(* Title: HOL/Auth/OtwayRees_AN
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "otway" for the Otway-Rees protocol.
Abadi-Needham version: minimal encryption, explicit messages
From page 11 of
Abadi and Needham. Prudent Engineering Practice for Cryptographic Protocols.
IEEE Trans. SE 22 (1), 1996
*)
AddEs knows_Spy_partsEs;
AddDs [impOfSubs analz_subset_parts];
AddDs [impOfSubs Fake_parts_insert];
(*A "possibility property": there are traces that reach the end*)
Goal "[| B ~= Server |] \
\ ==> EX K. EX NA. EX evs: otway. \
\ Says B A (Crypt (shrK A) {|Nonce NA, Agent A, Agent B, Key K|}) \
\ : set evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (otway.Nil RS
otway.OR1 RS otway.Reception RS
otway.OR2 RS otway.Reception RS
otway.OR3 RS otway.Reception RS otway.OR4) 2);
by possibility_tac;
result();
Goal "[| Gets B X : set evs; evs : otway |] ==> EX A. Says A B X : set evs";
by (etac rev_mp 1);
by (etac otway.induct 1);
by Auto_tac;
qed"Gets_imp_Says";
(*Must be proved separately for each protocol*)
Goal "[| Gets B X : set evs; evs : otway |] ==> X : knows Spy evs";
by (blast_tac (claset() addSDs [Gets_imp_Says, Says_imp_knows_Spy]) 1);
qed"Gets_imp_knows_Spy";
AddDs [Gets_imp_knows_Spy RS parts.Inj];
(**** Inductive proofs about otway ****)
(** For reasoning about the encrypted portion of messages **)
Goal "[| Gets B {|X, Crypt(shrK B) X'|} : set evs; evs : otway |] ==> \
\ X : analz (knows Spy evs)";
by (blast_tac (claset() addSDs [Gets_imp_knows_Spy RS analz.Inj]) 1);
qed "OR4_analz_knows_Spy";
Goal "Says Server B {|X, Crypt K' {|NB, a, Agent B, K|}|} : set evs \
\ ==> K : parts (knows Spy evs)";
by (Blast_tac 1);
qed "Oops_parts_knows_Spy";
bind_thm ("OR4_parts_knows_Spy",
OR4_analz_knows_Spy RS (impOfSubs analz_subset_parts));
(*For proving the easier theorems about X ~: parts (knows Spy evs).*)
fun parts_induct_tac i =
etac otway.induct i THEN
ftac Oops_parts_knows_Spy (i+7) THEN
ftac OR4_parts_knows_Spy (i+6) THEN
prove_simple_subgoals_tac i;
(** Theorems of the form X ~: parts (knows Spy evs) imply that NOBODY
sends messages containing X! **)
(*Spy never sees a good agent's shared key!*)
Goal "evs : otway ==> (Key (shrK A) : parts (knows Spy evs)) = (A : bad)";
by (parts_induct_tac 1);
by (ALLGOALS Blast_tac);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
Goal "evs : otway ==> (Key (shrK A) : analz (knows Spy 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 "evs : otway ==> Key K ~: used evs --> K ~: keysFor (parts (knows Spy evs))";
by (parts_induct_tac 1);
(*Fake*)
by (blast_tac (claset() addSDs [keysFor_parts_insert]) 1);
(*OR3*)
by (Blast_tac 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.*)
Goal "[| Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, 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 Asm_simp_tac);
by (Blast_tac 1);
qed "Says_Server_message_form";
(*For proofs involving analz.*)
val analz_knows_Spy_tac =
dtac OR4_analz_knows_Spy 7 THEN assume_tac 7 THEN
ftac Says_Server_message_form 8 THEN assume_tac 8 THEN
REPEAT ((eresolve_tac [exE, conjE] ORELSE' hyp_subst_tac) 8);
(****
The following is to prove theorems of the form
Key K : analz (insert (Key KAB) (knows Spy evs)) ==>
Key K : analz (knows Spy 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 "evs : otway ==> \
\ ALL K KK. KK <= -(range shrK) --> \
\ (Key K : analz (Key`KK Un (knows Spy evs))) = \
\ (K : KK | Key K : analz (knows Spy evs))";
by (etac otway.induct 1);
by analz_knows_Spy_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 "[| evs : otway; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (knows Spy evs)) = \
\ (K = KAB | Key K : analz (knows Spy 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 "[| Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, K|}|} \
\ : set evs; \
\ Says Server B' \
\ {|Crypt (shrK A') {|NA', Agent A', Agent B', K|}, \
\ Crypt (shrK B') {|NB', Agent A', Agent B', K|}|} \
\ : set evs; \
\ evs : otway |] \
\ ==> A=A' & B=B' & NA=NA' & NB=NB'";
by (etac rev_mp 1);
by (etac rev_mp 1);
by (etac otway.induct 1);
by (ALLGOALS Asm_simp_tac);
(*Remaining cases: OR3 and OR4*)
by (REPEAT (Blast_tac 1));
qed "unique_session_keys";
(**** Authenticity properties relating to NA ****)
(*If the encrypted message appears then it originated with the Server!*)
Goal "[| A ~: bad; A ~= B; evs : otway |] \
\ ==> Crypt (shrK A) {|NA, Agent A, Agent B, Key K|} : parts (knows Spy evs) \
\ --> (EX NB. Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs)";
by (parts_induct_tac 1);
by (Blast_tac 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib])));
(*OR3*)
by (Blast_tac 1);
qed_spec_mp "NA_Crypt_imp_Server_msg";
(*Corollary: if A receives B's OR4 message then it originated with the Server.
