# Theory Recur

theory Recur
imports Public
```(*  Title:      HOL/Auth/Recur.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section‹The Otway-Bull Recursive Authentication Protocol›

theory Recur imports Public begin

text‹End marker for message bundles›
abbreviation
END :: "msg" where
"END == Number 0"

(*Two session keys are distributed to each agent except for the initiator,
Perhaps the two session keys could be bundled into a single message.
*)
inductive_set (*Server's response to the nested message*)
respond :: "event list => (msg*msg*key)set"
for evs :: "event list"
where
One:  "Key KAB ∉ used evs
==> (Hash[Key(shrK A)] ⦃Agent A, Agent B, Nonce NA, END⦄,
⦃Crypt (shrK A) ⦃Key KAB, Agent B, Nonce NA⦄, END⦄,
KAB)   ∈ respond evs"

(*The most recent session key is passed up to the caller*)
| Cons: "[| (PA, RA, KAB) ∈ respond evs;
Key KBC ∉ used evs;  Key KBC ∉ parts {RA};
PA = Hash[Key(shrK A)] ⦃Agent A, Agent B, Nonce NA, P⦄ |]
==> (Hash[Key(shrK B)] ⦃Agent B, Agent C, Nonce NB, PA⦄,
⦃Crypt (shrK B) ⦃Key KBC, Agent C, Nonce NB⦄,
Crypt (shrK B) ⦃Key KAB, Agent A, Nonce NB⦄,
RA⦄,
KBC)
∈ respond evs"

(*Induction over "respond" can be difficult due to the complexity of the
subgoals.  Set "responses" captures the general form of certificates.
*)
inductive_set
responses :: "event list => msg set"
for evs :: "event list"
where
(*Server terminates lists*)
Nil:  "END ∈ responses evs"

| Cons: "[| RA ∈ responses evs;  Key KAB ∉ used evs |]
==> ⦃Crypt (shrK B) ⦃Key KAB, Agent A, Nonce NB⦄,
RA⦄  ∈ responses evs"

inductive_set recur :: "event list set"
where
(*Initial trace is empty*)
Nil:  "[] ∈ recur"

(*The spy MAY say anything he CAN say.  Common to
all similar protocols.*)
| Fake: "[| evsf ∈ recur;  X ∈ synth (analz (knows Spy evsf)) |]
==> Says Spy B X  # evsf ∈ recur"

(*Alice initiates a protocol run.
END is a placeholder to terminate the nesting.*)
| RA1:  "[| evs1 ∈ recur;  Nonce NA ∉ used evs1 |]
==> Says A B (Hash[Key(shrK A)] ⦃Agent A, Agent B, Nonce NA, END⦄)
# evs1 ∈ recur"

(*Bob's response to Alice's message.  C might be the Server.
We omit PA = ⦃XA, Agent A, Agent B, Nonce NA, P⦄ because
it complicates proofs, so B may respond to any message at all!*)
| RA2:  "[| evs2 ∈ recur;  Nonce NB ∉ used evs2;
Says A' B PA ∈ set evs2 |]
==> Says B C (Hash[Key(shrK B)] ⦃Agent B, Agent C, Nonce NB, PA⦄)
# evs2 ∈ recur"

(*The Server receives Bob's message and prepares a response.*)
| RA3:  "[| evs3 ∈ recur;  Says B' Server PB ∈ set evs3;
(PB,RB,K) ∈ respond evs3 |]
==> Says Server B RB # evs3 ∈ recur"

(*Bob receives the returned message and compares the Nonces with
those in the message he previously sent the Server.*)
| RA4:  "[| evs4 ∈ recur;
Says B  C ⦃XH, Agent B, Agent C, Nonce NB,
XA, Agent A, Agent B, Nonce NA, P⦄ ∈ set evs4;
Says C' B ⦃Crypt (shrK B) ⦃Key KBC, Agent C, Nonce NB⦄,
Crypt (shrK B) ⦃Key KAB, Agent A, Nonce NB⦄,
RA⦄ ∈ set evs4 |]
==> Says B A RA # evs4 ∈ recur"

(*No "oops" message can easily be expressed.  Each session key is
associated--in two separate messages--with two nonces.  This is
one try, but it isn't that useful.  Re domino attack, note that
Recur.thy proves that each session key is secure provided the two
peers are, even if there are compromised agents elsewhere in
the chain.  Oops cases proved using parts_cut, Key_in_keysFor_parts,
etc.

