(* Title: HOL/Auth/Yahalom2
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "yahalom" for the Yahalom protocol, Variant 2.
This version trades encryption of NB for additional explicitness in YM3.
From page 259 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open Yahalom2;
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 X NB K. EX evs: yahalom. \
\ Says A B {|X, Crypt K (Nonce NB)|} : set evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (yahalom.Nil RS yahalom.YM1 RS yahalom.YM2 RS yahalom.YM3 RS
yahalom.YM4) 2);
by possibility_tac;
result();
(**** Inductive proofs about yahalom ****)
(*Nobody sends themselves messages*)
goal thy "!!evs. evs: yahalom ==> ALL A X. Says A A X ~: set evs";
by (etac yahalom.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 **)
(*Lets us treat YM4 using a similar argument as for the Fake case.*)
goal thy "!!evs. Says S A {|NB, Crypt (shrK A) Y, X|} : set evs ==> \
\ X : analz (sees Spy evs)";
by (blast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "YM4_analz_sees_Spy";
bind_thm ("YM4_parts_sees_Spy",
YM4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
(*Relates to both YM4 and Oops*)
goal thy "!!evs. Says S A {|NB, Crypt (shrK A) {|B,K,NA|}, X|} : set evs ==> \
\ K : parts (sees Spy evs)";
by (blast_tac (!claset addSEs partsEs
addSDs [Says_imp_sees_Spy RS parts.Inj]) 1);
qed "YM4_Key_parts_sees_Spy";
(*For proving the easier theorems about X ~: parts (sees Spy evs).*)
fun parts_sees_tac i =
forward_tac [YM4_Key_parts_sees_Spy] (i+6) THEN
forward_tac [YM4_parts_sees_Spy] (i+5) THEN
prove_simple_subgoals_tac i;
(*Induction for regularity theorems. If induction formula has the form
X ~: analz (sees Spy evs) --> ... then it shortens the proof by discarding
needless information about analz (insert X (sees Spy evs)) *)
fun parts_induct_tac i =
etac yahalom.induct i
THEN
REPEAT (FIRSTGOAL analz_mono_contra_tac)
THEN parts_sees_tac i;
(** Theorems of the form X ~: parts (sees 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 : yahalom ==> (Key (shrK A) : parts (sees Spy evs)) = (A : lost)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (Blast_tac 1);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : yahalom ==> (Key (shrK A) : analz (sees 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 Spy evs); \
\ evs : yahalom |] ==> A:lost";
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! Needed to apply analz_insert_Key*)
goal thy "!!evs. evs : yahalom ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (sees Spy evs))";
by (parts_induct_tac 1);
(*YM4: Key K is not fresh!*)
by (blast_tac (!claset addSEs sees_Spy_partsEs) 3);
(*YM3*)
by (blast_tac (!claset addss (!simpset)) 2);
(*Fake*)
by (best_tac
(!claset addIs [impOfSubs analz_subset_parts]
addDs [impOfSubs (analz_subset_parts RS keysFor_mono),
impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
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];
(*Describes the form of K when the Server sends this message. Useful for
Oops as well as main secrecy property.*)
goal thy
"!!evs. [| Says Server A {|nb', Crypt (shrK A) {|Agent B, Key K, na|}, X|} \
\ : set evs; \
\ evs : yahalom |] \
\ ==> K ~: range shrK & A ~= B";
by (etac rev_mp 1);
by (etac yahalom.induct 1);
by (ALLGOALS Asm_simp_tac);
qed "Says_Server_message_form";
(*For proofs involving analz.*)
val analz_sees_tac =
dtac YM4_analz_sees_Spy 6 THEN
forward_tac [Says_Server_message_form] 7 THEN
assume_tac 7 THEN
REPEAT ((etac conjE ORELSE' hyp_subst_tac) 7);
(****
The following is to prove theorems of the form
Key K : analz (insert (Key KAB) (sees Spy evs)) ==>
Key K : analz (sees Spy evs)
A more general formula must be proved inductively.
