(* 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;
set proof_timing;
HOL_quantifiers := false;
(*A "possibility property": 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 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 possibility_tac;
result();
(**** Inductive proofs about otway ****)
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : otway ==> ALL A X. Says A A X ~: set 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 evs ==> \
\ X : analz (spies evs)";
by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1);
qed "OR2_analz_spies";
goal thy "!!evs. Says S' B {|N, X, Crypt (shrK B) X'|} : set evs ==> \
\ X : analz (spies evs)";
by (blast_tac (claset() addSDs [Says_imp_spies RS analz.Inj]) 1);
qed "OR4_analz_spies";
goal thy "!!evs. Says Server B {|NA, X, Crypt K' {|NB,K|}|} : set evs \
\ ==> K : parts (spies evs)";
by (blast_tac (claset() addSEs spies_partsEs) 1);
qed "Oops_parts_spies";
(*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_spies",
OR2_analz_spies RS (impOfSubs analz_subset_parts));
bind_thm ("OR4_parts_spies",
OR4_analz_spies RS (impOfSubs analz_subset_parts));
(*For proving the easier theorems about X ~: parts (spies evs).*)
fun parts_induct_tac i =
etac otway.induct i THEN
forward_tac [Oops_parts_spies] (i+6) THEN
forward_tac [OR4_parts_spies] (i+5) THEN
forward_tac [OR2_parts_spies] (i+3) THEN
prove_simple_subgoals_tac i;
(** Theorems of the form X ~: parts (spies evs) imply that NOBODY
sends messages containing X! **)
(*Spy never sees another agent's shared key! (unless it's bad at start)*)
goal thy
"!!evs. evs : otway ==> (Key (shrK A) : parts (spies evs)) = (A : bad)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (ALLGOALS Blast_tac);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : otway ==> (Key (shrK A) : analz (spies evs)) = (A : bad)";
by (auto_tac(claset() addDs [impOfSubs analz_subset_parts], simpset()));
qed "Spy_analz_shrK";
Addsimps [Spy_analz_shrK];
AddSDs [Spy_see_shrK RSN (2, rev_iffD1),
Spy_analz_shrK RSN (2, rev_iffD1)];
(*Nobody can have used non-existent keys!*)
goal thy "!!evs. evs : otway ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (spies evs))";
by (parts_induct_tac 1);
(*Fake*)
by (best_tac
(claset() addSDs [impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
addIs [impOfSubs analz_subset_parts]
addDs [impOfSubs (analz_subset_parts RS keysFor_mono)]
addss (simpset())) 1);
(*OR1-3*)
by (ALLGOALS Blast_tac);
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, Key K|}|} : set evs; \
\ evs : otway |] \
\ ==> K ~: range shrK & (EX i. NA = Nonce i) & (EX j. NB = Nonce j)";
by (etac rev_mp 1);
by (etac otway.induct 1);
by (prove_simple_subgoals_tac 1);
by (Blast_tac 1);
qed "Says_Server_message_form";
(*For proofs involving analz.*)
val analz_spies_tac =
dtac OR2_analz_spies 4 THEN
dtac OR4_analz_spies 6 THEN
forward_tac [Says_Server_message_form] 7 THEN assume_tac 7 THEN
REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 7);
(****
The following is to prove theorems of the form
Key K : analz (insert (Key KAB) (spies evs)) ==>
Key K : analz (spies evs)
A more general formula must be proved inductively.
****)
(** Session keys are not used to encrypt other session keys **)
(*The equality makes the induction hypothesis easier to apply*)
goal thy
"!!evs. evs : otway ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (spies evs))) = \
\ (K : KK | Key K : analz (spies evs))";
by (etac otway.induct 1);
by analz_spies_tac;
by (REPEAT_FIRST (resolve_tac [allI, impI]));
by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
(*Fake*)
by (spy_analz_tac 1);
qed_spec_mp "analz_image_freshK";
goal thy
"!!evs. [| evs : otway; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (spies evs)) = \
\ (K = KAB | Key K : analz (spies evs))";
by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
qed "analz_insert_freshK";
(*** 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 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 (ALLGOALS Clarify_tac);
(*Remaining cases: OR3 and OR4*)
by (ex_strip_tac 2);
by (Blast_tac 2);
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
(*...we assume X is a recent message, and handle this case by contradiction*)
by (blast_tac (claset() addSEs spies_partsEs
delrules [conjI] (*no split-up to 4 subgoals*)) 1);
val lemma = result();
goal thy
"!!evs. [| Says Server B {|NA, X, Crypt (shrK B) {|NB, K|}|} : set evs; \
\ Says Server B' {|NA',X',Crypt (shrK B') {|NB',K|}|} : set evs; \
\ evs : otway |] ==> X=X' & B=B' & NA=NA' & NB=NB'";
by (prove_unique_tac lemma 1);
qed "unique_session_keys";
(*Crucial security property, but not itself enough to guarantee correctness!*)
goal thy
"!!evs. [| A ~: bad; B ~: bad; evs : otway |] \
\ ==> Says Server B \
\ {|NA, Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} : set evs --> \
\ Says B Spy {|NA, NB, Key K|} ~: set evs --> \
\ Key K ~: analz (spies evs)";
by (etac otway.induct 1);
by analz_spies_tac;
by (ALLGOALS
(asm_simp_tac (simpset() addcongs [conj_cong]
addsimps [analz_insert_eq, analz_insert_freshK]
addsimps (pushes@expand_ifs))));
(*Oops*)
by (blast_tac (claset() addSDs [unique_session_keys]) 4);
(*OR4*)
by (Blast_tac 3);
(*OR3*)
by (blast_tac (claset() addSEs spies_partsEs
addIs [impOfSubs analz_subset_parts]) 2);
(*Fake*)
by (spy_analz_tac 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);
goal thy
"!!evs. [| Says Server B \
\ {|NA, Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} : set evs; \
\ Says B Spy {|NA, NB, Key K|} ~: set evs; \
\ A ~: bad; B ~: bad; evs : otway |] \
\ ==> Key K ~: analz (spies evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (blast_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 ~: bad; A ~= B; evs : otway |] \
\ ==> Crypt (shrK A) {|NA, Agent A, Agent B|} : parts (spies evs) --> \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} : set evs";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (Blast_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 ~: bad; A ~= Spy; evs : otway |] \
\ ==> Crypt (shrK A) {|NA, Key K|} : parts (spies evs) --> \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} \
\ : set evs --> \
\ (EX B NB. Says Server B \
\ {|NA, \
\ Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} : set evs)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
(*OR1: it cannot be a new Nonce, contradiction.*)
by (blast_tac (claset() addSIs [parts_insertI]
addSEs spies_partsEs) 1);
(*OR3 and OR4*)
by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib])));
by (ALLGOALS Clarify_tac);
(*OR4*)
by (blast_tac (claset() addSIs [Crypt_imp_OR1]
addEs spies_partsEs) 2);
(*OR3*) (** LEVEL 5 **)
(*The hypotheses at this point suggest an attack in which nonce NB is used
in two different roles:
Says B' Server
{|Nonce NA, Agent Aa, Agent A,
Crypt (shrK Aa) {|Nonce NA, Agent Aa, Agent A|}, Nonce NB,
Crypt (shrK A) {|Nonce NA, Agent Aa, Agent A|}|}
: set evs3;
Says A B
{|Nonce NB, Agent A, Agent B,
Crypt (shrK A) {|Nonce NB, Agent A, Agent B|}|}
: set evs3;
*)
writeln "GIVE UP! on NA_Crypt_imp_Server_msg";
(*Thus the key property A_can_trust probably fails too.*)