(* Title: HOL/Auth/Recur
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "recur" for the Recursive Authentication protocol.
*)
open Recur;
proof_timing:=true;
HOL_quantifiers := false;
Pretty.setdepth 30;
(** Possibility properties: traces that reach the end
ONE theorem would be more elegant and faster!
By induction on a list of agents (no repetitions)
**)
(*Simplest case: Alice goes directly to the server*)
goal thy
"!!A. A ~= Server \
\ ==> EX K NA. EX evs: recur lost. \
\ Says Server A {|Crypt (shrK A) {|Key K, Agent Server, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS
(respond.One RSN (4,recur.RA3))) 2);
by possibility_tac;
result();
(*Case two: Alice, Bob and the server*)
goal thy
"!!A B. [| A ~= B; A ~= Server; B ~= Server |] \
\ ==> EX K. EX NA. EX evs: recur lost. \
\ Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (cut_facts_tac [Nonce_supply2, Key_supply2] 1);
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS recur.RA2 RS
(respond.One RS respond.Cons RSN (4,recur.RA3)) RS
recur.RA4) 2);
by basic_possibility_tac;
by (DEPTH_SOLVE (eresolve_tac [asm_rl, less_not_refl2,
less_not_refl2 RS not_sym] 1));
result();
(*Case three: Alice, Bob, Charlie and the server
TOO SLOW to run every time!
goal thy
"!!A B. [| A ~= B; B ~= C; A ~= Server; B ~= Server; C ~= Server |] \
\ ==> EX K. EX NA. EX evs: recur lost. \
\ Says B A {|Crypt (shrK A) {|Key K, Agent B, Nonce NA|}, \
\ Agent Server|} \
\ : set_of_list evs";
by (cut_facts_tac [Nonce_supply3, Key_supply3] 1);
by (REPEAT (eresolve_tac [exE, conjE] 1));
by (REPEAT (resolve_tac [exI,bexI] 1));
by (rtac (recur.Nil RS recur.RA1 RS recur.RA2 RS recur.RA2 RS
(respond.One RS respond.Cons RS respond.Cons RSN
(4,recur.RA3)) RS recur.RA4 RS recur.RA4) 2);
(*SLOW: 70 seconds*)
by basic_possibility_tac;
by (DEPTH_SOLVE (swap_res_tac [refl, conjI, disjCI] 1
ORELSE
eresolve_tac [asm_rl, less_not_refl2,
less_not_refl2 RS not_sym] 1));
result();
****************)
(**** Inductive proofs about recur ****)
(*Monotonicity*)
goal thy "!!evs. lost' <= lost ==> recur lost' <= recur lost";
by (rtac subsetI 1);
by (etac recur.induct 1);
by (REPEAT_FIRST
(blast_tac (!claset addIs (impOfSubs (sees_mono RS analz_mono RS synth_mono)
:: recur.intrs))));
qed "recur_mono";
(*Nobody sends themselves messages*)
goal thy "!!evs. evs : recur lost ==> ALL A X. Says A A X ~: set_of_list evs";
by (etac recur.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)];
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> Key KAB : parts{RB}";
by (etac respond.induct 1);
by (ALLGOALS Simp_tac);
qed "respond_Key_in_parts";
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> Key KAB ~: used evs";
by (etac respond.induct 1);
by (REPEAT (assume_tac 1));
qed "respond_imp_not_used";
goal thy
"!!evs. [| Key K : parts {RB}; (PB,RB,K') : respond evs |] \
\ ==> Key K ~: used evs";
by (etac rev_mp 1);
by (etac respond.induct 1);
by (auto_tac(!claset addDs [Key_not_used, respond_imp_not_used],
!simpset));
qed_spec_mp "Key_in_parts_respond";
(*Simple inductive reasoning about responses*)
goal thy "!!evs. (PA,RB,KAB) : respond evs ==> RB : responses evs";
by (etac respond.induct 1);
by (REPEAT (ares_tac (respond_imp_not_used::responses.intrs) 1));
qed "respond_imp_responses";
(** For reasoning about the encrypted portion of messages **)
val RA2_analz_sees_Spy = Says_imp_sees_Spy RS analz.Inj |> standard;
goal thy "!!evs. Says C' B {|Crypt K X, X', RA|} : set_of_list evs \
\ ==> RA : analz (sees lost Spy evs)";
by (blast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj]) 1);
qed "RA4_analz_sees_Spy";
(*RA2_analz... and RA4_analz... let us treat those cases using the same
argument as for the Fake case. This is possible for most, but not all,
proofs: Fake does not invent new nonces (as in RA2), and of course Fake
messages originate from the Spy. *)
bind_thm ("RA2_parts_sees_Spy",
RA2_analz_sees_Spy RS (impOfSubs analz_subset_parts));
bind_thm ("RA4_parts_sees_Spy",
RA4_analz_sees_Spy RS (impOfSubs analz_subset_parts));
(*For proving the easier theorems about X ~: parts (sees lost Spy evs).
