Many minor speedups:
1. Some use of rewriting with expand_ifs instead of addsplits[expand_if]
2. Faster proof of new_keys_not_used
3. New version of shrK_neq (no longer refers to "range")
(* Title: HOL/Auth/OtwayRees
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "otway" for the Otway-Rees protocol.
Version that encrypts Nonce NB
From page 244 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open OtwayRees;
proof_timing:=true;
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];
goal thy "!!A. [| Key (shrK A) : parts (spies evs); evs : otway |] ==> A:bad";
by (blast_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];
(*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);
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 (ALLGOALS Simp_tac);
by (ALLGOALS Blast_tac);
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 2);
(*Base*)
by (Blast_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 (Best_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 into 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";
(**** Authenticity properties relating to NA ****)
(*Only OR1 can have caused such a part of a message to appear.*)
goal thy
"!!evs. [| A ~: bad; 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);
qed_spec_mp "Crypt_imp_OR1";
(** The Nonce NA uniquely identifies A's message. **)
goal thy
"!!evs. [| evs : otway; A ~: bad |] \
\ ==> EX B'. ALL B. \
\ Crypt (shrK A) {|NA, Agent A, Agent B|} : parts (spies evs) \
\ --> B = B'";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (simp_tac (!simpset addsimps [all_conj_distrib]) 1);
(*OR1: creation of new Nonce. Move assertion into global context*)
by (expand_case_tac "NA = ?y" 1);
by (blast_tac (!claset addSEs spies_partsEs) 1);
val lemma = result();
goal thy
"!!evs.[| Crypt (shrK A) {|NA, Agent A, Agent B|}: parts (spies evs); \
\ Crypt (shrK A) {|NA, Agent A, Agent C|}: parts (spies evs); \
\ evs : otway; A ~: bad |] \
\ ==> B = C";
by (prove_unique_tac lemma 1);
qed "unique_NA";
(*It is impossible to re-use a nonce in both OR1 and OR2. This holds because
OR2 encrypts Nonce NB. It prevents the attack that can occur in the
over-simplified version of this protocol: see OtwayRees_Bad.*)
goal thy
"!!evs. [| A ~: bad; evs : otway |] \
\ ==> Crypt (shrK A) {|NA, Agent A, Agent B|} : parts (spies evs) --> \
\ Crypt (shrK A) {|NA', NA, Agent A', Agent A|} \
\ ~: parts (spies evs)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (REPEAT (blast_tac (!claset addSEs spies_partsEs
addSDs [impOfSubs parts_insert_subset_Un]) 1));
qed_spec_mp"no_nonce_OR1_OR2";
(*Crucial property: If the encrypted message appears, and A has used NA
to start a run, then it originated with the Server!*)
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 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 (asm_simp_tac (!simpset addsimps [ex_disj_distrib]) 1);
by (rtac conjI 1);
by (ALLGOALS Clarify_tac);
(*OR4*)
by (blast_tac (!claset addSIs [Crypt_imp_OR1]
addEs spies_partsEs) 3);
(*OR3, two cases*) (** LEVEL 5 **)
by (blast_tac (!claset addSEs [MPair_parts]
addSDs [Says_imp_spies RS parts.Inj]
addEs [no_nonce_OR1_OR2 RSN (2, rev_notE)]
delrules [conjI] (*stop split-up into 4 subgoals*)) 2);
by (blast_tac (!claset addSDs [Says_imp_spies RS parts.Inj]
addSEs [MPair_parts]
addIs [unique_NA]) 1);
qed_spec_mp "NA_Crypt_imp_Server_msg";
(*Corollary: if A receives B's OR4 message and the nonce NA agrees
then the key really did come from the Server! CANNOT prove this of the
bad form of this protocol, even though we can prove
Spy_not_see_encrypted_key*)
goal thy
"!!evs. [| Says B' A {|NA, Crypt (shrK A) {|NA, Key K|}|} : set evs; \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} : set evs; \
\ A ~: bad; A ~= Spy; evs : otway |] \
\ ==> EX NB. Says Server B \
\ {|NA, \
\ Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} \
\ : set evs";
by (blast_tac (!claset addSIs [NA_Crypt_imp_Server_msg]
addEs spies_partsEs) 1);
qed "A_trusts_OR4";
(** Crucial secrecy property: Spy does not see the keys sent in msg OR3
Does not in itself guarantee security: an attack could violate
