(* Title: HOL/Auth/NS_Shared
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "ns_shared" for Needham-Schroeder Shared-Key protocol.
From page 247 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open NS_Shared;
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 N K. EX evs: ns_shared. \
\ Says A B (Crypt K {|Nonce N, Nonce N|}) : set evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (ns_shared.Nil RS ns_shared.NS1 RS ns_shared.NS2 RS
ns_shared.NS3 RS ns_shared.NS4 RS ns_shared.NS5) 2);
by possibility_tac;
result();
goal thy
"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \
\ ==> EX evs: ns_shared. \
\ Says A B (Crypt ?K {|Nonce ?N, Nonce ?N|}) : set evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (ns_shared.Nil RS ns_shared.NS1 RS ns_shared.NS2 RS
ns_shared.NS3 RS ns_shared.NS4 RS ns_shared.NS5) 2);
by possibility_tac;
(**** Inductive proofs about ns_shared ****)
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : ns_shared ==> ALL A X. Says A A X ~: set evs";
by (etac ns_shared.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 message NS3*)
goal thy "!!evs. Says S A (Crypt KA {|N, B, K, X|}) : set evs \
\ ==> X : parts (spies evs)";
by (blast_tac (!claset addSEs spies_partsEs) 1);
qed "NS3_msg_in_parts_spies";
goal thy
"!!evs. Says Server A (Crypt (shrK A) {|NA, B, K, X|}) : set evs \
\ ==> K : parts (spies evs)";
by (blast_tac (!claset addSEs spies_partsEs) 1);
qed "Oops_parts_spies";
(*For proving the easier theorems about X ~: parts (spies evs).*)
fun parts_induct_tac i =
etac ns_shared.induct i THEN
forward_tac [Oops_parts_spies] (i+7) THEN
dtac NS3_msg_in_parts_spies (i+4) 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 : ns_shared ==> (Key (shrK A) : parts (spies evs)) = (A : bad)";
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 : ns_shared ==> (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 : ns_shared |] ==> 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 : ns_shared ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (spies evs))";
by (parts_induct_tac 1);
(*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);
(*NS2, NS4, NS5*)
by (ALLGOALS (blast_tac (!claset addSEs spies_partsEs)));
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];
(** Lemmas concerning the form of items passed in messages **)
(*Describes the form of K, X and K' when the Server sends this message.*)
goal thy
"!!evs. [| Says Server A (Crypt K' {|N, Agent B, Key K, X|}) : set evs; \
\ evs : ns_shared |] \
\ ==> K ~: range shrK & \
\ X = (Crypt (shrK B) {|Key K, Agent A|}) & \
\ K' = shrK A";
by (etac rev_mp 1);
by (etac ns_shared.induct 1);
by (Auto_tac());
qed "Says_Server_message_form";
(*If the encrypted message appears then it originated with the Server*)
goal thy
"!!evs. [| Crypt (shrK A) {|NA, Agent B, Key K, X|} : parts (spies evs); \
\ A ~: bad; evs : ns_shared |] \
\ ==> Says Server A (Crypt (shrK A) {|NA, Agent B, Key K, X|}) \
\ : set evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
qed "A_trusts_NS2";
(*EITHER describes the form of X when the following message is sent,
OR reduces it to the Fake case.
Use Says_Server_message_form if applicable.*)
goal thy
"!!evs. [| Says S A (Crypt (shrK A) {|Nonce NA, Agent B, Key K, X|}) \
\ : set evs; \
\ evs : ns_shared |] \
\ ==> (K ~: range shrK & X = (Crypt (shrK B) {|Key K, Agent A|})) \
\ | X : analz (spies evs)";
by (case_tac "A : bad" 1);
by (fast_tac (!claset addSDs [Says_imp_spies RS analz.Inj]
addss (!simpset)) 1);
by (forward_tac [Says_imp_spies RS parts.Inj] 1);
by (blast_tac (!claset addSDs [A_trusts_NS2, Says_Server_message_form]) 1);
qed "Says_S_message_form";
(*For proofs involving analz.*)
val analz_spies_tac =
forward_tac [Says_Server_message_form] 8 THEN
forward_tac [Says_S_message_form] 5 THEN
REPEAT_FIRST (eresolve_tac [asm_rl, conjE, disjE] ORELSE' hyp_subst_tac);
(****
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.