Freshness may be inferred from nonce NA.*)
Goal "[| Gets A (Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}) \
\ : set evs; \
\ A ~: bad; A ~= B; evs : otway |] \
\ ==> EX NB. Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs";
by (blast_tac (claset() addSIs [NA_Crypt_imp_Server_msg]) 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 "[| A ~: bad; B ~: bad; evs : otway |] \
\ ==> Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs --> \
\ Notes Spy {|NA, NB, Key K|} ~: set evs --> \
\ Key K ~: analz (knows Spy evs)";
by (etac otway.induct 1);
by analz_knows_Spy_tac;
by (ALLGOALS
(asm_simp_tac (simpset() addcongs [conj_cong]
addsimps [analz_insert_eq, analz_insert_freshK]
@ pushes @ split_ifs)));
(*Oops*)
by (blast_tac (claset() addSDs [unique_session_keys]) 4);
(*OR4*)
by (Blast_tac 3);
(*OR3*)
by (Blast_tac 2);
(*Fake*)
by (spy_analz_tac 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);
Goal "[| Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs; \
\ Notes Spy {|NA, NB, Key K|} ~: set evs; \
\ A ~: bad; B ~: bad; evs : otway |] \
\ ==> Key K ~: analz (knows Spy evs)";
by (ftac Says_Server_message_form 1 THEN assume_tac 1);
by (blast_tac (claset() addSEs [lemma]) 1);
qed "Spy_not_see_encrypted_key";
(*A's guarantee. The Oops premise quantifies over NB because A cannot know
what it is.*)
Goal "[| Gets A (Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}) \
\ : set evs; \
\ ALL NB. Notes Spy {|NA, NB, Key K|} ~: set evs; \
\ A ~: bad; B ~: bad; A ~= B; evs : otway |] \
\ ==> Key K ~: analz (knows Spy evs)";
by (blast_tac (claset() addSDs [A_trusts_OR4, Spy_not_see_encrypted_key]) 1);
qed "A_gets_good_key";
(**** Authenticity properties relating to NB ****)
(*If the encrypted message appears then it originated with the Server!*)
Goal "[| B ~: bad; A ~= B; evs : otway |] \
\ ==> Crypt (shrK B) {|NB, Agent A, Agent B, Key K|} : parts (knows Spy evs) \
\ --> (EX NA. Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs)";
by (parts_induct_tac 1);
by (Blast_tac 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps [ex_disj_distrib])));
(*OR3*)
by (Blast_tac 1);
qed_spec_mp "NB_Crypt_imp_Server_msg";
(*Guarantee for B: if it gets a well-formed certificate then the Server
has sent the correct message in round 3.*)
Goal "[| Gets B {|X, Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs; \
\ B ~: bad; A ~= B; evs : otway |] \
\ ==> EX NA. Says Server B \
\ {|Crypt (shrK A) {|NA, Agent A, Agent B, Key K|}, \
\ Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs";
by (blast_tac (claset() addSIs [NB_Crypt_imp_Server_msg]) 1);
qed "B_trusts_OR3";
(*The obvious combination of B_trusts_OR3 with Spy_not_see_encrypted_key*)
Goal "[| Gets B {|X, Crypt (shrK B) {|NB, Agent A, Agent B, Key K|}|} \
\ : set evs; \
\ ALL NA. Notes Spy {|NA, NB, Key K|} ~: set evs; \
\ A ~: bad; B ~: bad; A ~= B; evs : otway |] \
\ ==> Key K ~: analz (knows Spy evs)";
by (blast_tac (claset() addDs [B_trusts_OR3, Spy_not_see_encrypted_key]) 1);
qed "B_gets_good_key";