****)
(** Session keys are not used to encrypt other session keys **)
goal thy
"!!evs. evs : yahalom ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (sees Spy evs))) = \
\ (K : KK | Key K : analz (sees Spy evs))";
by (etac yahalom.induct 1);
by analz_sees_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 : yahalom; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (sees Spy evs)) = \
\ (K = KAB | Key K : analz (sees 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 thy
"!!evs. evs : yahalom ==> \
\ EX A' B' na' nb' X'. ALL A B na nb X. \
\ Says Server A \
\ {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, X|} \
\ : set evs --> A=A' & B=B' & na=na' & nb=nb' & X=X'";
by (etac yahalom.induct 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
by (Step_tac 1);
(*Remaining case: YM3*)
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 sees_Spy_partsEs
delrules [conjI] (*prevent split-up into 4 subgoals*)
addss (!simpset addsimps [parts_insertI])) 1);
val lemma = result();
goal thy
"!!evs. [| Says Server A \
\ {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, X|} \
\ : set evs; \
\ Says Server A' \
\ {|nb', Crypt (shrK A') {|Agent B', Key K, na'|}, X'|} \
\ : set evs; \
\ evs : yahalom |] \
\ ==> A=A' & B=B' & na=na' & nb=nb'";
by (prove_unique_tac lemma 1);
qed "unique_session_keys";
(** Crucial secrecy property: Spy does not see the keys sent in msg YM3 **)
goal thy
"!!evs. [| A ~: lost; B ~: lost; A ~= B; \
\ evs : yahalom |] \
\ ==> Says Server A \
\ {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, \
\ Crypt (shrK B) {|nb, Key K, Agent A|}|} \
\ : set evs --> \
\ Says A Spy {|na, nb, Key K|} ~: set evs --> \
\ Key K ~: analz (sees Spy evs)";
by (etac yahalom.induct 1);
by analz_sees_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps [analz_insert_eq, analz_insert_freshK]
setloop split_tac [expand_if])));
(*Oops*)
by (blast_tac (!claset addDs [unique_session_keys]) 3);
(*YM3*)
by (blast_tac (!claset delrules [impCE]
addSEs sees_Spy_partsEs
addIs [impOfSubs analz_subset_parts]) 2);
(*Fake*)
by (spy_analz_tac 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);
(*Final version*)
goal thy
"!!evs. [| Says Server A \
\ {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, \
\ Crypt (shrK B) {|nb, Key K, Agent A|}|} \
\ : set evs; \
\ Says A Spy {|na, nb, Key K|} ~: set evs; \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> Key K ~: analz (sees Spy 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";
(** Security Guarantee for A upon receiving YM3 **)
(*If the encrypted message appears then it originated with the Server.
May now apply Spy_not_see_encrypted_key, subject to its conditions.*)
goal thy
"!!evs. [| Crypt (shrK A) {|Agent B, Key K, na|} \
\ : parts (sees Spy evs); \
\ A ~: lost; evs : yahalom |] \
\ ==> EX nb. Says Server A \
\ {|nb, Crypt (shrK A) {|Agent B, Key K, na|}, \
\ Crypt (shrK B) {|nb, Key K, Agent A|}|} \
\ : set evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
by (Blast_tac 1);
qed "A_trusts_YM3";
(** Security Guarantee for B upon receiving YM4 **)
(*B knows, by the first part of A's message, that the Server distributed
the key for A and B, and has associated it with NB. *)
goal thy
"!!evs. [| Crypt (shrK B) {|Nonce NB, Key K, Agent A|} \
\ : parts (sees Spy evs); \
\ B ~: lost; evs : yahalom |] \
\ ==> EX NA. Says Server A \
\ {|Nonce NB, \
\ Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, \
\ Crypt (shrK B) {|Nonce NB, Key K, Agent A|}|} \
\ : set evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
(*YM3*)
by (Blast_tac 1);
qed "B_trusts_YM4_shrK";
(*With this protocol variant, we don't need the 2nd part of YM4 at all:
Nonce NB is available in the first part.*)
(*What can B deduce from receipt of YM4? Stronger and simpler than Yahalom
because we do not have to show that NB is secret. *)
goal thy