We instantiate the variable to "lost" since leaving it as a Var would
interfere with simplification.*)
val parts_induct_tac =
let val tac = forw_inst_tac [("lost","lost")]
in etac recur.induct 1 THEN
tac RA2_parts_sees_Spy 4 THEN
etac subst 4 (*RA2: DELETE needless definition of PA!*) THEN
forward_tac [respond_imp_responses] 5 THEN
tac RA4_parts_sees_Spy 6 THEN
prove_simple_subgoals_tac 1
end;
(** Theorems of the form X ~: parts (sees lost Spy evs) imply that NOBODY
sends messages containing X! **)
(** Spy never sees another agent's long-term key (unless initially lost) **)
goal thy
"!!evs. evs : recur lost \
\ ==> (Key (shrK A) : parts (sees lost Spy evs)) = (A : lost)";
by parts_induct_tac;
by (Fake_parts_insert_tac 1);
by (ALLGOALS
(asm_simp_tac (!simpset addsimps [parts_insert2, parts_insert_sees])));
(*RA3*)
by (blast_tac (!claset addDs [Key_in_parts_respond]) 2);
(*RA2*)
by (blast_tac (!claset addSEs partsEs addDs [parts_cut]) 1);
qed "Spy_see_shrK";
Addsimps [Spy_see_shrK];
goal thy
"!!evs. evs : recur lost \
\ ==> (Key (shrK A) : analz (sees lost 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 lost Spy evs); \
\ evs : recur lost |] ==> 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! **)
goal thy
"!!evs. [| K : keysFor (parts {RB}); (PB,RB,K') : respond evs |] \
\ ==> K : range shrK";
by (etac rev_mp 1);
by (etac (respond_imp_responses RS responses.induct) 1);
by (Auto_tac());
qed_spec_mp "Key_in_keysFor_parts";
goal thy "!!evs. evs : recur lost ==> \
\ Key K ~: used evs --> K ~: keysFor (parts (sees lost Spy evs))";
by parts_induct_tac;
(*RA3*)
by (best_tac (!claset addDs [Key_in_keysFor_parts]
unsafe_addss (!simpset addsimps [parts_insert_sees])) 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)]
unsafe_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];
(*** Proofs involving analz ***)
(*For proofs involving analz. We again instantiate the variable to "lost".*)
val analz_sees_tac =
etac subst 4 (*RA2: DELETE needless definition of PA!*) THEN
dres_inst_tac [("lost","lost")] RA2_analz_sees_Spy 4 THEN
forward_tac [respond_imp_responses] 5 THEN
dres_inst_tac [("lost","lost")] RA4_analz_sees_Spy 6;
(** Session keys are not used to encrypt other session keys **)
(*Version for "responses" relation. Handles case RA3 in the theorem below.
