
+(*Good for talking about Server's initial state*)
+goal thy "!!H. H <= Key``E ==> parts H = H";
+by (Auto_tac ());
+be parts.induct 1;
+by (ALLGOALS (fast_tac (!claset addss (!simpset))));
+qed "parts_image_subset";
+
+bind_thm ("parts_image_Key", subset_refl RS parts_image_subset);
+
+goal thy "!!H. H <= Key``E ==> analz H = H";
+by (Auto_tac ());
+be analz.induct 1;
+by (ALLGOALS (fast_tac (!claset addss (!simpset))));
+qed "analz_image_subset";
+
+bind_thm ("analz_image_Key", subset_refl RS analz_image_subset);
+
+Addsimps [parts_image_Key, analz_image_Key];
+
+goalw thy [keysFor_def] "keysFor (parts (initState C)) = {}";
+by (agent.induct_tac "C" 1);
+by (auto_tac (!claset addIs [range_eqI] delrules partsEs, !simpset));
+qed "keysFor_parts_initState";
+Addsimps [keysFor_parts_initState];
+
+goalw thy [keysFor_def] "keysFor (Key``E) = {}";
+by (Auto_tac ());
+qed "keysFor_image_Key";
+Addsimps [keysFor_image_Key];
+
+goal thy "serverKey A ~: newK``E";
+by (agent.induct_tac "A" 1);
+by (Auto_tac ());
+qed "serverKey_notin_image_newK";
+Addsimps [serverKey_notin_image_newK];
+
+
+(*Agents see their own serverKeys!*)
+goal thy "Key (serverKey A) : analz (sees A evs)";
+by (list.induct_tac "evs" 1);
+by (asm_simp_tac (!simpset addsimps [impOfSubs(Un_upper2 RS analz_mono)]) 2);
+by (agent.induct_tac "A" 1);
+by (auto_tac (!claset addIs [range_eqI], !simpset));
+qed "analz_own_serverKey";
+
+bind_thm ("parts_own_serverKey",
+ [analz_subset_parts, analz_own_serverKey] MRS subsetD);
+
+Addsimps [analz_own_serverKey, parts_own_serverKey];
+
+
+
+(** Specialized rewrite rules for (sees A (Says...#evs)) **)
+
+goal thy "sees A (Says A B X # evs) = insert X (sees A evs)";
+by (Simp_tac 1);
+qed "sees_own";
+
+goal thy "!!A. Server ~= A ==> \
+\ sees Server (Says A B X # evs) = sees Server evs";
+by (Asm_simp_tac 1);
+qed "sees_Server";
+
+goal thy "!!A. Friend i ~= A ==> \
+\ sees (Friend i) (Says A B X # evs) = sees (Friend i) evs";
+by (Asm_simp_tac 1);
+qed "sees_Friend";
+
+goal thy "sees Enemy (Says A B X # evs) = insert X (sees Enemy evs)";
+by (Simp_tac 1);
+qed "sees_Enemy";
+
+goal thy "sees A (Says A' B X # evs) <= insert X (sees A evs)";
+by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
+by (Fast_tac 1);
+qed "sees_Says_subset_insert";
+
+goal thy "sees A evs <= sees A (Says A' B X # evs)";
+by (simp_tac (!simpset setloop split_tac [expand_if]) 1);
+by (Fast_tac 1);
+qed "sees_subset_sees_Says";
+
+(*Pushing Unions into parts; one of the A's equals B, and thus sees Y*)
+goal thy "(UN A. parts (sees A (Says B C Y # evs))) = \
+\ parts {Y} Un (UN A. parts (sees A evs))";
+by (Step_tac 1);
+be rev_mp 1; (*for some reason, split_tac does not work on assumptions*)
+val ss = (!simpset addsimps [parts_Un, sees_Cons]
+ setloop split_tac [expand_if]);
+by (ALLGOALS (fast_tac (!claset addss ss)));
+qed "UN_parts_sees_Says";
+
+goal thy "Says A B X : set_of_list evs --> X : sees Enemy evs";
+by (list.induct_tac "evs" 1);
+by (Auto_tac ());
+qed_spec_mp "Says_imp_sees_Enemy";
+
+Addsimps [Says_imp_sees_Enemy];
+AddIs [Says_imp_sees_Enemy];
+
+goal thy "initState C <= Key `` range serverKey";
+by (agent.induct_tac "C" 1);
+by (Auto_tac ());
+qed "initState_subset";
+
+goal thy "X : sees C evs --> \
+\ (EX A B. Says A B X : set_of_list evs) | \
