(* Title: HOL/Auth/NSP_Bad
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
ML{*add_path "$ISABELLE_HOME/src/HOL/Auth"*}
Original file is ../Auth/NS_Public_Bad
*)
header{*Analyzing the Needham-Schroeder Public-Key Protocol in UNITY*}
theory NSP_Bad imports "../Auth/Public" "../UNITY_Main" begin
text{*This is the flawed version, vulnerable to Lowe's attack.
From page 260 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989).
*}
types state = "event list"
constdefs
(*The spy MAY say anything he CAN say. We do not expect him to
invent new nonces here, but he can also use NS1. Common to
all similar protocols.*)
Fake :: "(state*state) set"
"Fake == {(s,s').
\<exists>B X. s' = Says Spy B X # s
& X \<in> synth (analz (spies s))}"
(*The numeric suffixes on A identify the rule*)
(*Alice initiates a protocol run, sending a nonce to Bob*)
NS1 :: "(state*state) set"
"NS1 == {(s1,s').
\<exists>A1 B NA.
s' = Says A1 B (Crypt (pubK B) {|Nonce NA, Agent A1|}) # s1
& Nonce NA \<notin> used s1}"
(*Bob responds to Alice's message with a further nonce*)
NS2 :: "(state*state) set"
"NS2 == {(s2,s').
\<exists>A' A2 B NA NB.
s' = Says B A2 (Crypt (pubK A2) {|Nonce NA, Nonce NB|}) # s2
& Says A' B (Crypt (pubK B) {|Nonce NA, Agent A2|}) \<in> set s2
& Nonce NB \<notin> used s2}"
(*Alice proves her existence by sending NB back to Bob.*)
NS3 :: "(state*state) set"
"NS3 == {(s3,s').
\<exists>A3 B' B NA NB.
s' = Says A3 B (Crypt (pubK B) (Nonce NB)) # s3
& Says A3 B (Crypt (pubK B) {|Nonce NA, Agent A3|}) \<in> set s3
& Says B' A3 (Crypt (pubK A3) {|Nonce NA, Nonce NB|}) \<in> set s3}"
constdefs
Nprg :: "state program"
(*Initial trace is empty*)
"Nprg == mk_total_program({[]}, {Fake, NS1, NS2, NS3}, UNIV)"
declare spies_partsEs [elim]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]
text{*For other theories, e.g. Mutex and Lift, using [iff] slows proofs down.
Here, it facilitates re-use of the Auth proofs.*}
declare Fake_def [THEN def_act_simp, iff]
declare NS1_def [THEN def_act_simp, iff]
declare NS2_def [THEN def_act_simp, iff]
declare NS3_def [THEN def_act_simp, iff]
declare Nprg_def [THEN def_prg_Init, simp]
text{*A "possibility property": there are traces that reach the end.
Replace by LEADSTO proof!*}
lemma "A \<noteq> B ==>
\<exists>NB. \<exists>s \<in> reachable Nprg. Says A B (Crypt (pubK B) (Nonce NB)) \<in> set s"
apply (intro exI bexI)
apply (rule_tac [2] act = "totalize_act NS3" in reachable.Acts)
apply (rule_tac [3] act = "totalize_act NS2" in reachable.Acts)
apply (rule_tac [4] act = "totalize_act NS1" in reachable.Acts)
apply (rule_tac [5] reachable.Init)
apply (simp_all (no_asm_simp) add: Nprg_def totalize_act_def)
apply auto
done
subsection{*Inductive Proofs about @{term ns_public}*}
lemma ns_constrainsI:
"(!!act s s'. [| act \<in> {Id, Fake, NS1, NS2, NS3};
(s,s') \<in> act; s \<in> A |] ==> s' \<in> A')
==> Nprg \<in> A co A'"
apply (simp add: Nprg_def mk_total_program_def)
apply (rule constrainsI, auto)
done
text{*This ML code does the inductions directly.*}
ML{*
val ns_constrainsI = thm "ns_constrainsI";
val impCE = thm "impCE";
fun ns_constrains_tac(cs,ss) i =
SELECT_GOAL
(EVERY [REPEAT (etac Always_ConstrainsI 1),
REPEAT (resolve_tac [StableI, stableI,
constrains_imp_Constrains] 1),
rtac ns_constrainsI 1,
full_simp_tac ss 1,
REPEAT (FIRSTGOAL (etac disjE)),
ALLGOALS (clarify_tac (cs delrules [impI,impCE])),
REPEAT (FIRSTGOAL analz_mono_contra_tac),
ALLGOALS (asm_simp_tac ss)]) i;
(*Tactic for proving secrecy theorems*)
fun ns_induct_tac(cs,ss) =
(SELECT_GOAL o EVERY)
[rtac AlwaysI 1,
force_tac (cs,ss) 1,
(*"reachable" gets in here*)
