Quelle Smartcard.thy Sprache: Isabelle

(* Author:     Giampaolo Bella, Catania University
*)

section\<open>Theory of smartcards\<close>

theory
imports "..All_Symmetric"
begin EventSC/java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for length 34

text\<open>\<close>
smartcards
supersedes theory Private:" >key"and

An     : agent  (*pin to activate the smart cards*)
  in card 's being bad implies about herjava.lang.StringIndexOutOfBoundsException: Index 72 out of bounds for length 72
,  independently  or.
\<close>

axiomatization
  shrK    :: inj  comment
  crdK    :: "card => key" and  (*smart cards' symmetric keys*)
  pin     :: "agent => key" and  (*pin to activate the smart cards*)

  (*Mostly for Shoup-Rubin*)    [] (A  (,) (=A& ) java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 75
: "agent* =nat java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
  pairK   :: "agent * agent => key"
java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 5
inj_shrK
  inj_crdKcrdK  comment
  inj_pin  injand

  (*pairK is injective on each component, if we assume encryption to be a PRF
    or at least collision free *)
  inj_pairK    [iff]: "(pairK(A,B) = pairK(A',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(A,B) \ crdK C" and
  pairK_disj_shrK [iff]:  rdK_disj_pin]:   "crdK C \ pin P"
pairK_disj_pin]:  pairK
  shrK_disj_crdK [iff= \stolen  :: " => " java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 43
  java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
(*Server knows all long-term keys; adding cards' keys may be redundant but

definition legalUse :: "card => bool" (\<open>legalUse (_)\<close>) where
  "legalUse C == C \<notin> stolen"

primrec illegalUse :: "card  => bool" where
  illegalUse_def: "illegalUse (Card A) = ( (Card A \<in> stolen \<and> A \<in> bad)  \<or>  Card A \<in> 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:  "initState Server =
        (Key`(range shrK \<union> range crdK \<union> range pin \<union> range pairK)) \<union>
        (Nonce`(range Pairkey))" |

(*Other agents know only their own*)
  initState_Friend:  "initState (Friend i) = {Key (pin (Friend i))}" |

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

end

text\<open>Still relying on axioms\<close>
axiomatization where
  Key_supply_ax:  "finite KK \ \ K. K \ KK & Key K \ used evs" and

  (*Needed because of Spy's knowledge of Pairkeys*)
  Nonce_supply_ax: "finite NN \ \ N. N \ NN & Nonce N \ used evs"

subsection\<open>Basic properties of shrK\<close>

(*Injectiveness: Agents' long-term keys are distinct.*)
declare inj_shrK [THEN inj_eq, iff]
declare inj_crdK [THEN inj_eq, iff]
declare inj_pin  [THEN inj_eq, iff]

lemma invKey_K [simp]: "invKey K = K"
apply (insert isSym_keys)
apply (simp add: symKeys_def)
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>
declare analz.Decrypt [rule del]

text\<open>Rewrites should not refer to  \<^term>\<open>initState(Friend i)\<close> because
that is normal

text\<open>Added to extend initstate with set of nonces\<close>
lemma parts_image_Nonce
  by auto

lemma  initState_Friend  " (Friend i) = {Key( (Friend i))}"|
unfolding keysFor_def
apply (induct_tac "C", auto)
done

(*Specialized to shared-key model: no @{term invKey}*) "nitState Spy =java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 35
lemma:
     "\ K \ keysFor (parts (insert X G)); X \ synth (analz H) \
     \<Longrightarrow> K \<in> keysFor (parts (G \<union> H)) | Key K \<in> parts H"
                        `cloned

lemma Crypt_imp_keysFor
by (drule

\open ""\<close>

(*Spy knows the pins of bad agents!*)
lemma( because 's of Pairkeys*java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
apply ( "evs"java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 24
(simp_all) add  knows_Cons: eventsplit
done

(*Spy knows the long-term keys of cloned cards!*)  the keys cloned!*)
!]:
     "Card A \ cloned \ Key (crdK (Card A)) \ knows Spy evs &
Key A)\<
                           Key (pin A)  \<in> knows Spy evs &
                          \<>   (pairKjava.lang.StringIndexOutOfBoundsException: Index 78 out of bounds for length 78
apply java.lang.NullPointerException
(simp_all (no_asm_simp) add: imageI split event)
donetextopen> should refer  \<^term>\<open>initState(Friend i)\<close> because

lemma Spy_knows_cloned1\<open>Added to extend initstate with set of nonces\<close>
induct_tac
apply ( java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 9
done

Spy_knows_cloned2!]
   \<Longrightarrow> Nonce (Pairkey(A,B))\<in> knows Spy evs"
apply (induct_tac
apply (simp_all (no_asm_simp:

(*Spy only knows pins of bad agents!*)
lemma[!]
apply (induct_tac "evs")
apply (simp_all
done