"!!evs. [| Says A' B {|Crypt (shrK B) {|Nonce NB, Key K, Agent A|}, X|} \
\ : set evs; \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> EX NA. Says Server A \
\ {|Nonce NB, \
\ Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, \
\ Crypt (shrK B) {|Nonce NB, Key K, Agent A|}|} \
\ : set evs";
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1);
by (blast_tac (!claset addSDs [B_trusts_YM4_shrK]) 1);
qed "B_trusts_YM4";
(*** Authenticating B to A ***)
(*The encryption in message YM2 tells us it cannot be faked.*)
goal thy
"!!evs. evs : yahalom \
\ ==> Crypt (shrK B) {|Agent A, Nonce NA|} : parts (sees Spy evs) --> \
\ B ~: lost --> \
\ (EX NB. Says B Server {|Agent B, Nonce NB, \
\ Crypt (shrK B) {|Agent A, Nonce NA|}|} \
\ : set evs)";
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
(*YM2*)
by (Blast_tac 1);
bind_thm ("B_Said_YM2", result() RSN (2, rev_mp) RS mp);
(*If the server sends YM3 then B sent YM2, perhaps with a different NB*)
goal thy
"!!evs. evs : yahalom \
\ ==> Says Server A {|nb, Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, X|} \
\ : set evs --> \
\ B ~: lost --> \
\ (EX nb'. Says B Server {|Agent B, nb', \
\ Crypt (shrK B) {|Agent A, Nonce NA|}|} \
\ : set evs)";
by (etac yahalom.induct 1);
by (ALLGOALS Asm_simp_tac);
(*YM3*)
by (blast_tac (!claset addSDs [B_Said_YM2]
addSEs [MPair_parts]
addDs [Says_imp_sees_Spy RS parts.Inj]) 3);
(*Fake, YM2*)
by (ALLGOALS Blast_tac);
val lemma = result() RSN (2, rev_mp) RS mp |> standard;
(*If A receives YM3 then B has used nonce NA (and therefore is alive)*)
goal thy
"!!evs. [| Says S A {|nb, Crypt (shrK A) {|Agent B, Key K, Nonce NA|}, X|} \
\ : set evs; \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> EX nb'. Says B Server \
\ {|Agent B, nb', Crypt (shrK B) {|Agent A, Nonce NA|}|} \
\ : set evs";
by (blast_tac (!claset addSDs [A_trusts_YM3, lemma]
addEs sees_Spy_partsEs) 1);
qed "YM3_auth_B_to_A";
(*** Authenticating A to B using the certificate Crypt K (Nonce NB) ***)
(*Assuming the session key is secure, if both certificates are present then
A has said NB. We can't be sure about the rest of A's message, but only
NB matters for freshness.*)
goal thy
"!!evs. evs : yahalom \
\ ==> Key K ~: analz (sees Spy evs) --> \
\ Crypt K (Nonce NB) : parts (sees Spy evs) --> \
\ Crypt (shrK B) {|Nonce NB, Key K, Agent A|} \
\ : parts (sees Spy evs) --> \
\ B ~: lost --> \
\ (EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs)";
by (parts_induct_tac 1);
(*Fake*)
by (Fake_parts_insert_tac 1);
(*YM3: by new_keys_not_used we note that Crypt K (Nonce NB) could not exist*)
by (fast_tac (!claset addSDs [Crypt_imp_invKey_keysFor] addss (!simpset)) 1);
(*YM4: was Crypt K (Nonce NB) the very last message? If not, use ind. hyp.*)
by (asm_simp_tac (!simpset addsimps [ex_disj_distrib]) 1);
(*yes: apply unicity of session keys*)
by (not_lost_tac "Aa" 1);
by (blast_tac (!claset addSEs [MPair_parts]
addSDs [A_trusts_YM3, B_trusts_YM4_shrK]
addDs [Says_imp_sees_Spy RS parts.Inj,
unique_session_keys]) 1);
val lemma = normalize_thm [RSspec, RSmp] (result()) |> standard;
(*If B receives YM4 then A has used nonce NB (and therefore is alive).
Moreover, A associates K with NB (thus is talking about the same run).
Other premises guarantee secrecy of K.*)
goal thy
"!!evs. [| Says A' B {|Crypt (shrK B) {|Nonce NB, Key K, Agent A|}, \
\ Crypt K (Nonce NB)|} : set evs; \
\ (ALL NA. Says A Spy {|Nonce NA, Nonce NB, Key K|} ~: set evs); \
\ A ~: lost; B ~: lost; evs : yahalom |] \
\ ==> EX X. Says A B {|X, Crypt K (Nonce NB)|} : set evs";
by (etac (Says_imp_sees_Spy RS parts.Inj RS MPair_parts) 1);
by (dtac B_trusts_YM4_shrK 1);
by (safe_tac (!claset));
by (rtac lemma 1);
by (rtac Spy_not_see_encrypted_key 2);
by (REPEAT_FIRST assume_tac);
by (ALLGOALS (blast_tac (!claset addSEs [MPair_parts]
addDs [Says_imp_sees_Spy RS parts.Inj])));
qed_spec_mp "YM4_imp_A_Said_YM3";