Note that it holds for *any* set H (not just "sees lost Spy evs")
satisfying the inductive hypothesis.*)
goal thy
"!!evs. [| RB : responses evs; \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un H)) = \
\ (K : KK | Key K : analz H) |] \
\ ==> ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (insert RB (Key``KK Un H))) = \
\ (K : KK | Key K : analz (insert RB H))";
by (etac responses.induct 1);
by (ALLGOALS (asm_simp_tac analz_image_freshK_ss));
qed "resp_analz_image_freshK_lemma";
(*Version for the protocol. Proof is almost trivial, thanks to the lemma.*)
goal thy
"!!evs. evs : recur lost ==> \
\ ALL K KK. KK <= Compl (range shrK) --> \
\ (Key K : analz (Key``KK Un (sees lost Spy evs))) = \
\ (K : KK | Key K : analz (sees lost Spy evs))";
by (etac recur.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 addsimps [resp_analz_image_freshK_lemma])));
(*Base*)
by (Blast_tac 1);
(*RA4, RA2, Fake*)
by (REPEAT (spy_analz_tac 1));
val raw_analz_image_freshK = result();
qed_spec_mp "analz_image_freshK";
(*Instance of the lemma with H replaced by (sees lost Spy evs):
[| RB : responses evs; evs : recur lost; |]
==> KK <= Compl (range shrK) -->
Key K : analz (insert RB (Key``KK Un sees lost Spy evs)) =
(K : KK | Key K : analz (insert RB (sees lost Spy evs)))
*)
bind_thm ("resp_analz_image_freshK",
raw_analz_image_freshK RSN
(2, resp_analz_image_freshK_lemma) RS spec RS spec);
goal thy
"!!evs. [| evs : recur lost; KAB ~: range shrK |] ==> \
\ Key K : analz (insert (Key KAB) (sees lost Spy evs)) = \
\ (K = KAB | Key K : analz (sees lost Spy evs))";
by (asm_simp_tac (analz_image_freshK_ss addsimps [analz_image_freshK]) 1);
qed "analz_insert_freshK";
(*Everything that's hashed is already in past traffic. *)
goal thy "!!evs. [| Hash {|Key(shrK A), X|} : parts (sees lost Spy evs); \
\ evs : recur lost; A ~: lost |] \
\ ==> X : parts (sees lost Spy evs)";
by (etac rev_mp 1);
by parts_induct_tac;
(*RA3 requires a further induction*)
by (etac responses.induct 2);
by (ALLGOALS Asm_simp_tac);
(*Fake*)
by (simp_tac (!simpset addsimps [parts_insert_sees]) 1);
by (Fake_parts_insert_tac 1);
qed "Hash_imp_body";
(** The Nonce NA uniquely identifies A's message.
This theorem applies to steps RA1 and RA2!
Unicity is not used in other proofs but is desirable in its own right.
**)
goal thy
"!!evs. [| evs : recur lost; A ~: lost |] \
\ ==> EX B' P'. ALL B P. \
\ Hash {|Key(shrK A), Agent A, B, NA, P|} : parts (sees lost Spy evs) \
\ --> B=B' & P=P'";
by parts_induct_tac;
by (Fake_parts_insert_tac 1);
by (etac responses.induct 3);
by (ALLGOALS (simp_tac (!simpset addsimps [all_conj_distrib])));
by (step_tac (!claset addSEs partsEs) 1);
(*RA1,2: creation of new Nonce. Move assertion into global context*)
by (ALLGOALS (expand_case_tac "NA = ?y"));
by (REPEAT_FIRST (ares_tac [exI]));
by (REPEAT (blast_tac (!claset addSDs [Hash_imp_body]
addSEs sees_Spy_partsEs) 1));
val lemma = result();
goalw thy [HPair_def]
"!!evs.[| Hash[Key(shrK A)] {|Agent A,B,NA,P|} : parts (sees lost Spy evs); \
\ Hash[Key(shrK A)] {|Agent A,B',NA,P'|} : parts (sees lost Spy evs);\
\ evs : recur lost; A ~: lost |] \
\ ==> B=B' & P=P'";
by (REPEAT (eresolve_tac partsEs 1));
by (prove_unique_tac lemma 1);
qed "unique_NA";
(*** Lemmas concerning the Server's response
(relations "respond" and "responses")
***)
goal thy
"!!evs. [| RB : responses evs; evs : recur lost |] \
\ ==> (Key (shrK B) : analz (insert RB (sees lost Spy evs))) = (B:lost)";
by (etac responses.induct 1);
by (ALLGOALS
(asm_simp_tac
(analz_image_freshK_ss addsimps [Spy_analz_shrK,
resp_analz_image_freshK])));
qed "shrK_in_analz_respond";