(* Title: HOL/Auth/NS_Public_Bad
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
Flawed version, vulnerable to Lowe's attack.
From page 260 of
Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989)
*)
open NS_Public_Bad;
proof_timing:=true;
HOL_quantifiers := false;
Pretty.setdepth 20;
AddIffs [Spy_in_lost];
(*Replacing the variable by a constant improves search speed by 50%!*)
val Says_imp_sees_Spy' =
read_instantiate_sg (sign_of thy) [("lost","lost")] Says_imp_sees_Spy;
(*A "possibility property": there are traces that reach the end*)
goal thy
"!!A B. [| A ~= B |] \
\ ==> EX NB. EX evs: ns_public. \
\ Says A B (Crypt (pubK B) (Nonce NB)) : set_of_list evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (ns_public.Nil RS ns_public.NS1 RS ns_public.NS2 RS ns_public.NS3) 2);
by (ALLGOALS (simp_tac (!simpset setsolver safe_solver)));
by (REPEAT_FIRST (resolve_tac [refl, conjI]));
by (REPEAT_FIRST (fast_tac (!claset addss (!simpset setsolver safe_solver))));
result();
(**** Inductive proofs about ns_public ****)
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : ns_public ==> ALL A X. Says A A X ~: set_of_list evs";
by (etac ns_public.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 proving the easier theorems about X ~: parts (sees lost Spy evs) *)
fun parts_induct_tac i = SELECT_GOAL
(DETERM (etac ns_public.induct 1 THEN
(*Fake message*)
TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
addss (!simpset)) 2)) THEN
(*Base case*)
fast_tac (!claset addss (!simpset)) 1 THEN
ALLGOALS Asm_simp_tac) i;
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
sends messages containing X! **)
(*Spy never sees another agent's private key! (unless it's lost at start)*)
goal thy
"!!evs. evs : ns_public \
\ ==> (Key (priK A) : parts (sees lost Spy evs)) = (A : lost)";
by (parts_induct_tac 1);
by (Auto_tac());
qed "Spy_see_priK";
Addsimps [Spy_see_priK];
goal thy
"!!evs. evs : ns_public \
\ ==> (Key (priK A) : analz (sees lost Spy evs)) = (A : lost)";
by (auto_tac(!claset addDs [impOfSubs analz_subset_parts], !simpset));
qed "Spy_analz_priK";
Addsimps [Spy_analz_priK];
goal thy "!!A. [| Key (priK A) : parts (sees lost Spy evs); \
\ evs : ns_public |] ==> A:lost";
by (fast_tac (!claset addDs [Spy_see_priK]) 1);
qed "Spy_see_priK_D";
bind_thm ("Spy_analz_priK_D", analz_subset_parts RS subsetD RS Spy_see_priK_D);
AddSDs [Spy_see_priK_D, Spy_analz_priK_D];
(*** Future nonces can't be seen or used! ***)
goal thy "!!evs. evs : ns_public ==> \
\ length evs <= length evt --> \
\ Nonce (newN evt) ~: parts (sees lost Spy evs)";
by (parts_induct_tac 1);
by (REPEAT_FIRST (fast_tac (!claset
addSEs partsEs
addSDs [Says_imp_sees_Spy' RS parts.Inj]
addEs [leD RS notE]
addDs [impOfSubs analz_subset_parts,
impOfSubs parts_insert_subset_Un,
Suc_leD]
addss (!simpset))));
qed_spec_mp "new_nonces_not_seen";
Addsimps [new_nonces_not_seen];
val nonce_not_seen_now = le_refl RSN (2, new_nonces_not_seen) RSN (2,rev_notE);
(**** Authenticity properties obtained from NS2 ****)
(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
is secret. (Honest users generate fresh nonces.)*)
goal thy
"!!evs. evs : ns_public \
\ ==> Nonce NA ~: analz (sees lost Spy evs) --> \
\ Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
\ Crypt (pubK C) {|NA', Nonce NA|} ~: parts (sees lost Spy evs)";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS3*)
by (fast_tac (!claset addSEs partsEs
addEs [nonce_not_seen_now]) 4);
(*NS2*)
by (fast_tac (!claset addSEs partsEs
addEs [nonce_not_seen_now]) 3);
(*Fake*)
by (best_tac (!claset addIs [impOfSubs (subset_insertI RS analz_mono)]
addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
addss (!simpset)) 2);
(*Base*)
by (fast_tac (!claset addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 1);
bind_thm ("no_nonce_NS1_NS2", result() RSN (2, rev_mp) RSN (2, rev_mp));
(*Uniqueness for NS1: nonce NA identifies agents A and B*)
goal thy
"!!evs. evs : ns_public \
\ ==> Nonce NA ~: analz (sees lost Spy evs) --> \
\ (EX A' B'. ALL A B. \
\ Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
\ A=A' & B=B')";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS1*)
by (expand_case_tac "NA = ?y" 3 THEN
REPEAT (fast_tac (!claset addSEs (nonce_not_seen_now::partsEs)) 3));
(*Base*)
by (fast_tac (!claset addss (!simpset)) 1);
(*Fake*)
by (simp_tac (!simpset addsimps [all_conj_distrib,parts_insert_sees]) 1);
