(* 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 (shrK A) {|Nonce NA, Key K|}|} \
\ : 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 K' {|NB,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 addEs [leD RS notE]
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]
addEs [leD RS notE]
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 (shrK B) {|NB, K|}|} : 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 (shrK B) {|NB, K|}|} : 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 (shrK B) {|NB, K|}|} \
\ : set_of_list evs; \
\ Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} \
\ : 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 |] \
\ ==> Says Server B \
\ {|NA, Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} : 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_simp_tac (!simpset addsimps ([not_parts_not_analz,
analz_insert_Key_newK] @ pushes)
setloop split_tac [expand_if])));
(*OR3*)
by (fast_tac (!claset addEs [Says_imp_old_keys RS less_irrefl]
addss (!simpset addsimps [parts_insert2])) 3);
(*OR4, OR2, Fake*)
by (REPEAT_FIRST 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 (shrK A) {|NA, K|}, \
\ Crypt (shrK B) {|NB, K|}|} : 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 (shrK A) {|NA, Agent A, Agent B|} \
\ : parts (sees lost Spy evs) --> \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} \
\ : 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 (shrK A) {|NA, Key K|} : parts (sees lost Spy evs) --> \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} \
\ : set_of_list evs --> \
\ (EX B NB. Says Server B \
\ {|NA, \
\ Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} \
\ : 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 (shrK Aa) {|Nonce NAa, Agent Aa, Agent A|}, Nonce NA,
Crypt (shrK A) {|Nonce NAa, Agent Aa, Agent A|}|}
: set_of_list evsa;
Says A B
{|Nonce NA, Agent A, Agent B,
Crypt (shrK A) {|Nonce NA, Agent A, Agent B|}|}
: set_of_list evsa
*)
writeln "GIVE UP! on NA_Crypt_imp_Server_msg";
(*Thus the key property A_can_trust probably fails too.*)