src/HOL/Auth/OtwayRees_Bad.ML

Simplified new_keys_not_seen, etc.: replaced the
union over all
agents by the Spy alone. Proofs run faster and they do not have to be
set up in terms of a previous lemma.

(*  Title:      HOL/Auth/OtwayRees_Bad
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1996  University of Cambridge

Inductive relation "otway" for the Otway-Rees protocol.

The FAULTY version omitting encryption of Nonce NB, as suggested on page 247 of
  Burrows, Abadi and Needham.  A Logic of Authentication.
  Proc. Royal Soc. 426 (1989)

This file illustrates the consequences of such errors.  We can still prove
impressive-looking properties such as Spy_not_see_encrypted_key, yet the
protocol is open to a middleperson attack.  Attempting to prove some key lemmas
indicates the possibility of this attack.
*)

open OtwayRees_Bad;

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 K. EX NA. EX evs: otway.          \
\               Says B A {|Nonce NA, Crypt {|Nonce NA, Key K|} (shrK A)|} \
\                 : set_of_list evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (otway.Nil RS otway.OR1 RS otway.OR2 RS otway.OR3 RS otway.OR4) 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 otway ****)

(*The Spy can see more than anybody else, except for their initial state*)
goal thy 
 "!!evs. evs : otway ==> \
\     sees lost A evs <= initState lost A Un sees lost Spy evs";
by (etac otway.induct 1);
by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD] 
                                addss (!simpset))));
qed "sees_agent_subset_sees_Spy";


(*Nobody sends themselves messages*)
goal thy "!!evs. evs : otway ==> ALL A X. Says A A X ~: set_of_list evs";
by (etac otway.induct 1);
by (Auto_tac());
qed_spec_mp "not_Says_to_self";
Addsimps [not_Says_to_self];
AddSEs   [not_Says_to_self RSN (2, rev_notE)];


(** For reasoning about the encrypted portion of messages **)

goal thy "!!evs. Says A' B {|N, Agent A, Agent B, X|} : set_of_list evs ==> \
\                X : analz (sees lost Spy evs)";
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "OR2_analz_sees_Spy";

goal thy "!!evs. Says S B {|N, X, X'|} : set_of_list evs ==> \
\                X : analz (sees lost Spy evs)";
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "OR4_analz_sees_Spy";

goal thy "!!evs. Says Server B {|NA, X, Crypt {|NB,K|} K'|} : set_of_list evs \
\                 ==> K : parts (sees lost Spy evs)";
by (fast_tac (!claset addSEs partsEs
                      addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
qed "Oops_parts_sees_Spy";

(*OR2_analz... and OR4_analz... let us treat those cases using the same 
  argument as for the Fake case.  This is possible for most, but not all,
  proofs: Fake does not invent new nonces (as in OR2), and of course Fake
  messages originate from the Spy. *)

bind_thm ("OR2_parts_sees_Spy",
          OR2_analz_sees_Spy RS (impOfSubs analz_subset_parts));
bind_thm ("OR4_parts_sees_Spy",
          OR4_analz_sees_Spy RS (impOfSubs analz_subset_parts));

val parts_Fake_tac = 
    forward_tac [OR2_parts_sees_Spy] 4 THEN 
    forward_tac [OR4_parts_sees_Spy] 6 THEN
    forward_tac [Oops_parts_sees_Spy] 7;

(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
fun parts_induct_tac i = SELECT_GOAL
    (DETERM (etac otway.induct 1 THEN parts_Fake_tac THEN
	     (*Fake message*)
	     TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
					   impOfSubs Fake_parts_insert]
                                    addss (!simpset)) 2)) THEN
     (*Base case*)
     fast_tac (!claset addss (!simpset)) 1 THEN
     ALLGOALS Asm_simp_tac) i;


(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
    sends messages containing X! **)