by (step_tac (!claset addSIs [impOfSubs (subset_insertI RS analz_mono)]) 1);
by (ex_strip_tac 1);
by (best_tac (!claset delrules [conjI]
addSDs [impOfSubs Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 1);
val lemma = result();
goal thy
"!!evs. [| Crypt(pubK B) {|Nonce NA, Agent A|} : parts(sees lost Spy evs); \
\ Crypt(pubK B') {|Nonce NA, Agent A'|} : parts(sees lost Spy evs); \
\ Nonce NA ~: analz (sees lost Spy evs); \
\ evs : ns_public |] \
\ ==> A=A' & B=B'";
by (dtac lemma 1);
by (mp_tac 1);
by (REPEAT (etac exE 1));
(*Duplicate the assumption*)
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
by (fast_tac (!claset addSDs [spec]) 1);
qed "unique_NA";
(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs \
\ --> Nonce NA ~: analz (sees lost Spy evs)";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS3*)
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 4);
(*NS1*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
(*Fake*)
by (REPEAT_FIRST (rtac conjI ORELSE' spy_analz_tac));
(*NS2*)
by (Step_tac 1);
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addDs [unique_NA]) 1);
bind_thm ("Spy_not_see_NA", result() RSN (2, rev_mp));
(*Authentication for A: if she receives message 2 and has used NA
to start a run, then B has sent message 2.*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Crypt (pubK A) {|Nonce NA, Nonce NB|} : parts (sees lost Spy evs) \
\ --> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs \
\ --> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs";
by (etac ns_public.induct 1);
by (ALLGOALS Asm_simp_tac);
(*NS1*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
(*Fake*)
by (REPEAT_FIRST (resolve_tac [impI, conjI]));
by (fast_tac (!claset addss (!simpset)) 1);
by (forward_tac [Spy_not_see_NA] 1 THEN REPEAT (assume_tac 1));
by (best_tac (!claset addSIs [disjI2]
addSDs [impOfSubs Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 1);
(*NS2*)
by (Step_tac 1);
by (forward_tac [Spy_not_see_NA] 1 THEN REPEAT (assume_tac 1));
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addDs [unique_NA]) 1);
qed_spec_mp "NA_decrypt_imp_B_msg";
(*Corollary: if A receives B's message NS2 and the nonce NA agrees
then that message really originated with B.*)
goal thy
"!!evs. [| Says B' A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set_of_list evs;\
\ Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs;\
\ A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Says B A (Crypt(pubK A) {|Nonce NA, Nonce NB|}): set_of_list evs";
by (fast_tac (!claset addSIs [NA_decrypt_imp_B_msg]
addEs partsEs
addDs [Says_imp_sees_Spy' RS parts.Inj]) 1);
qed "A_trusts_NS2";
(*If the encrypted message appears then it originated with Alice in NS1*)
goal thy
"!!evs. evs : ns_public \
\ ==> Nonce NA ~: analz (sees lost Spy evs) --> \
\ Crypt (pubK B) {|Nonce NA, Agent A|} : parts (sees lost Spy evs) --> \
\ Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set_of_list evs";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*Fake*)
by (best_tac (!claset addSIs [disjI2]
addSDs [impOfSubs Fake_parts_insert]
addIs [impOfSubs (subset_insertI RS analz_mono)]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 2);
(*Base*)
by (fast_tac (!claset addss (!simpset)) 1);
qed_spec_mp "B_trusts_NS1";
(**** Authenticity properties obtained from NS2 ****)
(*Uniqueness for NS2: nonce NB identifies agent A and nonce NA
[proof closely follows that for unique_NA] *)
goal thy
"!!evs. evs : ns_public \
\ ==> Nonce NB ~: analz (sees lost Spy evs) --> \
\ (EX A' NA'. ALL A NA. \
\ Crypt (pubK A) {|Nonce NA, Nonce NB|} : parts (sees lost Spy evs) --> \
\ A=A' & NA=NA')";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS2*)
by (expand_case_tac "NB = ?y" 3 THEN
REPEAT (fast_tac (!claset addSEs (nonce_not_seen_now::partsEs)) 3));
(*Base*)
by (fast_tac (!claset addss (!simpset)) 1);
(*Fake*)
by (simp_tac (!simpset addsimps [all_conj_distrib,parts_insert_sees]) 1);
by (step_tac (!claset addSIs [impOfSubs (subset_insertI RS analz_mono)]) 1);
by (ex_strip_tac 1);
by (best_tac (!claset delrules [conjI]
addSDs [impOfSubs Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 1);
val lemma = result();
goal thy
"!!evs. [| Crypt(pubK A) {|Nonce NA, Nonce NB|} : parts(sees lost Spy evs); \
\ Crypt(pubK A'){|Nonce NA', Nonce NB|} : parts(sees lost Spy evs); \