+\ (EX A. Notes A X : set_of_list evs) | \
+\ (EX A. X = Key (serverKey A))";
+by (list.induct_tac "evs" 1);
+by (ALLGOALS Asm_simp_tac);
+by (fast_tac (!claset addDs [impOfSubs initState_subset]) 1);
+br conjI 1;
+by (Fast_tac 2);
+by (event.induct_tac "a" 1);
+by (ALLGOALS (asm_simp_tac (!simpset addsimps [mem_if])));
+by (ALLGOALS Fast_tac);
+qed_spec_mp "seesD";
+
+
+Addsimps [UN_parts_sees_Says, sees_own, sees_Server, sees_Friend, sees_Enemy];
+Delsimps [sees_Cons]; (**** NOTE REMOVAL -- laws above are cleaner ****)
+
+
+goal thy "!!K. newK evs = invKey K ==> newK evs = K";
+br (invKey_eq RS iffD1) 1;
+by (Simp_tac 1);
+val newK_invKey = result();
+
+
+(** Rewrites to push in Key and Crypt messages, so that other messages can
+ be pulled out using the analz_insert rules **)
+
+fun insComm x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)]
+ insert_commute;
+
+val pushKeys = map (insComm "Key ?K")
+ ["Agent ?C", "Nonce ?N", "MPair ?X ?Y", "Crypt ?X ?K'"];
+
+val pushCrypts = map (insComm "Crypt ?X ?K")
+ ["Agent ?C", "Nonce ?N", "MPair ?X' ?Y"];
+
+(*Cannot be added with Addsimps -- we don't always want to re-order messages*)
+val pushes = pushKeys@pushCrypts;
+
+val pushKey_newK = insComm "Key (newK ?evs)" "Key (serverKey ?C)";
+
+
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Auth/Shared.thy Wed Aug 21 13:25:27 1996 +0200
@@ -0,0 +1,67 @@
+(* Title: HOL/Auth/Shared
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1996 University of Cambridge
+
+Theory of Shared Keys (common to all symmetric-key protocols)
+
+Server keys; initial states of agents; new nonces and keys; function "sees"
+*)
+
+Shared = Message + List +
+
+consts
+ serverKey :: agent => key (*symmetric keys*)
+
+rules
+ isSym_serverKey "isSymKey (serverKey A)"
+
+consts (*Initial states of agents -- parameter of the construction*)
+ initState :: agent => msg set
+
+primrec initState agent
+ (*Server knows all keys; other agents know only their own*)
+ initState_Server "initState Server = Key `` range serverKey"
+ initState_Friend "initState (Friend i) = {Key (serverKey (Friend i))}"
+ initState_Enemy "initState Enemy = {Key (serverKey Enemy)}"
+
+datatype (*Messages, and components of agent stores*)
+ event = Says agent agent msg
+ | Notes agent msg
+
+consts
+ sees1 :: [agent, event] => msg set
+
+primrec sees1 event
+ (*First agent recalls all that it says, but NOT everything
+ that is sent to it; it must note such things if/when received*)
+ sees1_Says "sees1 A (Says A' B X) = (if A:{A',Enemy} then {X} else {})"
+ (*part of A's internal state*)
+ sees1_Notes "sees1 A (Notes A' X) = (if A=A' then {X} else {})"
+
+consts
+ sees :: [agent, event list] => msg set
+
+primrec sees list
+ (*Initial knowledge includes all public keys and own private key*)
+ sees_Nil "sees A [] = initState A"
+ sees_Cons "sees A (ev#evs) = sees1 A ev Un sees A evs"
+
+
+(*Agents generate "random" nonces. Different traces always yield
+ different nonces. Same applies for keys.*)
+consts
+ newN :: "event list => nat"
+ newK :: "event list => key"
+
+rules
+ inj_serverKey "inj serverKey"
+
+ inj_newN "inj newN"
+ fresh_newN "Nonce (newN evs) ~: parts (initState B)"
+
+ inj_newK "inj newK"
+ fresh_newK "Key (newK evs) ~: parts (initState B)"
+ isSym_newK "isSymKey (newK evs)"
+
+end