src/HOL/Auth/Recur.ML
 author paulson Mon, 27 Jan 1997 15:29:39 +0100 changeset 2560 c57b585eecd9 parent 2550 8d8344bcf98a child 2637 e9b203f854ae permissions -rw-r--r--
Tidied unicity theorems
```
(*  Title:      HOL/Auth/Recur
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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
(best_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";

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);
!simpset));
qed_spec_mp "Key_in_parts_respond";

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 (fast_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));

(*We instantiate the variable to "lost".  Leaving it as a Var makes proofs
harder to complete, since simplification does less for us.*)
val parts_Fake_tac =
let val tac = forw_inst_tac [("lost","lost")]
in  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
end;

(*For proving the easier theorems about X ~: parts (sees lost Spy evs) *)
fun parts_induct_tac i = SELECT_GOAL
(DETERM (etac recur.induct 1 THEN parts_Fake_tac THEN
(*Fake message*)
TRY (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
(*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 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 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [parts_insert_sees])));
(*RA3*)
(*RA2*)
qed "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";

goal thy  "!!A. [| Key (shrK A) : parts (sees lost Spy evs);       \
\                  evs : recur lost |] ==> A:lost";
by (fast_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);

(** 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 1);
(*RA3*)
(*Fake*)
by (best_tac
impOfSubs (parts_insert_subset_Un RS keysFor_mono)]
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);

(*** Proofs involving analz ***)

(*For proofs involving analz.  We again instantiate the variable to "lost".*)
val analz_Fake_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_Fake_tac;
by (REPEAT_FIRST (resolve_tac [allI, impI]));
by (REPEAT_FIRST (rtac analz_image_freshK_lemma ));
by (ALLGOALS
(asm_simp_tac
(*Base*)
(*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 (etac recur.induct 1);
by parts_Fake_tac;
(*RA3 requires a further induction*)
by (etac responses.induct 5);
by (ALLGOALS Asm_simp_tac);
(*Fake*)
by (best_tac (!claset addDs [impOfSubs analz_subset_parts,
impOfSubs Fake_parts_insert]
(*Two others*)
by (REPEAT (fast_tac (!claset addss (!simpset)) 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 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 (best_tac (!claset addSDs [Hash_imp_body]
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
resp_analz_image_freshK])));
qed "shrK_in_analz_respond";

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) |]         \
\       ==> (Key K : analz (insert RB H)) --> \
\           (Key K : parts{RB} | Key K : analz H)";
by (etac responses.induct 1);
by (ALLGOALS
(asm_simp_tac
(*Simplification using two distinct treatments of "image"*)
by (simp_tac (!simpset addsimps [parts_insert2]) 1);
by (fast_tac (!claset delrules [allE]) 1);
qed "resp_analz_insert_lemma";

bind_thm ("resp_analz_insert",
raw_analz_image_freshK RSN
(2, resp_analz_insert_lemma) RSN(2, rev_mp));

(*The Server does not send such messages.  This theorem lets us avoid
assuming B~=Server in RA4.*)
goal thy
"!!evs. evs : recur lost       \
\ ==> ALL C X Y P. Says Server C {|X, Agent Server, Agent C, Y, P|} \
\                  ~: set_of_list evs";
by (etac recur.induct 1);
by (etac (respond.induct) 5);
by (Auto_tac());
qed_spec_mp "Says_Server_not";

(*The last key returned by respond indeed appears in a certificate*)
goal thy
"!!K. (Hash[Key(shrK A)] {|Agent A, B, NA, P|}, RA, K) : respond evs \
\ ==> Crypt (shrK A) {|Key K, B, NA|} : parts {RA}";
by (etac respond.elim 1);
by (ALLGOALS Asm_full_simp_tac);
qed "respond_certificate";

goal thy
"!!K'. (PB,RB,KXY) : respond evs                          \
\  ==> EX A' B'. ALL A B N.                                \
\        Crypt (shrK A) {|Key K, Agent B, N|} : parts {RB} \
\          -->   (A'=A & B'=B) | (A'=B & B'=A)";
by (etac respond.induct 1);
by (ALLGOALS (asm_full_simp_tac (!simpset addsimps [all_conj_distrib])));
(*Base case*)
by (Fast_tac 1);
by (Step_tac 1);
by (expand_case_tac "K = KBC" 1);
by (dtac respond_Key_in_parts 1);
by (expand_case_tac "K = KAB" 1);
by (REPEAT (ares_tac [exI] 2));
by (ex_strip_tac 1);
by (dtac respond_certificate 1);
by (Fast_tac 1);
val lemma = result();

goal thy
"!!RB. [| Crypt (shrK A) {|Key K, Agent B, N|} : parts {RB};      \