Oops:  "[| evso ∈ recur;  Says Server B RB ∈ set evso;
RB ∈ responses evs';  Key K ∈ parts {RB} |]
==> Notes Spy ⦃Key K, RB⦄ # evso ∈ recur"
*)

declare Says_imp_knows_Spy [THEN analz.Inj, dest]
declare parts.Body  [dest]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un  [dest]

(** Possibility properties: traces that reach the end
ONE theorem would be more elegant and faster!
By induction on a list of agents (no repetitions)
**)

text‹Simplest case: Alice goes directly to the server›
lemma "Key K ∉ used []
==> ∃NA. ∃evs ∈ recur.
Says Server A ⦃Crypt (shrK A) ⦃Key K, Agent Server, Nonce NA⦄,
END⦄  ∈ set evs"
apply (intro exI bexI)
apply (rule_tac [2] recur.Nil [THEN recur.RA1,
THEN recur.RA3 [OF _ _ respond.One]])
done

text‹Case two: Alice, Bob and the server›
lemma "[| Key K ∉ used []; Key K' ∉ used []; K ≠ K';
Nonce NA ∉ used []; Nonce NB ∉ used []; NA < NB |]
==> ∃NA. ∃evs ∈ recur.
Says B A ⦃Crypt (shrK A) ⦃Key K, Agent B, Nonce NA⦄,
END⦄  ∈ set evs"
apply (intro exI bexI)
apply (rule_tac [2]
recur.Nil
[THEN recur.RA1 [of _ NA],
THEN recur.RA2 [of _ NB],
THEN recur.RA3 [OF _ _ respond.One
[THEN respond.Cons [of _ _ K _ K']]],
THEN recur.RA4], possibility)
done

(*Case three: Alice, Bob, Charlie and the server Rather slow (5 seconds)*)
lemma "[| Key K ∉ used []; Key K' ∉ used [];
Key K'' ∉ used []; K ≠ K'; K' ≠ K''; K ≠ K'';
Nonce NA ∉ used []; Nonce NB ∉ used []; Nonce NC ∉ used [];
NA < NB; NB < NC |]
==> ∃K. ∃NA. ∃evs ∈ recur.
Says B A ⦃Crypt (shrK A) ⦃Key K, Agent B, Nonce NA⦄,
END⦄  ∈ set evs"
apply (intro exI bexI)
apply (rule_tac [2]
recur.Nil [THEN recur.RA1,
THEN recur.RA2, THEN recur.RA2,
THEN recur.RA3
[OF _ _ respond.One
[THEN respond.Cons, THEN respond.Cons]],
THEN recur.RA4, THEN recur.RA4])
apply basic_possibility
apply (tactic "DEPTH_SOLVE (swap_res_tac @{context} [refl, conjI, disjCI] 1)")
done

lemma respond_imp_not_used: "(PA,RB,KAB) ∈ respond evs ==> Key KAB ∉ used evs"
by (erule respond.induct, simp_all)

lemma Key_in_parts_respond [rule_format]:
"[| Key K ∈ parts {RB};  (PB,RB,K') ∈ respond evs |] ==> Key K ∉ used evs"
apply (erule rev_mp, erule respond.induct)
apply (auto dest: Key_not_used respond_imp_not_used)
done

lemma respond_imp_responses:
"(PA,RB,KAB) ∈ respond evs ==> RB ∈ responses evs"
apply (erule respond.induct)
apply (blast intro!: respond_imp_not_used responses.intros)+
done

(** For reasoning about the encrypted portion of messages **)

lemmas RA2_analz_spies = Says_imp_spies [THEN analz.Inj]

lemma RA4_analz_spies:
"Says C' B ⦃Crypt K X, X', RA⦄ ∈ set evs ==> RA ∈ analz (spies evs)"
by blast

(*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. *)

lemmas RA2_parts_spies =  RA2_analz_spies [THEN analz_into_parts]
lemmas RA4_parts_spies =  RA4_analz_spies [THEN analz_into_parts]

(** 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) **)

lemma Spy_see_shrK [simp]:
"evs ∈ recur ==> (Key (shrK A) ∈ parts (spies evs)) = (A ∈ bad)"
apply (erule recur.induct, auto)
txt‹RA3.  It's ugly to call auto twice, but it seems necessary.›
apply (auto dest: Key_in_parts_respond simp add: parts_insert_spies)
done

lemma Spy_analz_shrK [simp]:
"evs ∈ recur ==> (Key (shrK A) ∈ analz (spies evs)) = (A ∈ bad)"
by auto

lemma Spy_see_shrK_D [dest!]:
"[|Key (shrK A) ∈ parts (knows Spy evs);  evs ∈ recur|] ==> A ∈ bad"
by (blast dest: Spy_see_shrK)

(*** Proofs involving analz ***)

(** 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 "spies evs")
