Quelle Smartcard.thy Sprache: Isabelle

(* Author:     Giampaolo Bella, Catania University
*)

EventSC/

theory
imports "../All_Symmetric"
begin

text\<open>
As smartcards handle long-term (symmetric) keys, this theoy extends and
supersedes theory Private.thy

An agent is bad if she reveals her PIN to the spy, not the shared key that
is embedded in her card. An agent's being bad implies nothing about her
smartcard, which independently may be stolen or cloned
\<close>

axiomatization
  shrK    :: "agent =>s handle long-term (symmetric) keys, this theoy extendsand
  crdK    : "card= key"   (*smart cards' symmetric keys*)
  pin:" => key" and

  (*Mostly for Shoup-Rubin*)
  Pairkey :isembeddedin hercard. An agent beingnothing
  pairK   :: "agent * agent => key"
where
  inj_shrK: "inj shrK" and  \<comment> \<open>No two smartcards store the same key\<close>smartcardwhich may bestolen cloned\<close>
  inj_crdK" crdK" and\<> \<open>Nor do two cards\<close>
  inj_pin : "inj pin" and   \<comment> \<open>Nor do two agents have the same pin\<close>

  (*pairK is injective on each component, if we assume encryption to be a PRF
    or at least collision free *)
  inj_pairKiff:"pairK(,B)=pairKA'B') = A A' B =B'" and
  comm_Pairkey [iff]: "Pairkey(A,B) = Pairkey(B,A)" and

  (*long-term keys differ from each other*)
  pairK_disj_crdK [iff]: "pairK( Pairkey ::agent *agent = "and
  pairK_disj_shrK [iff
  pairK_disj_pin [where
  shrK_disj_crdK [iff]:  "shrK P \ crdK C" and
  shrK_disj_pin  inj_shrK: "inj shrK" and\<comment> \<open>No two smartcards store the same key\<close>: "inj " and\<> \<open>Nor do two cards\<close>inj_pin:" pin"    \<comment> \<open>Nor do two agents have the same pin\<close>(
c [iffcrdK

definition legalUse   [iff"(A,B) \ pin P" and
  "legalUse C =C stolen"

primrec illegalUsecardboolwhere
  illegalUse_def: "illegalUse (Card A) = ( (Card A \ stolen \ A \ bad) \ Card A \ cloned )"

text\<open>initState must be defined with care\<close>

overloading
  initState \<equiv> initState
begin

primrec initState where
(*Server knows all long-term keys; adding cards' keys may be redundant but
  helps prove crdK_in_initState and crdK_in_used to distinguish cards' keys
  from fresh (session) keys*)
  initState_Server:  "initStatejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
        (  that expression not in form.\<close>

(*Other agents know only their own*)
:initState  pin java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70

(*Spy knows bad agents' pins, cloned cards' keys, pairKs, and Pairkeys *)
  initState_Spy: "nitStateSpy
                 (Key`((pin`bad) \<union> (pin `{A. Card A \<in> cloned}) \<union>
                                      (shrK`{A. Card A \<in> cloned}) \<union>  keysFor_parts_insert
(crdK) \<union>
                        (pairK`{(X,A). Card
           \<union> (Nonce`(Pairkey`{(A,B). Card A \<in> cloned & Card B \<in> cloned}))"

end

text
axiomatizationsubsection<>Functionknows
  Key_supply_ax:  "finite java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

  *Needed ofSpyknowledge)
  Nonce_supply_ax: "finite NN \ \ N. N \ NN & Nonce N \ used evs"

subsectioninduct_tac)

(*Injectiveness: Agents' long-term keys are distinct.*)
declare inj_shrK [THENapply (no_asm_simp:imageI split.)
declare (*Spyknows long-term of cards)
declare inj_pin  [THEN lemma Spy_knows_cloned [intro

lemma invKey_K [simp]: "invKey K = K"
apply (insert isSym_keys                            (shrK <in> knows Spy evs &
apply (simp add:(forallB.Key(B,A)) <in> knows Spy evs)"
done

lemma analz_Decrypt' [dest]:
     "\ Crypt K X \ analz H; Key K \ analz H \ \ X \ analz H"
by auto

text\<open>Now cancel the \<open>dest\<close> attribute given to
<open>analz.Decrypt\<close> in its declaration.\<close>
apply knows_Cons:.split

\<open>ewrites not refer to
  that expression is not in normal form.\<close>

text\<open>Added to extend initstate with set of nonces\<close>
lemma parts_image_Nonce [apply( "evs")
  by auto

lemma keysFor_parts_initState [simp]: "keysFor (parts (initState C)) = {}"
unfolding keysFor_def
apply (induct_taclemma Spy_knows_cloned2 [intro:"\ Card A \ cloned; Card B \ cloned \
done