****)
(*NOT useful in this form, but it says that session keys are not used
to encrypt messages containing other keys, in the actual protocol.
We require that agents should behave like this subsequently also.*)
goal thy
"!!evs. [| evs : ns_shared; Kab ~: range shrK |] ==> \
\ (Crypt KAB X) : parts (spies evs) & \
\ Key K : parts {X} --> Key K : parts (spies evs)";
by (parts_induct_tac 1);
(*Deals with Faked messages*)
by (blast_tac (!claset addSEs partsEs
addDs [impOfSubs parts_insert_subset_Un]) 1);
(*Base, NS4 and NS5*)
by (Auto_tac());
result();
(** Session keys are not used to encrypt other session keys **)
(*The equality makes the induction hypothesis easier to apply*)
goal thy
"!!evs. evs : ns_shared ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (spies evs))) = \
\ (K : KK | Key K : analz (spies evs))";
by (etac ns_shared.induct 1);
by analz_spies_tac;
by (REPEAT_FIRST (resolve_tac [allI, impI]));
by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
(*Takes 24 secs*)
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 : ns_shared; 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 session key K uniquely identifies the message **)
goal thy
"!!evs. evs : ns_shared ==> \
\ EX A' NA' B' X'. ALL A NA B X. \
\ Says Server A (Crypt (shrK A) {|NA, Agent B, Key K, X|}) : set evs \
\ --> A=A' & NA=NA' & B=B' & X=X'";
by (etac ns_shared.induct 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
by Safe_tac;
(*NS3*)
by (ex_strip_tac 2);
by (Blast_tac 2);
(*NS2: it can't be a new key*)
by (expand_case_tac "K = ?y" 1);
by (REPEAT (ares_tac [refl,exI,impI,conjI] 2));
by (blast_tac (!claset delrules [conjI] (*prevent split-up into 4 subgoals*)
addSEs spies_partsEs) 1);
val lemma = result();
(*In messages of this form, the session key uniquely identifies the rest*)
goal thy
"!!evs. [| Says Server A \
\ (Crypt (shrK A) {|NA, Agent B, Key K, X|}) : set evs; \
\ Says Server A' \
\ (Crypt (shrK A') {|NA', Agent B', Key K, X'|}) : set evs; \
\ evs : ns_shared |] ==> A=A' & NA=NA' & B=B' & X = X'";
by (prove_unique_tac lemma 1);
qed "unique_session_keys";
(** Crucial secrecy property: Spy does not see the keys sent in msg NS2 **)
goal thy
"!!evs. [| A ~: bad; B ~: bad; evs : ns_shared |] \
\ ==> Says Server A \
\ (Crypt (shrK A) {|NA, Agent B, Key K, \
\ Crypt (shrK B) {|Key K, Agent A|}|}) \
\ : set evs --> \
\ (ALL NB. Says A Spy {|NA, NB, Key K|} ~: set evs) --> \
\ Key K ~: analz (spies evs)";
by (etac ns_shared.induct 1);
by analz_spies_tac;
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_insert_eq, analz_insert_freshK] @ pushes)
addsplits [expand_if])));
(*Oops*)
by (blast_tac (!claset addDs [unique_session_keys]) 5);
(*NS3, replay sub-case*)
by (Blast_tac 4);
(*NS2*)
by (blast_tac (!claset addSEs spies_partsEs
addIs [parts_insertI, impOfSubs analz_subset_parts]) 2);
(*Fake*)
by (spy_analz_tac 1);
(*NS3, Server sub-case*) (**LEVEL 6 **)
by (clarify_tac (!claset delrules [impCE]) 1);