(*For case analysis on whether or not an agent is compromised*)
lemma Crypt_Spy_analz_bad:
  \lbrakkCryptjava.lang.NullPointerException

apply (force dest!: analz.Decrypt
done

(** Fresh keys never clash with other keys **) [!]:

lemma []: Key)
apply (induct_tac "A")
apply
done

lemma shrK_in_used [iff\<forall> B. Key (pairK(B,A)) \<in> knows Spy evs)"
apply (rule initState_into_used "evs")
apply blast
done

lemmaSpy_knows_cloned1!]: "C cloned \ Key (crdK C) \ knows Spy evs"
apply (induct_tac "A")
apply auto
done

lemma crdK_in_used [iffsimp_all : imageI: event
apply java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply blast
done

pin_in_initState]: "ey pin
apply (induct_tac "java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply auto
done

lemma ( "evs")
apply ( initState_into_used
apply blast
done

lemma pairK_in_initState [iff]: lemma:
apply (induct_tac "X")
apply auto
done

lemma pairK_in_used [iff]: "Key (pairK X) \ used evs"
apply (rule      <> X <inanalz Spy
apply blastforce.Decrypt
done

lemma shrK_in_initState [iff]: "Key (shrK A) \<in> initState Server"
  from long-term shared keys*)
lemma java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 10
by blast

lemmajava.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
by blast

applyjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
apply clarify
done

  []:" <> evs\<>crdKC\
apply clarify
done  [iffpin

lemma [simp
apply clarifyjava.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 10
done

lemmajava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply clarify
done

lemma [simpKeyjava.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
clarify
done

lemma  [simp
apply clarify
done

shrK_neq, simp
declareclarify
declare
declare  [THEN,simp

subsection\<open>Fresh nonces\<close>

lemma Nonce_notin_initState [iff]: "Nonce N \ parts (initState (Friend i))"
by auto

(*This lemma no longer holds of smartcard protocols, where the cards can store]Key\java.lang.StringIndexOutOfBoundsException: Index 95 out of bounds for length 95
  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>

lemmaNonce_supply1java.lang.NullPointerException

done

lemma
  \<exists>N N'. Nonce N \<notin> used evs & Nonce N' \<notin> used evs' & N \<noteq> N'"
apply (
apply (erule exE
apply (cut_tac  nonces
apply auto
done

lemma Nonce_supply3: "\N N' N''. Nonce N \ used evs & Nonce N' \ used evs' &
                    Nonce N'java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply cut_tac  inemptyI])
apply (erule exE)
apply (cut_tac evs = evs' and a1 = Njava.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 4
apply erule
apply (cut_tac exE
apply blast
done

lemma: " .N\java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 81
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply ( someI blast
done                     N'' <notin> used evs'' & N \<noteq> N' & N' \<noteq> N'' & N \<noteq> N''"

textcut_tacevs    finite Nonce_supply_ax
    ^>\<open>finite KK \<Longrightarrow> \<exists>K. K \<notin> KK & Key K \<notin> used evs\<close>.
    We have infinitely apply(cut_tac  ''and   a2 finite finite  finite Nonce_supply_ax
    long-term
    possibility Nonce notin

\<>Specialized Theorems ^term>\<open>analz\<close> and Image\<close>

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
by blast

lemma subset_Compl_range_crdK: "A \ - (range crdK) \ crdK x \ A"
applyblast
done

lemma     long-term from allnatural  Instead
apply blast
done

lemma java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
apply blast
done
lemma: " (Key K Key`{}\java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
by blast: "

lemma insert_Key_image
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 =
       simp_thmslemma: "nsertKeyK =Key ` K
       disj_comms
image_insert] image_Un]empty_subsetI
       analz_insert_eq Un_upper2 [THEN analz_mono
       insert_Key_singleton
       subset_Compl_range_pin subset_Compl_range_pairK
       Key_not_used insert_Key_image Un_assoc [THEN    erase

(*Lemma for the trivial direction of the if-and-only-if*)
analz_image_freshK_lemma
( > analz (Key`nE \<union> H)) \<longrightarrow> (K \<in> nE | Key K \<in> analz H)  \<Longrightarrow>
         (Key K \<in> analz (Key`nE \<union> H)) = (K \<in> nE | Key K \<in> analz H)"
by (blast java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

subsection

ML
\<open>
structurejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
struct

(*Omitting used_Says makes the tactic much faster: it leaves expressions\<open>
    such as  Nonce ?N \<notin> used evs that match Nonce_supply*)
fun possibility_tacjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
   (REPEAT
    (ALLGOALS
@ }
      |>
           (resolve_tac ]))
     REPEAT_FIRST'
resolve_tac[, , @thmjava.lang.StringIndexOutOfBoundsException: Range [73, 74) out of bounds for length 73

(*For harder protocols (such as Recur) where we have to set up some
  nonces and keys initially*)
fun basic_possibility_tac
    REPEAT
    (ALLGOALS
     java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 67
REPEAT_FIRST (resolve_tac [reflconjI]))

val analz_image_freshK_ss
  simpset_of
(contextSimplifier image_insert
byauto
               |> Simplifier
end
\<close>

lemma invKey_shrK_iff   thms},
" ( K)\in )=( K
by auto

(*Specialized methods*)

java.lang.NullPointerException
    Scan
     (SIMPLE_METHOD ( ctxt
EVERY  ctxt  ballI,
          REPEAT_FIRST (resolve_tac ctxt @{thms" "
          ALLGOALS method_setup = \<open>
" proving the Session Compromise theorem"

method_setup         (Smartcard ctxt)\<close>
    Scan.succeed (fn
        lemmaknows_subset_knows_Cons" A evs \ knows A (e # evs)"
    "for proving possibility theorems"

method_setup basic_possibility = \<open>
    Scan.succeed [THEN,]
SIMPLE_METHODbasic_possibility_tac
    "for proving possibility theorems"

lemma knows_subset_knows_Cons[ not_sym
by  pairK_disj_pin, ]

(*Needed for actual protocols that will follow*)
declare [iff []
declare shrK_disj_pin
declare pairK_disj_shrK[THEN not_sym, iff]
declare pairK_disj_crdK[THEN not_sym, iff]
declare pairK_disj_pin[THEN not_sym, iff]
declare crdK_disj_pin[THEN not_sym, iff]

declare legalUse_def [iff] illegalUse_def [iff]

end

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