\          Crypt (shrK A') {|Key K, Agent B', N'|} : parts {RB};   \
\          (PB,RB,KXY) : respond evs |]                            \
\ ==>   (A'=A & B'=B) | (A'=B & B'=A)";
by (prove_unique_tac lemma 1);
qed "unique_session_keys";

(** Crucial secrecy property: Spy does not see the keys sent in msg RA3
Does not in itself guarantee security: an attack could violate
the premises, e.g. by having A=Spy **)

goal thy
"!!evs. [| (PB,RB,KAB) : respond evs;  evs : recur lost |]         \
\        ==> ALL A A' N. A ~: lost & A' ~: lost -->                 \
\            Crypt (shrK A) {|Key K, Agent A', N|} : parts{RB} -->  \
\            Key K ~: analz (insert RB (sees lost Spy evs))";
by (etac respond.induct 1);
by (forward_tac [respond_imp_responses] 2);
by (forward_tac [respond_imp_not_used] 2);
by (ALLGOALS (*23 seconds*)
(asm_simp_tac
[shrK_in_analz_respond, resp_analz_image_freshK, parts_insert2])));
by (ALLGOALS Simp_tac);
by (fast_tac (!claset addIs [impOfSubs analz_subset_parts]) 1);
by (step_tac (!claset addSEs [MPair_parts]) 1);
(** LEVEL 7 **)
by (fast_tac (!claset addSDs [resp_analz_insert, Key_in_parts_respond]
by (fast_tac (!claset addSDs [respond_certificate]) 3);
by (dtac unique_session_keys 1);
by (etac respond_certificate 1);
by (assume_tac 1);
by (Fast_tac 1);
qed_spec_mp "respond_Spy_not_see_session_key";

goal thy
"!!evs. [| Crypt (shrK A) {|Key K, Agent A', N|}          \
\              : parts (sees lost Spy evs);                \
\           A ~: lost;  A' ~: lost;  evs : recur lost |]   \
\        ==> Key K ~: analz (sees lost Spy evs)";
by (etac rev_mp 1);
by (etac recur.induct 1);
by analz_Fake_tac;
by (ALLGOALS
(asm_simp_tac
setloop split_tac [expand_if])));
(*RA4*)
by (spy_analz_tac 5);
(*RA2*)
by (spy_analz_tac 3);
(*Fake*)
by (spy_analz_tac 2);
(*Base*)
by (fast_tac (!claset addss (!simpset)) 1);
(*RA3 remains*)
by (step_tac (!claset delrules [impCE]) 1);
(*RA3, case 2: K is an old key*)
(*RA3, case 1: use lemma previously proved by induction*)
by (fast_tac (!claset addSEs [respond_Spy_not_see_session_key RSN
(2,rev_notE)]) 1);
qed "Spy_not_see_session_key";

goal thy
"!!evs. [| Crypt (shrK A) {|Key K, Agent A', N|}         \
\              : parts(sees lost Spy evs);                \
\           C ~: {A,A',Server};                           \
\           A ~: lost;  A' ~: lost;  evs : recur lost |]  \
\        ==> Key K ~: analz (sees lost C evs)";
by (rtac (subset_insertI RS sees_mono RS analz_mono RS contra_subsetD) 1);
by (rtac (sees_lost_agent_subset_sees_Spy RS analz_mono RS contra_subsetD) 1);
by (FIRSTGOAL (rtac Spy_not_see_session_key));
by (REPEAT_FIRST
(deepen_tac
(!claset addIs (map impOfSubs [recur_mono, parts_mono, sees_mono])) 0));
qed "Agent_not_see_session_key";

(**** Authenticity properties for Agents ****)

(*The response never contains Hashes*)
goal thy
"!!evs. [| Hash {|Key (shrK B), M|} : parts (insert RB H); \
\           (PB,RB,K) : respond evs |]                      \
\        ==> Hash {|Key (shrK B), M|} : parts H";
by (etac rev_mp 1);
by (etac (respond_imp_responses RS responses.induct) 1);
by (Auto_tac());
qed "Hash_in_parts_respond";

(*Only RA1 or RA2 can have caused such a part of a message to appear.
This result is of no use to B, who cannot verify the Hash.  Moreover,
it can say nothing about how recent A's message is.  It might later be
used to prove B's presence to A at the run's conclusion.*)
goalw thy [HPair_def]
"!!evs. [| Hash {|Key(shrK A), Agent A, Agent B, NA, P|}         \
\             : parts (sees lost Spy evs);                        \
\            A ~: lost;  evs : recur lost |]                        \
\        ==> Says A B (Hash[Key(shrK A)] {|Agent A, Agent B, NA, P|})  \
\             : set_of_list evs";
by (etac rev_mp 1);
by (parts_induct_tac 1);
(*RA3*)
by (fast_tac (!claset addSDs [Hash_in_parts_respond]) 1);
qed_spec_mp "Hash_auth_sender";

(** These two results subsume (for all agents) the guarantees proved
separately for A and B in the Otway-Rees protocol.
**)

(*Certificates can only originate with the Server.*)
goal thy
"!!evs. [| Crypt (shrK A) Y : parts (sees lost Spy evs);    \
\           A ~: lost;  A ~= Spy;  evs : recur lost |]       \
\        ==> EX C RC. Says Server C RC : set_of_list evs &   \
\                     Crypt (shrK A) Y : parts {RC}";
by (etac rev_mp 1);
by (parts_induct_tac 1);
(*RA4*)
by (Fast_tac 4);
(*RA3*)
by (full_simp_tac (!simpset addsimps [parts_insert_sees]) 3
THEN Fast_tac 3);
(*RA1*)
by (Fast_tac 1);
(*RA2: it cannot be a new Nonce, contradiction.*)
by (deepen_tac (!claset delrules [impCE]