rtac (Always_reachable RS Always_ConstrainsI RS StableI) 1,
ns_constrains_tac(cs,ss) 1];
*}
method_setup ns_induct = {*
Method.ctxt_args (fn ctxt =>
Method.SIMPLE_METHOD' (ns_induct_tac (local_clasimpset_of ctxt))) *}
"for inductive reasoning about the Needham-Schroeder protocol"
text{*Converts invariants into statements about reachable states*}
lemmas Always_Collect_reachableD =
Always_includes_reachable [THEN subsetD, THEN CollectD]
text{*Spy never sees another agent's private key! (unless it's bad at start)*}
lemma Spy_see_priK:
"Nprg \<in> Always {s. (Key (priK A) \<in> parts (spies s)) = (A \<in> bad)}"
apply ns_induct
apply blast
done
declare Spy_see_priK [THEN Always_Collect_reachableD, simp]
lemma Spy_analz_priK:
"Nprg \<in> Always {s. (Key (priK A) \<in> analz (spies s)) = (A \<in> bad)}"
by (rule Always_reachable [THEN Always_weaken], auto)
declare Spy_analz_priK [THEN Always_Collect_reachableD, simp]
subsection{*Authenticity properties obtained from NS2*}
text{*It is impossible to re-use a nonce in both NS1 and NS2 provided the
nonce is secret. (Honest users generate fresh nonces.)*}
lemma no_nonce_NS1_NS2:
"Nprg
\<in> Always {s. Crypt (pubK C) {|NA', Nonce NA|} \<in> parts (spies s) -->
Crypt (pubK B) {|Nonce NA, Agent A|} \<in> parts (spies s) -->
Nonce NA \<in> analz (spies s)}"
apply ns_induct
apply (blast intro: analz_insertI)+
done
text{*Adding it to the claset slows down proofs...*}
lemmas no_nonce_NS1_NS2_reachable =
no_nonce_NS1_NS2 [THEN Always_Collect_reachableD, rule_format]
text{*Unicity for NS1: nonce NA identifies agents A and B*}
lemma unique_NA_lemma:
"Nprg
\<in> Always {s. Nonce NA \<notin> analz (spies s) -->
Crypt(pubK B) {|Nonce NA, Agent A|} \<in> parts(spies s) -->
Crypt(pubK B') {|Nonce NA, Agent A'|} \<in> parts(spies s) -->
A=A' & B=B'}"
apply ns_induct
apply auto
txt{*Fake, NS1 are non-trivial*}
done
text{*Unicity for NS1: nonce NA identifies agents A and B*}
lemma unique_NA:
"[| Crypt(pubK B) {|Nonce NA, Agent A|} \<in> parts(spies s);
Crypt(pubK B') {|Nonce NA, Agent A'|} \<in> parts(spies s);
Nonce NA \<notin> analz (spies s);
s \<in> reachable Nprg |]
==> A=A' & B=B'"
by (blast dest: unique_NA_lemma [THEN Always_Collect_reachableD])
text{*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure*}
lemma Spy_not_see_NA:
"[| A \<notin> bad; B \<notin> bad |]
==> Nprg \<in> Always
{s. Says A B (Crypt(pubK B) {|Nonce NA, Agent A|}) \<in> set s
--> Nonce NA \<notin> analz (spies s)}"
apply ns_induct
txt{*NS3*}
prefer 4 apply (blast intro: no_nonce_NS1_NS2_reachable)
txt{*NS2*}
prefer 3 apply (blast dest: unique_NA)
txt{*NS1*}
prefer 2 apply blast
txt{*Fake*}
apply spy_analz
done
text{*Authentication for A: if she receives message 2 and has used NA
to start a run, then B has sent message 2.*}
lemma A_trusts_NS2:
"[| A \<notin> bad; B \<notin> bad |]
==> Nprg \<in> Always
{s. Says A B (Crypt(pubK B) {|Nonce NA, Agent A|}) \<in> set s &
Crypt(pubK A) {|Nonce NA, Nonce NB|} \<in> parts (knows Spy s)
--> Says B A (Crypt(pubK A) {|Nonce NA, Nonce NB|}) \<in> set s}"
(*insert an invariant for use in some of the subgoals*)
apply (insert Spy_not_see_NA [of A B NA], simp, ns_induct)
apply (auto dest: unique_NA)
done
text{*If the encrypted message appears then it originated with Alice in NS1*}
lemma B_trusts_NS1:
"Nprg \<in> Always
{s. Nonce NA \<notin> analz (spies s) -->
Crypt (pubK B) {|Nonce NA, Agent A|} \<in> parts (spies s)
--> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) \<in> set s}"
apply ns_induct
apply blast
done
subsection{*Authenticity properties obtained from NS2*}
text{*Unicity for NS2: nonce NB identifies nonce NA and agent A.