(*Spy never sees another agent's shared key! (unless it's lost at start)*)
goal thy 
 "!!evs. evs : otway \
\        ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
by (parts_induct_tac 1);
by (Auto_tac());
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];

goal thy 
 "!!evs. evs : otway \
\        ==> (Key (shrK A) : analz (sees lost Spy evs)) = (A : lost)";
by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset));
qed "Spy_analz_shrK";
Addsimps [Spy_analz_shrK];

goal thy  "!!A. [| Key (shrK A) : parts (sees lost Spy evs);       \
\                  evs : otway |] ==> A:lost";
by (fast_tac (!claset addDs [Spy_see_shrK]) 1);
qed "Spy_see_shrK_D";

bind_thm ("Spy_analz_shrK_D", analz_subset_parts RS subsetD RS Spy_see_shrK_D);
AddSDs [Spy_see_shrK_D, Spy_analz_shrK_D];


(*** Future keys can't be seen or used! ***)

(*Nobody can have SEEN keys that will be generated in the future. *)
goal thy "!!evs. evs : otway ==>               \
\                length evs <= length evs' --> \
\                Key (newK evs') ~: parts (sees lost Spy evs)";
by (parts_induct_tac 1);
by (REPEAT_FIRST (best_tac (!claset addDs [impOfSubs analz_subset_parts,
					   impOfSubs parts_insert_subset_Un,
					   Suc_leD]
                                addss (!simpset))));
qed_spec_mp "new_keys_not_seen";
Addsimps [new_keys_not_seen];

(*Variant: old messages must contain old keys!*)
goal thy 
 "!!evs. [| Says A B X : set_of_list evs;  \
\           Key (newK evt) : parts {X};    \
\           evs : otway                 \
\        |] ==> length evt < length evs";
by (rtac ccontr 1);
by (dtac leI 1);
by (fast_tac (!claset addSDs [new_keys_not_seen, Says_imp_sees_Spy]
                      addIs  [impOfSubs parts_mono]) 1);
qed "Says_imp_old_keys";


(*** Future nonces can't be seen or used! ***)

goal thy "!!evs. evs : otway ==> \
\                length evs <= length evs' --> \
\                Nonce (newN evs') ~: parts (sees lost Spy evs)";
by (parts_induct_tac 1);
by (REPEAT_FIRST
    (fast_tac (!claset addSEs partsEs
	               addSDs  [Says_imp_sees_Spy RS parts.Inj]
		       addDs  [impOfSubs analz_subset_parts,
			       impOfSubs parts_insert_subset_Un, Suc_leD]
		       addss (!simpset))));
qed_spec_mp "new_nonces_not_seen";
Addsimps [new_nonces_not_seen];

(*Variant: old messages must contain old nonces!*)
goal thy 
 "!!evs. [| Says A B X : set_of_list evs;    \
\           Nonce (newN evt) : parts {X};    \
\           evs : otway                 \
\        |] ==> length evt < length evs";
by (rtac ccontr 1);
by (dtac leI 1);
by (fast_tac (!claset addSDs [new_nonces_not_seen, Says_imp_sees_Spy]
                      addIs  [impOfSubs parts_mono]) 1);
qed "Says_imp_old_nonces";


(*Nobody can have USED keys that will be generated in the future.
  ...very like new_keys_not_seen*)
goal thy "!!evs. evs : otway ==> \
\                length evs <= length evs' --> \
\                newK evs' ~: keysFor (parts (sees lost Spy evs))";
by (parts_induct_tac 1);
(*OR1 and OR3*)
by (EVERY (map (fast_tac (!claset addDs [Suc_leD] addss (!simpset))) [4,2]));
(*Fake, OR2, OR4: these messages send unknown (X) components*)
by (REPEAT
    (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)) 1));
qed_spec_mp "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];



(*** Proofs involving analz ***)

(*Describes the form of K and NA when the Server sends this message.  Also
  for Oops case.*)
goal thy 
 "!!evs. [| Says Server B \
\            {|NA, X, Crypt {|NB, K|} (shrK B)|} : set_of_list evs;  \
\           evs : otway |]                                   \
\        ==> (EX evt: otway. K = Key(newK evt)) &            \
\            (EX i. NA = Nonce i) & (EX j. NB = Nonce j)";
by (etac rev_mp 1);
by (etac otway.induct 1);
by (ALLGOALS (fast_tac (!claset addss (!simpset))));
qed "Says_Server_message_form";