satisfying the inductive hypothesis.*)
lemma resp_analz_image_freshK_lemma:
"[| RB ∈ responses evs;
∀K KK. KK ⊆ - (range shrK) -->
(Key K ∈ analz (Key`KK Un H)) =
(K ∈ KK | Key K ∈ analz H) |]
==> ∀K KK. KK ⊆ - (range shrK) -->
(Key K ∈ analz (insert RB (Key`KK Un H))) =
(K ∈ KK | Key K ∈ analz (insert RB H))"
apply (erule responses.induct)
apply (simp_all del: image_insert
done

text‹Version for the protocol.  Proof is easy, thanks to the lemma.›
lemma raw_analz_image_freshK:
"evs ∈ recur ==>
∀K KK. KK ⊆ - (range shrK) -->
(Key K ∈ analz (Key`KK Un (spies evs))) =
(K ∈ KK | Key K ∈ analz (spies evs))"
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
drule_tac [5] respond_imp_responses,
drule_tac [6] RA4_analz_spies, analz_freshK, spy_analz)
txt‹RA3›
done

(*Instance of the lemma with H replaced by (spies evs):
[| RB ∈ responses evs;  evs ∈ recur; |]
==> KK ⊆ - (range shrK) -->
Key K ∈ analz (insert RB (Key`KK Un spies evs)) =
(K ∈ KK | Key K ∈ analz (insert RB (spies evs)))
*)
lemmas resp_analz_image_freshK =
resp_analz_image_freshK_lemma [OF _ raw_analz_image_freshK]

lemma analz_insert_freshK:
"[| evs ∈ recur;  KAB ∉ range shrK |]
==> (Key K ∈ analz (insert (Key KAB) (spies evs))) =
(K = KAB | Key K ∈ analz (spies evs))"
by (simp del: image_insert

text‹Everything that's hashed is already in past traffic.›
lemma Hash_imp_body:
"[| Hash ⦃Key(shrK A), X⦄ ∈ parts (spies evs);
evs ∈ recur;  A ∉ bad |] ==> X ∈ parts (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct,
drule_tac [6] RA4_parts_spies,
drule_tac [5] respond_imp_responses,
drule_tac [4] RA2_parts_spies)
txt‹RA3 requires a further induction›
apply (erule_tac [5] responses.induct, simp_all)
txt‹Fake›
apply (blast intro: parts_insertI)
done

(** 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.
**)

lemma unique_NA:
"[| Hash ⦃Key(shrK A), Agent A, B, NA, P⦄ ∈ parts (spies evs);
Hash ⦃Key(shrK A), Agent A, B',NA, P'⦄ ∈ parts (spies evs);
evs ∈ recur;  A ∉ bad |]
==> B=B' & P=P'"
apply (erule rev_mp, erule rev_mp)
apply (erule recur.induct,
drule_tac [5] respond_imp_responses)
apply (force, simp_all)
txt‹Fake›
apply blast
apply (erule_tac [3] responses.induct)
txt‹RA1,2: creation of new Nonce›
apply simp_all
apply (blast dest!: Hash_imp_body)+
done

(*** Lemmas concerning the Server's response
(relations "respond" and "responses")
***)

lemma shrK_in_analz_respond [simp]:
"[| RB ∈ responses evs;  evs ∈ recur |]
==> (Key (shrK B) ∈ analz (insert RB (spies evs))) = (B:bad)"
apply (erule responses.induct)
apply (simp_all del: image_insert
done

lemma resp_analz_insert_lemma:
"[| Key K ∈ analz (insert RB H);
∀K KK. KK ⊆ - (range shrK) -->
(Key K ∈ analz (Key`KK Un H)) =
(K ∈ KK | Key K ∈ analz H);
RB ∈ responses evs |]
==> (Key K ∈ parts{RB} | Key K ∈ analz H)"
apply (erule rev_mp, erule responses.induct)
apply (simp_all del: image_insert parts_image
txt‹Simplification using two distinct treatments of "image"›
done

lemmas resp_analz_insert =
resp_analz_insert_lemma [OF _ raw_analz_image_freshK]

text‹The last key returned by respond indeed appears in a certificate›
lemma respond_certificate:
"(Hash[Key(shrK A)] ⦃Agent A, B, NA, P⦄, RA, K) ∈ respond evs
==> Crypt (shrK A) ⦃Key K, B, NA⦄ ∈ parts {RA}"
apply (ind_cases "(Hash[Key (shrK A)] ⦃Agent A, B, NA, P⦄, RA, K) ∈ respond evs")
apply simp_all
done

(*This unicity proof differs from all the others in the HOL/Auth directory.