Proof closely follows that of @{text unique_NA}.*}
lemma unique_NB_lemma:
"Nprg
\<in> Always {s. Nonce NB \<notin> analz (spies s) -->
Crypt (pubK A) {|Nonce NA, Nonce NB|} \<in> parts (spies s) -->
Crypt(pubK A'){|Nonce NA', Nonce NB|} \<in> parts(spies s) -->
A=A' & NA=NA'}"
apply ns_induct
apply auto
txt{*Fake, NS2 are non-trivial*}
done
lemma unique_NB:
"[| Crypt(pubK A) {|Nonce NA, Nonce NB|} \<in> parts(spies s);
Crypt(pubK A'){|Nonce NA', Nonce NB|} \<in> parts(spies s);
Nonce NB \<notin> analz (spies s);
s \<in> reachable Nprg |]
==> A=A' & NA=NA'"
apply (blast dest: unique_NB_lemma [THEN Always_Collect_reachableD])
done
text{*NB remains secret PROVIDED Alice never responds with round 3*}
lemma Spy_not_see_NB:
"[| A \<notin> bad; B \<notin> bad |]
==> Nprg \<in> Always
{s. Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s &
(ALL C. Says A C (Crypt (pubK C) (Nonce NB)) \<notin> set s)
--> Nonce NB \<notin> analz (spies s)}"
apply ns_induct
apply (simp_all (no_asm_simp) add: all_conj_distrib)
txt{*NS3: because NB determines A*}
prefer 4 apply (blast dest: unique_NB)
txt{*NS2: by freshness and unicity of NB*}
prefer 3 apply (blast intro: no_nonce_NS1_NS2_reachable)
txt{*NS1: by freshness*}
prefer 2 apply blast
txt{*Fake*}
apply spy_analz
done
text{*Authentication for B: if he receives message 3 and has used NB
in message 2, then A has sent message 3--to somebody....*}
lemma B_trusts_NS3:
"[| A \<notin> bad; B \<notin> bad |]
==> Nprg \<in> Always
{s. Crypt (pubK B) (Nonce NB) \<in> parts (spies s) &
Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s
--> (\<exists>C. Says A C (Crypt (pubK C) (Nonce NB)) \<in> set s)}"
(*insert an invariant for use in some of the subgoals*)
apply (insert Spy_not_see_NB [of A B NA NB], simp, ns_induct)
apply (simp_all (no_asm_simp) add: ex_disj_distrib)
apply auto
txt{*NS3: because NB determines A. This use of @{text unique_NB} is robust.*}
apply (blast intro: unique_NB [THEN conjunct1])
done
text{*Can we strengthen the secrecy theorem? NO*}
lemma "[| A \<notin> bad; B \<notin> bad |]
==> Nprg \<in> Always
{s. Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s
--> Nonce NB \<notin> analz (spies s)}"
apply ns_induct
apply auto
txt{*Fake*}
apply spy_analz
txt{*NS2: by freshness and unicity of NB*}
apply (blast intro: no_nonce_NS1_NS2_reachable)
txt{*NS3: unicity of NB identifies A and NA, but not B*}
apply (frule_tac A'=A in Says_imp_spies [THEN parts.Inj, THEN unique_NB])
apply (erule Says_imp_spies [THEN parts.Inj], auto)
apply (rename_tac s B' C)
txt{*This is the attack!
@{subgoals[display,indent=0,margin=65]}
*}
oops
(*
THIS IS THE ATTACK!
[| A \<notin> bad; B \<notin> bad |]
==> Nprg
\<in> Always
{s. Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s -->
Nonce NB \<notin> analz (knows Spy s)}
1. !!s B' C.
[| A \<notin> bad; B \<notin> bad; s \<in> reachable Nprg
Says A C (Crypt (pubK C) {|Nonce NA, Agent A|}) \<in> set s;
Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s;
C \<in> bad; Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \<in> set s;
Nonce NB \<notin> analz (knows Spy s) |]
==> False
*)
end