(*For proofs involving analz.*)
val analz_Fake_tac = 
    dtac OR2_analz_sees_Spy 4 THEN 
    dtac OR4_analz_sees_Spy 6 THEN
    forward_tac [Says_Server_message_form] 7 THEN
    assume_tac 7 THEN Full_simp_tac 7 THEN
    REPEAT ((eresolve_tac [bexE, exE, conjE] ORELSE' hyp_subst_tac) 7);


(****
 The following is to prove theorems of the form

          Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) ==>
          Key K : analz (sees lost Spy evs)

 A more general formula must be proved inductively.

****)


(** 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 : otway ==> \
\  ALL K E. (Key K : analz (Key``(newK``E) Un (sees lost Spy evs))) = \
\           (K : newK``E | Key K : analz (sees lost Spy evs))";
by (etac otway.induct 1);
by analz_Fake_tac;
by (REPEAT_FIRST (ares_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 7 **)
(*OR4, OR2, Fake*) 
by (EVERY (map spy_analz_tac [5,3,2]));
(*Oops, OR3, Base*)
by (REPEAT (fast_tac (!claset addIs [image_eqI] addss (!simpset)) 1));
qed_spec_mp "analz_image_newK";


goal thy
 "!!evs. evs : otway ==>                               \
\        Key K : analz (insert (Key (newK evt)) (sees lost Spy evs)) = \
\        (K = newK evt | Key K : analz (sees lost Spy evs))";
by (asm_simp_tac (HOL_ss addsimps [pushKey_newK, analz_image_newK, 
                                   insert_Key_singleton]) 1);
by (Fast_tac 1);
qed "analz_insert_Key_newK";


(*** The Key K uniquely identifies the Server's  message. **)

goal thy 
 "!!evs. evs : otway ==>                                                 \
\   EX B' NA' NB' X'. ALL B NA NB X.                                          \
\     Says Server B {|NA, X, Crypt {|NB, K|} (shrK B)|} : set_of_list evs --> \
\     B=B' & NA=NA' & NB=NB' & X=X'";
by (etac otway.induct 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
by (Step_tac 1);
(*Remaining cases: OR3 and OR4*)
by (ex_strip_tac 2);
by (Fast_tac 2);
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
(*...we assume X is a very new message, and handle this case by contradiction*)
by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
                      delrules [conjI]    (*prevent split-up into 4 subgoals*)
                      addss (!simpset addsimps [parts_insertI])) 1);
val lemma = result();

goal thy 
 "!!evs. [| Says Server B {|NA, X, Crypt {|NB, K|} (shrK B)|}      \
\            : set_of_list evs;                                    \ 
\           Says Server B' {|NA',X',Crypt {|NB',K|} (shrK B')|}    \
\            : set_of_list evs;                                    \
\           evs : otway |] ==> X=X' & B=B' & NA=NA' & NB=NB'";
by (dtac lemma 1);
by (REPEAT (etac exE 1));
(*Duplicate the assumption*)
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
by (fast_tac (!claset addSDs [spec]) 1);
qed "unique_session_keys";


(*Crucial security property, but not itself enough to guarantee correctness!*)
goal thy 
 "!!evs. [| A ~: lost;  B ~: lost;  evs : otway;  evt : otway |]        \
\    ==> Says Server B \
\          {|NA, Crypt {|NA, Key K|} (shrK A), \
\            Crypt {|NB, Key K|} (shrK B)|} : set_of_list evs --> \
\            Says B Spy {|NA, NB, Key K|} ~: set_of_list evs -->           \
\        Key K ~: analz (sees lost Spy evs)";
by (etac otway.induct 1);
by analz_Fake_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])));
(*OR3*)
by (fast_tac (!claset addSIs [parts_insertI]
                      addEs [Says_imp_old_keys RS less_irrefl]
                      addss (!simpset addsimps [parts_insert2])) 3);
(*OR4, OR2, Fake*) 
by (REPEAT_FIRST (resolve_tac [conjI, impI] ORELSE' spy_analz_tac));
(*Oops*) (** LEVEL 5 **)
by (fast_tac (!claset delrules [disjE]
		      addDs [unique_session_keys] addss (!simpset)) 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);