The conclusion isn't quite unicity but duplicity, in that there are two
possibilities.  Also, the presence of two different matching messages in
the inductive step complicates the case analysis.  Unusually for such proofs,
the quantifiers appear to be necessary.*)
lemma unique_lemma [rule_format]:
"(PB,RB,KXY) ∈ respond evs ==>
∀A B N. Crypt (shrK A) ⦃Key K, Agent B, N⦄ ∈ parts {RB} -->
(∀A' B' N'. Crypt (shrK A') ⦃Key K, Agent B', N'⦄ ∈ parts {RB} -->
(A'=A & B'=B) | (A'=B & B'=A))"
apply (erule respond.induct)
apply (blast dest: respond_certificate)
done

lemma unique_session_keys:
"[| 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 (rule unique_lemma, auto)

(** 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 **)

lemma respond_Spy_not_see_session_key [rule_format]:
"[| (PB,RB,KAB) ∈ respond evs;  evs ∈ recur |]
==> ∀A A' N. A ∉ bad & A' ∉ bad -->
Crypt (shrK A) ⦃Key K, Agent A', N⦄ ∈ parts{RB} -->
Key K ∉ analz (insert RB (spies evs))"
apply (erule respond.induct)
apply (frule_tac [2] respond_imp_responses)
apply (frule_tac [2] respond_imp_not_used)
apply (simp_all del: image_insert parts_image
resp_analz_image_freshK parts_insert2)
txt‹Base case of respond›
apply blast
txt‹Inductive step of respond›
apply (intro allI conjI impI, simp_all)
txt‹by unicity, either @{term "B=Aa"} or @{term "B=A'"}, a contradiction
apply (blast dest: unique_session_keys respond_certificate)
apply (blast dest!: respond_certificate)
apply (blast dest!: resp_analz_insert)
done

lemma Spy_not_see_session_key:
"[| Crypt (shrK A) ⦃Key K, Agent A', N⦄ ∈ parts (spies evs);
==> Key K ∉ analz (spies evs)"
apply (erule rev_mp)
apply (erule recur.induct)
apply (drule_tac [4] RA2_analz_spies,
frule_tac [5] respond_imp_responses,
drule_tac [6] RA4_analz_spies,
txt‹Fake›
apply spy_analz
txt‹RA2›
apply blast
txt‹RA3›
apply (metis Key_in_parts_respond parts.Body parts.Fst resp_analz_insert
respond_Spy_not_see_session_key usedI)
txt‹RA4›
apply blast
done

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

text‹The response never contains Hashes›
lemma Hash_in_parts_respond:
"[| Hash ⦃Key (shrK B), M⦄ ∈ parts (insert RB H);
(PB,RB,K) ∈ respond evs |]
==> Hash ⦃Key (shrK B), M⦄ ∈ parts H"
apply (erule rev_mp)
apply (erule respond_imp_responses [THEN responses.induct], auto)
done

text‹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.›
lemma Hash_auth_sender [rule_format]:
"[| Hash ⦃Key(shrK A), Agent A, Agent B, NA, P⦄ ∈ parts(spies evs);
A ∉ bad;  evs ∈ recur |]
==> Says A B (Hash[Key(shrK A)] ⦃Agent A, Agent B, NA, P⦄) ∈ set evs"
apply (unfold HPair_def)
apply (erule rev_mp)
apply (erule recur.induct,
drule_tac [6] RA4_parts_spies,
drule_tac [4] RA2_parts_spies,
simp_all)
txt‹Fake, RA3›
apply (blast dest: Hash_in_parts_respond)+
done

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

text‹Certificates can only originate with the Server.›
lemma Cert_imp_Server_msg:
"[| Crypt (shrK A) Y ∈ parts (spies evs);
A ∉ bad;  evs ∈ recur |]
==> ∃C RC. Says Server C RC ∈ set evs  &
Crypt (shrK A) Y ∈ parts {RC}"
apply (erule rev_mp, erule recur.induct, simp_all)
txt‹Fake›
apply blast
txt‹RA1›
apply blast
txt‹RA2: it cannot be a new Nonce, contradiction.›
apply blast
txt‹RA3.  Pity that the proof is so brittle: this step requires the rewriting,
which however would break all other steps.›