src/HOL/Auth/OtwayRees_Bad.ML
author paulson
Tue, 16 Dec 1997 15:17:26 +0100
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permissions -rw-r--r--
Simplified proofs using rewrites for f``A where f is injective
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(*  Title:      HOL/Auth/OtwayRees_Bad
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1996  University of Cambridge
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Inductive relation "otway" for the Otway-Rees protocol.
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The FAULTY version omitting encryption of Nonce NB, as suggested on page 247 of
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  Burrows, Abadi and Needham.  A Logic of Authentication.
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  Proc. Royal Soc. 426 (1989)
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This file illustrates the consequences of such errors.  We can still prove
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impressive-looking properties such as Spy_not_see_encrypted_key, yet the
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protocol is open to a middleperson attack.  Attempting to prove some key lemmas
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indicates the possibility of this attack.
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*)
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open OtwayRees_Bad;
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proof_timing:=true;
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HOL_quantifiers := false;
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(*A "possibility property": there are traces that reach the end*)
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goal thy 
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 "!!A B. [| A ~= B; A ~= Server; B ~= Server |]   \
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\        ==> EX K. EX NA. EX evs: otway.          \
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\               Says B A {|Nonce NA, Crypt (shrK A) {|Nonce NA, Key K|}|} \
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\                 : set evs";
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by (REPEAT (resolve_tac [exI,bexI] 1));
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by (rtac (otway.Nil RS otway.OR1 RS otway.OR2 RS otway.OR3 RS otway.OR4) 2);
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by possibility_tac;
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result();
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(**** Inductive proofs about otway ****)
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(*Nobody sends themselves messages*)
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goal thy "!!evs. evs : otway ==> ALL A X. Says A A X ~: set evs";
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by (etac otway.induct 1);
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by (Auto_tac());
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qed_spec_mp "not_Says_to_self";
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Addsimps [not_Says_to_self];
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AddSEs   [not_Says_to_self RSN (2, rev_notE)];
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(** For reasoning about the encrypted portion of messages **)
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goal thy "!!evs. Says A' B {|N, Agent A, Agent B, X|} : set evs ==> \
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\                X : analz (spies evs)";
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by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1);
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qed "OR2_analz_spies";
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goal thy "!!evs. Says S' B {|N, X, Crypt (shrK B) X'|} : set evs ==> \
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\                X : analz (spies evs)";
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by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1);
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qed "OR4_analz_spies";
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goal thy "!!evs. Says Server B {|NA, X, Crypt K' {|NB,K|}|} : set evs \
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\                 ==> K : parts (spies evs)";
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by (blast_tac (claset() addSEs spies_partsEs) 1);
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qed "Oops_parts_spies";
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(*OR2_analz... and OR4_analz... let us treat those cases using the same 
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  argument as for the Fake case.  This is possible for most, but not all,
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  proofs: Fake does not invent new nonces (as in OR2), and of course Fake
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  messages originate from the Spy. *)
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bind_thm ("OR2_parts_spies",
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          OR2_analz_spies RS (impOfSubs analz_subset_parts));
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bind_thm ("OR4_parts_spies",
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          OR4_analz_spies RS (impOfSubs analz_subset_parts));
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(*For proving the easier theorems about X ~: parts (spies evs).*)
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fun parts_induct_tac i = 
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    etac otway.induct i			THEN 
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    forward_tac [Oops_parts_spies] (i+6) THEN
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    forward_tac [OR4_parts_spies]  (i+5) THEN
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    forward_tac [OR2_parts_spies]  (i+3) THEN 
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    prove_simple_subgoals_tac  i;
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(** Theorems of the form X ~: parts (spies evs) imply that NOBODY
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    sends messages containing X! **)
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(*Spy never sees another agent's shared key! (unless it's bad at start)*)
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goal thy 
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 "!!evs. evs : otway ==> (Key (shrK A) : parts (spies evs)) = (A : bad)";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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by (ALLGOALS Blast_tac);
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qed "Spy_see_shrK";
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Addsimps [Spy_see_shrK];
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goal thy 
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 "!!evs. evs : otway ==> (Key (shrK A) : analz (spies evs)) = (A : bad)";
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by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset()));
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qed "Spy_analz_shrK";
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Addsimps [Spy_analz_shrK];
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goal thy  "!!A. [| Key (shrK A) : parts (spies evs); evs : otway |] ==> A:bad";
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by (blast_tac (claset() addDs [Spy_see_shrK]) 1);
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qed "Spy_see_shrK_D";
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bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
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AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];
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(*Nobody can have used non-existent keys!*)
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goal thy "!!evs. evs : otway ==>          \
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\         Key K ~: used evs --> K ~: keysFor (parts (spies evs))";
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by (parts_induct_tac 1);