goal thy 
 "!!evs. [| Says Server B                                         \
\            {|NA, Crypt {|NA, K|} (shrK A),                      \
\                  Crypt {|NB, K|} (shrK B)|} : set_of_list evs;  \
\            Says B Spy {|NA, NB, K|} ~: set_of_list evs;         \
\           A ~: lost;  B ~: lost;  evs : otway |]                \
\        ==> K ~: analz (sees lost Spy evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (fast_tac (!claset addSEs [lemma]) 1);
qed "Spy_not_see_encrypted_key";


(*** Attempting to prove stronger properties ***)

(*Only OR1 can have caused such a part of a message to appear.
  I'm not sure why A ~= B premise is needed: OtwayRees.ML doesn't need it.
  Perhaps it's because OR2 has two similar-looking encrypted messages in
	this version.*)
goal thy 
 "!!evs. [| A ~: lost;  A ~= B; evs : otway |]                 \
\        ==> Crypt {|NA, Agent A, Agent B|} (shrK A)           \
\             : parts (sees lost Spy evs) -->                  \
\            Says A B {|NA, Agent A, Agent B,                  \
\                       Crypt {|NA, Agent A, Agent B|} (shrK A)|}  \
\             : set_of_list evs";
by (parts_induct_tac 1);
by (Fast_tac 1);
qed_spec_mp "Crypt_imp_OR1";


(*Crucial property: If the encrypted message appears, and A has used NA
  to start a run, then it originated with the Server!*)
(*Only it is FALSE.  Somebody could make a fake message to Server
          substituting some other nonce NA' for NB.*)
goal thy 
 "!!evs. [| A ~: lost;  A ~= Spy;  evs : otway |]                        \
\        ==> Crypt {|NA, Key K|} (shrK A) : parts (sees lost Spy evs) --> \
\            Says A B {|NA, Agent A, Agent B,                  \
\                       Crypt {|NA, Agent A, Agent B|} (shrK A)|}  \
\             : set_of_list evs -->                            \
\            (EX B NB. Says Server B                           \
\                 {|NA,                                        \
\                   Crypt {|NA, Key K|} (shrK A),              \
\                   Crypt {|NB, Key K|} (shrK B)|}             \
\                   : set_of_list evs)";
by (parts_induct_tac 1);
(*OR1: it cannot be a new Nonce, contradiction.*)
by (fast_tac (!claset addSIs [parts_insertI]
                      addSEs partsEs
                      addEs [Says_imp_old_nonces RS less_irrefl]
                      addss (!simpset)) 1);
(*OR4*)
by (REPEAT (Safe_step_tac 2));
by (REPEAT (best_tac (!claset addSDs [parts_cut]) 3));
by (fast_tac (!claset addSIs [Crypt_imp_OR1]
                      addEs  partsEs
                      addDs [Says_imp_sees_Spy RS parts.Inj]) 2);
(*OR3*)  (** LEVEL 5 **)
by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
by (step_tac (!claset delrules [disjCI, impCE]) 1);
(*The hypotheses at this point suggest an attack in which nonce NA is used
  in two different roles:
          Says B' Server
           {|Nonce NAa, Agent Aa, Agent A,
             Crypt {|Nonce NAa, Agent Aa, Agent A|} (shrK Aa), Nonce NA,
             Crypt {|Nonce NAa, Agent Aa, Agent A|} (shrK A)|}
          : set_of_list evsa;
          Says A B
           {|Nonce NA, Agent A, Agent B,
             Crypt {|Nonce NA, Agent A, Agent B|} (shrK A)|}
          : set_of_list evsa 
*)
writeln "GIVE UP! on NA_Crypt_imp_Server_msg";


(*Thus the key property A_can_trust probably fails too.*)