Theory Shared

theory Shared
imports Event All_Symmetric
(*  Title:      HOL/Auth/Shared.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge

Theory of Shared Keys (common to all symmetric-key protocols)

Shared, long-term keys; initial states of agents

theory Shared
imports Event All_Symmetric

shrK :: "agent => key" (*symmetric keys*)

specification (shrK)
inj_shrK: "inj shrK"
--{*No two agents have the same long-term key*}
apply (rule exI [of _ "agent_case 0 (λn. n + 2) 1"])
apply (simp add: inj_on_def split: agent.split)

text{*Server knows all long-term keys; other agents know only their own*}

initState initState

primrec initState where
initState_Server: "initState Server = Key ` range shrK"
| initState_Friend: "initState (Friend i) = {Key (shrK (Friend i))}"
| initState_Spy: "initState Spy = Key`shrK`bad"


subsection{*Basic properties of shrK*}

(*Injectiveness: Agents' long-term keys are distinct.*)
lemmas shrK_injective = inj_shrK [THEN inj_eq]
declare shrK_injective [iff]

lemma invKey_K [simp]: "invKey K = K"
apply (insert isSym_keys)
apply (simp add: symKeys_def)

lemma analz_Decrypt' [dest]:
"[| Crypt K X ∈ analz H; Key K ∈ analz H |] ==> X ∈ analz H"
by auto

text{*Now cancel the @{text dest} attribute given to
@{text analz.Decrypt} in its declaration.*}

declare analz.Decrypt [rule del]

text{*Rewrites should not refer to @{term "initState(Friend i)"} because
that expression is not in normal form.*}

lemma keysFor_parts_initState [simp]: "keysFor (parts (initState C)) = {}"
apply (unfold keysFor_def)
apply (induct_tac "C", auto)

(*Specialized to shared-key model: no @{term invKey}*)
lemma keysFor_parts_insert:
"[| K ∈ keysFor (parts (insert X G)); X ∈ synth (analz H) |]
==> K ∈ keysFor (parts (G ∪ H)) | Key K ∈ parts H"

by (metis invKey_K keysFor_parts_insert)

lemma Crypt_imp_keysFor: "Crypt K X ∈ H ==> K ∈ keysFor H"
by (metis Crypt_imp_invKey_keysFor invKey_K)

subsection{*Function "knows"*}

(*Spy sees shared keys of agents!*)
lemma Spy_knows_Spy_bad [intro!]: "A: bad ==> Key (shrK A) ∈ knows Spy evs"
apply (induct_tac "evs")
apply (simp_all (no_asm_simp) add: imageI knows_Cons split add: event.split)

(*For case analysis on whether or not an agent is compromised*)
lemma Crypt_Spy_analz_bad: "[| Crypt (shrK A) X ∈ analz (knows Spy evs); A: bad |]
==> X ∈ analz (knows Spy evs)"

by (metis Spy_knows_Spy_bad analz.Inj analz_Decrypt')

(** Fresh keys never clash with long-term shared keys **)

(*Agents see their own shared keys!*)
lemma shrK_in_initState [iff]: "Key (shrK A) ∈ initState A"
by (induct_tac "A", auto)

lemma shrK_in_used [iff]: "Key (shrK A) ∈ used evs"
by (rule initState_into_used, blast)

(*Used in parts_induct_tac and analz_Fake_tac to distinguish session keys
from long-term shared keys*)

lemma Key_not_used [simp]: "Key K ∉ used evs ==> K ∉ range shrK"
by blast

lemma shrK_neq [simp]: "Key K ∉ used evs ==> shrK B ≠ K"
by blast

lemmas shrK_sym_neq = shrK_neq [THEN not_sym]
declare shrK_sym_neq [simp]

subsection{*Fresh nonces*}

lemma Nonce_notin_initState [iff]: "Nonce N ∉ parts (initState B)"
by (induct_tac "B", auto)

lemma Nonce_notin_used_empty [simp]: "Nonce N ∉ used []"
by (simp add: used_Nil)

subsection{*Supply fresh nonces for possibility theorems.*}

(*In any trace, there is an upper bound N on the greatest nonce in use.*)
lemma Nonce_supply_lemma: "∃N. ALL n. N<=n --> Nonce n ∉ used evs"
apply (induct_tac "evs")
apply (rule_tac x = 0 in exI)
apply (simp_all (no_asm_simp) add: used_Cons split add: event.split)
apply (metis le_sup_iff msg_Nonce_supply)

lemma Nonce_supply1: "∃N. Nonce N ∉ used evs"
by (metis Nonce_supply_lemma order_eq_iff)

lemma Nonce_supply2: "∃N N'. Nonce N ∉ used evs & Nonce N' ∉ used evs' & N ≠ N'"
apply (cut_tac evs = evs in Nonce_supply_lemma)
apply (cut_tac evs = "evs'" in Nonce_supply_lemma, clarify)
apply (metis Suc_n_not_le_n nat_le_linear)