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(*Fake*)
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by (best_tac
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      (claset() addSDs [impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
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               addIs  [impOfSubs analz_subset_parts]
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               addDs  [impOfSubs (analz_subset_parts RS keysFor_mono)]
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               addss  (simpset())) 1);
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(*OR1-3*)
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by (ALLGOALS Blast_tac);
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qed_spec_mp "new_keys_not_used";
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bind_thm ("new_keys_not_analzd",
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          [analz_subset_parts RS keysFor_mono,
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           new_keys_not_used] MRS contra_subsetD);
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Addsimps [new_keys_not_used, new_keys_not_analzd];
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(*** Proofs involving analz ***)
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(*Describes the form of K and NA when the Server sends this message.  Also
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  for Oops case.*)
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goal thy 
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 "!!evs. [| Says Server B                                                 \
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\            {|NA, X, Crypt (shrK B) {|NB, Key K|}|} : set evs;           \
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\           evs : otway |]                                                \
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\     ==> K ~: range shrK & (EX i. NA = Nonce i) & (EX j. NB = Nonce j)";
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by (etac rev_mp 1);
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by (etac otway.induct 1);
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by (prove_simple_subgoals_tac 1);
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by (Blast_tac 1); 
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qed "Says_Server_message_form";
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(*For proofs involving analz.*)
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val analz_spies_tac = 
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    dtac OR2_analz_spies 4 THEN 
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    dtac OR4_analz_spies 6 THEN
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    forward_tac [Says_Server_message_form] 7 THEN assume_tac 7 THEN 
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    REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 7);
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(****
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 The following is to prove theorems of the form
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  Key K : analz (insert (Key KAB) (spies evs)) ==>
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  Key K : analz (spies evs)
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 A more general formula must be proved inductively.
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****)
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(** Session keys are not used to encrypt other session keys **)
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(*The equality makes the induction hypothesis easier to apply*)
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goal thy  
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 "!!evs. evs : otway ==>                                    \
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\  ALL K KK. KK <= Compl (range shrK) -->                   \
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\            (Key K : analz (Key``KK Un (spies evs))) =  \
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\            (K : KK | Key K : analz (spies evs))";
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by (etac otway.induct 1);
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by analz_spies_tac;
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by (REPEAT_FIRST (resolve_tac [allI, impI]));
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by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
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by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
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(*Fake*) 
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by (spy_analz_tac 1);
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qed_spec_mp "analz_image_freshK";
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goal thy
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 "!!evs. [| evs : otway;  KAB ~: range shrK |] ==>          \
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\        Key K : analz (insert (Key KAB) (spies evs)) =  \
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\        (K = KAB | Key K : analz (spies evs))";
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by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
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qed "analz_insert_freshK";
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(*** The Key K uniquely identifies the Server's  message. **)
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goal thy 
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 "!!evs. evs : otway ==>                                                  \
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\   EX B' NA' NB' X'. ALL B NA NB X.                                      \
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\     Says Server B {|NA, X, Crypt (shrK B) {|NB, K|}|} : set evs -->     \
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\     B=B' & NA=NA' & NB=NB' & X=X'";
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by (etac otway.induct 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [all_conj_distrib])));
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by (ALLGOALS Clarify_tac);
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(*Remaining cases: OR3 and OR4*)
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by (ex_strip_tac 2);
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by (Blast_tac 2);
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by (expand_case_tac "K = ?y" 1);
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by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
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(*...we assume X is a recent message, and handle this case by contradiction*)
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by (blast_tac (claset() addSEs spies_partsEs
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                      delrules [conjI]    (*no split-up to 4 subgoals*)) 1);
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val lemma = result();
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goal thy 
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 "!!evs. [| Says Server B {|NA, X, Crypt (shrK B) {|NB, K|}|}   : set evs; \ 
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\           Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} : set evs; \
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\           evs : otway |] ==> X=X' & B=B' & NA=NA' & NB=NB'";
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by (prove_unique_tac lemma 1);
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qed "unique_session_keys";
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(*Crucial security property, but not itself enough to guarantee correctness!*)
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goal thy 
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 "!!evs. [| A ~: bad;  B ~: bad;  evs : otway |]                    \