by (forward_tac [Says_imp_spies RS parts.Inj RS A_trusts_NS2] 1);
by (assume_tac 2);
by (blast_tac (!claset addDs [Says_imp_spies RS analz.Inj RS
Crypt_Spy_analz_bad]) 1);
by (blast_tac (!claset addDs [unique_session_keys]) 1);
val lemma = result() RS mp RS mp RSN(2,rev_notE);
(*Final version: Server's message in the most abstract form*)
goal thy
"!!evs. [| Says Server A \
\ (Crypt K' {|NA, Agent B, Key K, X|}) : set evs; \
\ (ALL NB. Says A Spy {|NA, NB, Key K|} ~: set evs); \
\ A ~: bad; B ~: bad; evs : ns_shared \
\ |] ==> Key K ~: analz (spies evs)";
by (forward_tac [Says_Server_message_form] 1 THEN assume_tac 1);
by (blast_tac (!claset addSDs [lemma]) 1);
qed "Spy_not_see_encrypted_key";
(**** Guarantees available at various stages of protocol ***)
A_trusts_NS2 RS Spy_not_see_encrypted_key;
(*If the encrypted message appears then it originated with the Server*)
goal thy
"!!evs. [| Crypt (shrK B) {|Key K, Agent A|} : parts (spies evs); \
\ B ~: bad; evs : ns_shared |] \
\ ==> EX NA. Says Server A \
\ (Crypt (shrK A) {|NA, Agent B, Key K, \
\ Crypt (shrK B) {|Key K, Agent A|}|}) \
\ : set evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
by (Fake_parts_insert_tac 1);
(*Fake case*)
by (ALLGOALS Blast_tac);
qed "B_trusts_NS3";
goal thy
"!!evs. [| B ~: bad; evs : ns_shared |] \
\ ==> Key K ~: analz (spies evs) --> \
\ Says Server A (Crypt (shrK A) {|NA, Agent B, Key K, X|}) \
\ : set evs --> \
\ Crypt K (Nonce NB) : parts (spies evs) --> \
\ Says B A (Crypt K (Nonce NB)) : set evs";
by (etac ns_shared.induct 1);
by (forward_tac [Says_S_message_form] 5 THEN assume_tac 5);
by (dtac NS3_msg_in_parts_spies 5);
by (forward_tac [Oops_parts_spies] 8);
by (TRYALL (rtac impI));
by (REPEAT_FIRST
(dtac (spies_subset_spies_Says RS analz_mono RS contra_subsetD)));
by (ALLGOALS (asm_simp_tac (!simpset addsimps [all_conj_distrib])));
(**LEVEL 7**)
by (blast_tac (!claset addSDs [Crypt_Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]) 1);
by (Blast_tac 2);
by (REPEAT_FIRST (rtac impI ORELSE' etac conjE ORELSE' hyp_subst_tac));
(*Subgoal 1: contradiction from the assumptions
Key K ~: used evsa and Crypt K (Nonce NB) : parts (spies evsa) *)
by (dtac Crypt_imp_invKey_keysFor 1);
(**LEVEL 11**)
by (Asm_full_simp_tac 1);
by (rtac disjI1 1);
by (thin_tac "?PP-->?QQ" 1);
by (case_tac "Ba : bad" 1);
by (blast_tac (!claset addDs [Says_imp_spies RS parts.Inj RS B_trusts_NS3,
unique_session_keys]) 2);
by (blast_tac (!claset addDs [Says_imp_spies RS analz.Inj RS
Crypt_Spy_analz_bad]) 1);
val lemma = result();
goal thy
"!!evs. [| Crypt K (Nonce NB) : parts (spies evs); \
\ Says Server A (Crypt (shrK A) {|NA, Agent B, Key K, X|}) \
\ : set evs; \
\ ALL NB. Says A Spy {|NA, NB, Key K|} ~: set evs; \
\ A ~: bad; B ~: bad; evs : ns_shared |] \
\ ==> Says B A (Crypt K (Nonce NB)) : set evs";
by (blast_tac (!claset addSIs [lemma RS mp RS mp RS mp]
addSEs [Spy_not_see_encrypted_key RSN (2,rev_notE)]) 1);
qed "A_trusts_NS4";