lemma Nonce_supply3: "∃N N' N''. Nonce N ∉ used evs & Nonce N' ∉ used evs' &
Nonce N'' ∉ used evs'' & N ≠ N' & N' ≠ N'' & N ≠ N''"

apply (cut_tac evs = evs in Nonce_supply_lemma)
apply (cut_tac evs = "evs'" in Nonce_supply_lemma)
apply (cut_tac evs = "evs''" in Nonce_supply_lemma, clarify)
apply (rule_tac x = N in exI)
apply (rule_tac x = "Suc (N+Na)" in exI)
apply (rule_tac x = "Suc (Suc (N+Na+Nb))" in exI)
apply (simp (no_asm_simp) add: less_not_refl3 le_add1 le_add2 less_Suc_eq_le)

lemma Nonce_supply: "Nonce (@ N. Nonce N ∉ used evs) ∉ used evs"
apply (rule Nonce_supply_lemma [THEN exE])
apply (rule someI, blast)

text{*Unlike the corresponding property of nonces, we cannot prove
@{term "finite KK ==> ∃K. K ∉ KK & Key K ∉ used evs"}.
We have infinitely many agents and there is nothing to stop their
long-term keys from exhausting all the natural numbers. Instead,
possibility theorems must assume the existence of a few keys.*}

subsection{*Specialized Rewriting for Theorems About @{term analz} and Image*}

lemma subset_Compl_range: "A <= - (range shrK) ==> shrK x ∉ A"
by blast

lemma insert_Key_singleton: "insert (Key K) H = Key ` {K} ∪ H"
by blast

lemma insert_Key_image: "insert (Key K) (Key`KK ∪ C) = Key`(insert K KK) ∪ C"
by blast

(** Reverse the normal simplification of "image" to build up (not break down)
the set of keys. Use analz_insert_eq with (Un_upper2 RS analz_mono) to
erase occurrences of forwarded message components (X). **)

lemmas analz_image_freshK_simps =
simp_thms mem_simps --{*these two allow its use with @{text "only:"}*}
image_insert [THEN sym] image_Un [THEN sym] empty_subsetI insert_subset
analz_insert_eq Un_upper2 [THEN analz_mono, THEN [2] rev_subsetD]
insert_Key_singleton subset_Compl_range
Key_not_used insert_Key_image Un_assoc [THEN sym]

(*Lemma for the trivial direction of the if-and-only-if*)
lemma analz_image_freshK_lemma:
"(Key K ∈ analz (Key`nE ∪ H)) --> (K ∈ nE | Key K ∈ analz H) ==>
(Key K ∈ analz (Key`nE ∪ H)) = (K ∈ nE | Key K ∈ analz H)"

by (blast intro: analz_mono [THEN [2] rev_subsetD])

subsection{*Tactics for possibility theorems*}

structure Shared =

(*Omitting used_Says makes the tactic much faster: it leaves expressions
such as Nonce ?N ∉ used evs that match Nonce_supply*)
fun possibility_tac ctxt =
(ALLGOALS (simp_tac (ctxt
delsimps [@{thm used_Says}, @{thm used_Notes}, @{thm used_Gets}]
setSolver safe_solver))
REPEAT_FIRST (eq_assume_tac ORELSE'
resolve_tac [refl, conjI, @{thm Nonce_supply}])))

(*For harder protocols (such as Recur) where we have to set up some
nonces and keys initially*)
fun basic_possibility_tac ctxt =
(ALLGOALS (asm_simp_tac (ctxt setSolver safe_solver))
REPEAT_FIRST (resolve_tac [refl, conjI]))

val analz_image_freshK_ss =
(@{context} delsimps [image_insert, image_Un]
delsimps [@{thm imp_disjL}] (*reduces blow-up*)
addsimps @{thms analz_image_freshK_simps})


(*Lets blast_tac perform this step without needing the simplifier*)
lemma invKey_shrK_iff [iff]:
"(Key (invKey K) ∈ X) = (Key K ∈ X)"
by auto

(*Specialized methods*)

method_setup analz_freshK = {*
Scan.succeed (fn ctxt =>
(EVERY [REPEAT_FIRST (resolve_tac [allI, ballI, impI]),
REPEAT_FIRST (rtac @{thm analz_image_freshK_lemma}),
ALLGOALS (asm_simp_tac (put_simpset Shared.analz_image_freshK_ss ctxt))]))) *}

"for proving the Session Key Compromise theorem"

method_setup possibility = {*
Scan.succeed (fn ctxt => SIMPLE_METHOD (Shared.possibility_tac ctxt)) *}

"for proving possibility theorems"

method_setup basic_possibility = {*
Scan.succeed (fn ctxt => SIMPLE_METHOD (Shared.basic_possibility_tac ctxt)) *}

"for proving possibility theorems"

lemma knows_subset_knows_Cons: "knows A evs <= knows A (e # evs)"
by (cases e) (auto simp: knows_Cons)