# HG changeset patch # User paulson # Date 840626727 -7200 # Node ID 58573e7041b47d837175b21215c463a55a6f3821 # Parent 8b24773de6dbacc3159812c6cf8abba97e322bc0 Separation of theory Event into two parts: Shared for general shared-key material NS_Shared for the Needham-Schroeder shared-key protocol diff -r 8b24773de6db -r 58573e7041b4 src/HOL/Auth/NS_Shared.ML --- /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"; + + + + diff -r 8b24773de6db -r 58573e7041b4 src/HOL/Auth/NS_Shared.thy --- /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 diff -r 8b24773de6db -r 58573e7041b4 src/HOL/Auth/Shared.ML --- /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)"; + + diff -r 8b24773de6db -r 58573e7041b4 src/HOL/Auth/Shared.thy --- /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