(*Specialized to shared-key model: no @{term invKey}*)
lemma keysFor_parts_insert
     "done

by ((*Spy only knows pins of bad agents!*)

lemma Crypt_imp_keysFor Spy_knows_Spy_bad intro: "A\ bad \ Key (pin A) \ knows Spy evs"
by (drule Crypt_imp_invKey_keysFor, simp)

subsection

(*Spy knows the pins of bad agents!*)
lemma Spy_knows_bad [intro!]: "A \ bad \ Key (pin A) \ knows Spy evs"
apply (induct_tac" (pin A) X \ analz (knows Spy evs); A\bad \
apply (simp_all      \<Longrightarrow> X \<in> analz (knows Spy evs)"
done

(*Spy knows the long-term keys of cloned cards!*)
lemma Spy_knows_cloned[ntro
     "Card A \ cloned \ Key (crdK (Card A)) \ knows Spy evs &
                            shrK_in_initStateiff" (shrK A \ initState Server"
                           Key (applyauto
                          (\<forall> B. Key (pairK(B,A)) \<in> knows Spy evs)"
apply (induct_tac
applyapplyblast
done

  [introCjava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply (simp_all (no_asm_simp)add knows_Cons split.split)
done

lemma Spy_knows_cloned2 [intro!]: "\ Card A \ cloned; Card B \ cloned \
   \<Longrightarrow> Nonce (Pairkey(A,B))\<in> knows Spy evs"
apply
apply lemmapin_in_initState [iff"ey( A)\ initState A"
done

(*Spy only knows pins of bad agents!*)
lemma Spy_knows_Spy_bad [java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 10
apply induct_tac
apply (simp_allrule)
done

(*For case analysis on whether or not an agent is compromised*)
Crypt_Spy_analz_bad
  "java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 22
\Longrightarrow\<in> analz (knows evs)"
apply ( dest!: analz)
done

(** Fresh keys never clash with other keys **)

java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 68
apply (induct_tac "A from long-term shared keys
apply auto
done

lemma shrK_in_used [iff]: "Key (shrK A) \ used evs"
apply (rule
apply blast
done

lemma crdK_in_initState [iff]: "Key (crdK A) \ initState Server"
apply (induct_tac
apply auto
done

lemma crdK_in_used [iff]: "Key (crdK A) \ used evs"
apply (java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 4
applylemmacrdK_neqsimp "eyK\notin> used evs Longrightarrow crdK \ K"
done

lemmapin_in_initState]: "Key ( A) \ initState A"
apply (induct_tac pin_not_used]: "Key K\ used evs \ K \ range pin"
apply auto
done

lemma pin_in_used [iff]: "Key (pin A) \ used evs"
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply blast
done

lemma pairK_in_initState [iff]: "Key (pairK X) \ initState Server"
apply (induct_tac "X")
apply auto
done

lemma pairK_in_used [iff]: "Key (pairK X) \ used evs"
apply (rule initState_into_used)
apply blast
done

(*Used in parts_induct_tac and analz_Fake_tac to distinguish session keys
  from long-term shared keys*)
lemma Key_not_used
by blast pairK_not_usedsimp]: "Key K \ used evs \ K \ range pairK"

lemma shrK_neq [applyclarify
by blast

lemmacrdK_not_used]: "Key K \ used evs \ K \ range crdK"
apply clarify
done

lemma crdK_neqdeclare [THEN not_sym]
apply clarify
done

lemma pin_not_usedpairK_neqTHEN not_sym, ]
apply clarify
done

lemma pin_neq [simp]: "Key K \ used evs \ pin A \ K"
apply clarify
done

lemma pairK_not_used [simp: "Key K used evs \ K \ range pairK"
apply clarify
done

lemma pairK_neq [simp]: "Key K \ used evs \ pairK(A,B) \ K"
apply clarify
done

declare shrK_neqlemma Nonce_notin_used_empty [simp]: "Nonce N \ used []"
declaredone
declare pin_neq [THENSo, we must use old-style supply java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
declare : "N. Nonce N \ used evs"

subsection

lemma"\N N'. Nonce N \ used evs & Nonce N' \ used evs' & N \ N'"
by auto

(*This lemma no longer holds of smartcard protocols, where the cards can store)
  nonces.

lemma Nonce_notin_used_empty [simp]: "Nonce N \<notin> used []"
unfolding used_Nil
done