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\        ==> Says Server B                                            \
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\              {|NA, Crypt (shrK A) {|NA, Key K|},                    \
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\                Crypt (shrK B) {|NB, Key K|}|} : set evs -->         \
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\            Says B Spy {|NA, NB, Key K|} ~: set evs -->              \
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\            Key K ~: analz (spies evs)";
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by (etac otway.induct 1);
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by analz_spies_tac;
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by (ALLGOALS
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    (asm_simp_tac (simpset() addcongs [conj_cong] 
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                            addsimps [analz_insert_eq, analz_insert_freshK]
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                            addsimps (pushes@expand_ifs))));
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(*Oops*)
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by (blast_tac (claset() addSDs [unique_session_keys]) 4);
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(*OR4*) 
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by (Blast_tac 3);
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(*OR3*)
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by (blast_tac (claset() addSEs spies_partsEs
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                       addIs [impOfSubs analz_subset_parts]) 2);
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(*Fake*) 
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by (spy_analz_tac 1);
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val lemma = result() RS mp RS mp RSN(2,rev_notE);
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goal thy 
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 "!!evs. [| Says Server B                                         \
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\            {|NA, Crypt (shrK A) {|NA, Key K|},                  \
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\                  Crypt (shrK B) {|NB, Key K|}|} : set evs;      \
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\           Says B Spy {|NA, NB, Key K|} ~: set evs;              \
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\           A ~: bad;  B ~: bad;  evs : otway |]                \
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\        ==> Key K ~: analz (spies evs)";
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by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
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by (blast_tac (claset() addSEs [lemma]) 1);
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qed "Spy_not_see_encrypted_key";
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(*** Attempting to prove stronger properties ***)
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(*Only OR1 can have caused such a part of a message to appear.
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  I'm not sure why A ~= B premise is needed: OtwayRees.ML doesn't need it.
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  Perhaps it's because OR2 has two similar-looking encrypted messages in
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        this version.*)
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goal thy 
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 "!!evs. [| A ~: bad;  A ~= B;  evs : otway |]                \
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\        ==> Crypt (shrK A) {|NA, Agent A, Agent B|} : parts (spies evs) --> \
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\            Says A B {|NA, Agent A, Agent B,                  \
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\                       Crypt (shrK A) {|NA, Agent A, Agent B|}|}  : set evs";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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by (Blast_tac 1);
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qed_spec_mp "Crypt_imp_OR1";
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(*Crucial property: If the encrypted message appears, and A has used NA
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  to start a run, then it originated with the Server!*)
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(*Only it is FALSE.  Somebody could make a fake message to Server
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          substituting some other nonce NA' for NB.*)
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goal thy 
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 "!!evs. [| A ~: bad;  A ~= Spy;  evs : otway |]                    \
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\        ==> Crypt (shrK A) {|NA, Key K|} : parts (spies evs) --> \
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\            Says A B {|NA, Agent A, Agent B,                        \
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\                       Crypt (shrK A) {|NA, Agent A, Agent B|}|}    \
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\             : set evs -->                                          \
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\            (EX B NB. Says Server B                                 \
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\                 {|NA,                                              \
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\                   Crypt (shrK A) {|NA, Key K|},                    \
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\                   Crypt (shrK B) {|NB, Key K|}|}  : set evs)";
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by (parts_induct_tac 1);
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by (Fake_parts_insert_tac 1);
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(*OR1: it cannot be a new Nonce, contradiction.*)
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by (blast_tac (claset() addSIs [parts_insertI]
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                       addSEs spies_partsEs) 1);
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(*OR3 and OR4*)
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib])));
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by (ALLGOALS Clarify_tac);
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(*OR4*)
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by (blast_tac (claset() addSIs [Crypt_imp_OR1]
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                       addEs  spies_partsEs) 2);
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(*OR3*)  (** LEVEL 5 **)
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(*The hypotheses at this point suggest an attack in which nonce NB is used
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  in two different roles:
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          Says B' Server
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           {|Nonce NA, Agent Aa, Agent A,
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             Crypt (shrK Aa) {|Nonce NA, Agent Aa, Agent A|}, Nonce NB,
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             Crypt (shrK A) {|Nonce NA, Agent Aa, Agent A|}|}
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          : set evs3;
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          Says A B
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           {|Nonce NB, Agent A, Agent B,
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             Crypt (shrK A) {|Nonce NB, Agent A, Agent B|}|}
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          : set evs3;
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*)
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writeln "GIVE UP! on NA_Crypt_imp_Server_msg";
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(*Thus the key property A_can_trust probably fails too.*)