\ Nonce NB ~: analz (sees lost Spy evs); \
\ evs : ns_public |] \
\ ==> A=A' & NA=NA'";
by (dtac lemma 1);
by (mp_tac 1);
by (REPEAT (etac exE 1));
(*Duplicate the assumption*)
by (forw_inst_tac [("psi", "ALL C.?P(C)")] asm_rl 1);
by (fast_tac (!claset addSDs [spec]) 1);
qed "unique_NB";
(*NB remains secret PROVIDED Alice never responds with round 3*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs --> \
\ (ALL C. Says A C (Crypt (pubK C) (Nonce NB)) ~: set_of_list evs) --> \
\ Nonce NB ~: analz (sees lost Spy evs)";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS1*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
(*Fake*)
by (REPEAT_FIRST (rtac conjI ORELSE' spy_analz_tac));
(*NS2 and NS3*)
by (Step_tac 1);
(*NS2*)
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 3);
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
by (Fast_tac 1);
(*NS3*)
by (Fast_tac 2);
by (fast_tac (!claset addSEs partsEs
addSDs [Says_imp_sees_Spy' RS parts.Inj,
new_nonces_not_seen,
impOfSubs analz_subset_parts]) 1);
by (forw_inst_tac [("A'","A")] (Says_imp_sees_Spy' RS parts.Inj RS unique_NB) 1
THEN REPEAT (eresolve_tac [asm_rl, Says_imp_sees_Spy' RS parts.Inj] 1));
by (Fast_tac 1);
bind_thm ("Spy_not_see_NB", result() RSN (2, rev_mp) RS mp);
(*Authentication for B: if he receives message 3 and has used NB
in message 2, then A has sent message 3.*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Crypt (pubK B) (Nonce NB) : parts (sees lost Spy evs) \
\ --> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs \
\ --> (EX C. Says A C (Crypt (pubK C) (Nonce NB)) : set_of_list evs)";
by (etac ns_public.induct 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [ex_disj_distrib])));
by (ALLGOALS Asm_simp_tac);
(*NS1*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
(*Fake*)
by (REPEAT_FIRST (resolve_tac [impI, conjI]));
by (fast_tac (!claset addss (!simpset)) 1);
br (ccontr RS disjI2) 1;
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT (assume_tac 1));
by (Fast_tac 1);
by (best_tac (!claset addSDs [impOfSubs Fake_parts_insert]
addDs [impOfSubs analz_subset_parts]
addss (!simpset)) 1);
(*NS3*)
by (Step_tac 1);
by (forward_tac [Spy_not_see_NB] 1 THEN REPEAT1 (assume_tac 1));
by (Fast_tac 1);
by (best_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addDs [unique_NB]) 1);
qed_spec_mp "NB_decrypt_imp_A_msg";
(*Corollary: if B receives message NS3 and the nonce NB agrees
then A sent NB to somebody....*)
goal thy
"!!evs. [| Says A' B (Crypt (pubK B) (Nonce NB)): set_of_list evs; \
\ Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) \
\ : set_of_list evs; \
\ A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> EX C. Says A C (Crypt (pubK C) (Nonce NB)) : set_of_list evs";
by (fast_tac (!claset addSIs [NB_decrypt_imp_A_msg]
addEs partsEs
addDs [Says_imp_sees_Spy' RS parts.Inj]) 1);
qed "B_trusts_NS3";
(*Can we strengthen the secrecy theorem? NO*)
goal thy
"!!evs. [| A ~: lost; B ~: lost; evs : ns_public |] \
\ ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evs \
\ --> Nonce NB ~: analz (sees lost Spy evs)";
by (etac ns_public.induct 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps ([analz_subset_parts RS contra_subsetD] @ pushes)
setloop split_tac [expand_if])));
(*NS1*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
(*Fake*)
by (REPEAT_FIRST (rtac conjI ORELSE' spy_analz_tac));
(*NS2 and NS3*)
by (Step_tac 1);
(*NS2*)
by (fast_tac (!claset addSEs partsEs
addSDs [new_nonces_not_seen,
Says_imp_sees_Spy' RS parts.Inj]) 2);
by (fast_tac (!claset addSDs [Says_imp_sees_Spy' RS parts.Inj]
addEs [no_nonce_NS1_NS2 RSN (2, rev_notE)]) 1);
(*NS3*)
by (forw_inst_tac [("A'","A")] (Says_imp_sees_Spy' RS parts.Inj RS unique_NB) 1
THEN REPEAT (eresolve_tac [asm_rl, Says_imp_sees_Spy' RS parts.Inj] 1));
by (Step_tac 1);
(*
THIS IS THE ATTACK!
Level 9
!!evs. [| A ~: lost; B ~: lost; evs : ns_public |]
==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|})
: set_of_list evs -->
Nonce NB ~: analz (sees lost Spy evs)
1. !!evs Aa Ba B' NAa NBa evsa.
[| A ~: lost; B ~: lost; evsa : ns_public; A ~= Ba;
Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evsa;
Says A Ba (Crypt (pubK Ba) {|Nonce NA, Agent A|}) : set_of_list evsa;
Ba : lost;
Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB|}) : set_of_list evsa;
Nonce NB ~: analz (sees lost Spy evsa) |]
==> False
*)