the premises, e.g. by having A=Spy **)
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";
(**** Authenticity properties relating to NB ****)
(*Only OR2 can have caused such a part of a message to appear. We do not
know anything about X: it does NOT have to have the right form.*)
goal thy
"!!evs. [| B ~: bad; evs : otway |] \
\ ==> Crypt (shrK B) {|NA, NB, Agent A, Agent B|} \
\ : parts (spies evs) --> \
\ (EX X. Says B Server \
\ {|NA, Agent A, Agent B, X, \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|}|} \
\ : set evs)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (ALLGOALS Blast_tac);
bind_thm ("Crypt_imp_OR2", result() RSN (2,rev_mp) RS exE);
(** The Nonce NB uniquely identifies B's message. **)
goal thy
"!!evs. [| evs : otway; B ~: bad |] \
\ ==> EX NA' A'. ALL NA A. \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|} : parts(spies evs) \
\ --> NA = NA' & A = A'";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (simp_tac (!simpset addsimps [all_conj_distrib]) 1);
(*OR2: creation of new Nonce. Move assertion into global context*)
by (expand_case_tac "NB = ?y" 1);
by (blast_tac (!claset addSEs spies_partsEs) 1);
val lemma = result();
goal thy
"!!evs.[| Crypt (shrK B) {|NA, NB, Agent A, Agent B|} : parts(spies evs); \
\ Crypt (shrK B) {|NC, NB, Agent C, Agent B|} : parts(spies evs); \
\ evs : otway; B ~: bad |] \
\ ==> NC = NA & C = A";
by (prove_unique_tac lemma 1);
qed "unique_NB";
(*If the encrypted message appears, and B has used Nonce NB,
then it originated with the Server!*)
goal thy
"!!evs. [| B ~: bad; B ~= Spy; evs : otway |] \
\ ==> Crypt (shrK B) {|NB, Key K|} : parts (spies evs) \
\ --> (ALL X'. Says B Server \
\ {|NA, Agent A, Agent B, X', \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|}|} \
\ : set evs \
\ --> 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);
(*OR4*)
by (blast_tac (!claset addSEs [MPair_parts, Crypt_imp_OR2]) 2);
(*OR3*)
by (safe_tac (!claset delrules [disjCI, impCE]));
by (blast_tac (!claset delrules [conjI] (*stop split-up*)) 3);
by (blast_tac (!claset addSDs [Says_imp_spies RS parts.Inj]
addSEs [MPair_parts]
addDs [unique_NB]) 2);
by (blast_tac (!claset addSEs [MPair_parts, no_nonce_OR1_OR2 RSN (2, rev_notE)]
addSDs [Says_imp_spies RS parts.Inj]
delrules [conjI, impCE] (*stop split-up*)) 1);
qed_spec_mp "NB_Crypt_imp_Server_msg";
(*Guarantee for B: if it gets a message with matching NB then the Server
has sent the correct message.*)
goal thy
"!!evs. [| B ~: bad; B ~= Spy; evs : otway; \
\ Says S' B {|NA, X, Crypt (shrK B) {|NB, Key K|}|} : set evs; \
\ Says B Server {|NA, Agent A, Agent B, X', \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|} |} \
\ : set evs |] \
\ ==> Says Server B \
\ {|NA, \
\ Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} \
\ : set evs";
by (blast_tac (!claset addSIs [NB_Crypt_imp_Server_msg]
addSEs spies_partsEs) 1);
qed "B_trusts_OR3";
B_trusts_OR3 RS Spy_not_see_encrypted_key;
goal thy
"!!evs. [| B ~: bad; evs : otway |] \
\ ==> Says Server B \
\ {|NA, Crypt (shrK A) {|NA, Key K|}, \
\ Crypt (shrK B) {|NB, Key K|}|} : set evs --> \
\ (EX X. Says B Server {|NA, Agent A, Agent B, X, \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|} |} \
\ : set evs)";
by (etac otway.induct 1);
by (ALLGOALS Asm_simp_tac);
by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj]
addSEs [MPair_parts, Crypt_imp_OR2]) 3);
by (ALLGOALS Blast_tac);
bind_thm ("OR3_imp_OR2", result() RSN (2,rev_mp) RS exE);
(*After getting and checking OR4, agent A can trust that B has been active.
We could probably prove that X has the expected form, but that is not
strictly necessary for authentication.*)
goal thy
"!!evs. [| Says B' A {|NA, Crypt (shrK A) {|NA, Key K|}|} : set evs; \
\ Says A B {|NA, Agent A, Agent B, \
\ Crypt (shrK A) {|NA, Agent A, Agent B|}|} : set evs; \
\ A ~: bad; A ~= Spy; B ~: bad; evs : otway |] \
\ ==> EX NB X. Says B Server {|NA, Agent A, Agent B, X, \
\ Crypt (shrK B) {|NA, NB, Agent A, Agent B|} |}\
\ : set evs";
by (blast_tac (!claset addSDs [A_trusts_OR4]
addSEs [OR3_imp_OR2]) 1);
qed "A_auths_B";