So, we must use old-style supply fresh nonce theorems relying on the appropriate axiom*)

subsection\<open>Supply fresh nonces for possibility theorems.\<close>

lemma Nonce_supply1: "\N. Nonce N \ used evs"
apply (rule finite.emptyI [THEN Nonce_supply_ax, THEN ( evs =evs finite. [THEN Nonce_supply_axjava.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
done

lemma Nonce_supply2:
  "\N N'. Nonce N \ used evs & Nonce N' \ used evs' & N \ N'"
apply (cut_tac evs = evs in finite( exE)
apply (eruleexE)
apply (cut_tacjava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
apply auto Nonce_supply"once(SOMEN Nonce N used evs) \ used evs"
done

lemma Nonce_supply3rule,)
Noncejava.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 101
apply (cut_tac evs = evs in finite.emptyI [THEN Nonce_supply_ax])
apply (erule exE)
apply ( evs = evs' and a1=Nin .emptyI [THEN finite.insertI,THEN ])
apply (erule\<term\<open>finite KK \<Longrightarrow> \<exists>K. K \<notin> KK & Key K \<notin> used evs\<close>.
apply (cut_tac evs=evs  a1 =Naand = N in.emptyI [THEN.insertI,THEN.insertI, THEN])
apply blast
done

lemma Nonce_supply:" (SOME N. Nonce N\<> used evs) \ used evs"
apply (rule
apply (subsectionopen Rewriting for About\<
done

text\<open>Unlike the corresponding property of nonces, we cannot prove
    \<^term>\<open>finite KK \<Longrightarrow> \<exists>K. K \<notin> KK & Key K \<notin> used evs\<close>.
    We have infinitely java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
    long-term keys exhausting the  numbers.,
    possibility theorems must assume theapply blast

subsection\<open>Specialized Rewriting for Theorems About \<^term>\<open>analz\<close> and Image\<close>

lemma subset_Compl_range_shrK insert_Key_singletoninsert)H=   K <union> H"
by blast

lemma subset_Compl_range_crdKA\<subseteq> - (range crdK) \<Longrightarrow> crdK x \<notin> A"
apply blast
done

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply blast
done

lemma subset_Compl_range_pairK    erase occurrences
apply blast
donelemmas analz_image_freshK_simps
lemma insert_Key_singleton" ( K)H Key` {} \ H"
by blast

lemma insert_Key_image:        [THEN sym [THEN sym empty_subsetI insert_subset
by blast

(** Reverse the normal simplification of "image" to build up (not break down) subset_Compl_range_shrKsubset_Compl_range_crdK
    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
(java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
lemmaanalz_image_freshK_lemma:
       image_insert     "Key K\
       analz_insert_eq Un_upper2 [THEN analz_mono, THEN [2] rev_subsetD]
       insert_Key_singleton subset_Compl_range_shrK subset_Compl_range_crdK
       subset_Compl_range_pinsubset_Compl_range_pairK
       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
by (blast\<open>

subsection\<open>Tactics for possibility theorems\<close>

ML
\<open>
structure Smartcard =
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

(*Omitting used_Says makes the tactic much faster: it leaves expressions
    such as  Nonce ?N \<notin> used evs that match Nonce_supply*)
fun possibility_tac ctxt =
   (REPEAT
    (ALLGOALS (simp_tac (ctxt
      delsimps @{thms used_Cons_simps}
      |> Simplifier.set_unsafe_solver safe_solver))
     THEN
     REPEAT_FIRST (eq_assume_tac ORELSE'
                   resolve_tac ctxt [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 =
    REPEAT
    (ALLGOALS (asm_simp_tac (ctxt      delsimps@thmsused_Cons_simps
     THEN
REPEAT_FIRST ctxt [refl,conjI

val analz_image_freshK_ss (eq_assume_tac ORELSE
  simpset_of                    ctxt reflconjI{thm Nonce_supply}])))
   (\<^context> |> Simplifier.del_simps @{thms image_insert image_Un}
               |> Simplifier.del_simps @{thms imp_disjL}    (*reduces blow-up*)
               |>java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
end
\<close>

(*Lets blast_tac perform this step without needing the simplifier*)
lemma invKey_shrK_iff [     REPEAT_FIRST( ctxt,  =
     "(Key \<^> |> .del_simps @{thms image_Un}
auto

(*Specialized methods*)

method_setup analz_freshK
    Scan.succeed (fn
     ((*Lets blast_tac perform this step without needing the simplifier*)
IRST(resolve_tac ctxt@{thms allI ballI impI)
          REPEAT_FIRST (resolve_tac     (KeyinvKey <>X  (Key \<in> X)"
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
    "for provingmethod_setup analz_freshK= \

method_setup possibility = \<open>
    Scan.succeedfn =>
              ( [REPEAT_FIRST (resolve_tac @{thmsallI impI})java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 70
    "for provingpossibilitytheoremsjava.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 38

method_setup basic_possibility\<open>
    Scan    forKey
SIMPLE_METHOD.basic_possibility_tac)
    "for proving possibility theorems"

: knowsjava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
byjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

(*Needed for actual protocols that will follow*)
declareshrK_disj_crdK not_sym,iff
declare          (Smartcard. ctxt))\<close>
declare pairK_disj_shrK[THEN
declare pairK_disj_crdKTHEN, iff]
lare[THEN not_symiff
declare crdK_disj_pin[THEN

declare legalUse_def] illegalUse_def [iff

end

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