Quelle  Merchant_Registration.thy

(*  Title:      HOL/SET_Protocol/Merchant_Registration.thy
  Author: Giampaolo Bella
  Author: Fabio Massacci
  Author: Lawrence C Paulson
*)

section‹The SET Merchant Registration Protocol›

theory Merchant_Registration
imports Public_SET
begin

text‹Copmpared with Cardholder Reigstration, ‹KeyCryptKey› is not
  needed: no session key encrypts another. Instead we
  prove the "key compromise" theorems for sets KK that contain no private
  encryption keys (🍋‹priEK C›).›


inductive_set
  set_mr :: "event list set"
where

  Nil:    🍋 ‹Initial trace is empty›
           "[] ∈ set_mr"


| Fake:    🍋 ‹The spy MAY say anything he CAN say.›
           "[| evsf ∈ set_mr; X ∈ synth (analz (knows Spy evsf)) |]
            ==> Says Spy B X # evsf ∈ set_mr"
        

| Reception: 🍋 ‹If A sends a message X to B, then B might receive it›
             "[| evsr ∈ set_mr; Says A B X ∈ set evsr |]
              ==> Gets B X # evsr ∈ set_mr"


| SET_MR1: 🍋 ‹RegFormReq: M requires a registration form to a CA›
           "[| evs1 ∈ set_mr; M = Merchant k; Nonce NM1 ∉ used evs1 |]
            ==> Says M (CA i) {Agent M, Nonce NM1} # evs1 ∈ set_mr"


| SET_MR2: 🍋 ‹RegFormRes: CA replies with the registration form and the
  certificates for her keys›
  "[| evs2 ∈ set_mr; Nonce NCA ∉ used evs2;
      Gets (CA i) {Agent M, Nonce NM1} ∈ set evs2 |]
   ==> Says (CA i) M {sign (priSK (CA i)) {Agent M, Nonce NM1, Nonce NCA},
                       cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA),
                       cert (CA i) (pubSK (CA i)) onlySig (priSK RCA) }
         # evs2 ∈ set_mr"

| SET_MR3:
         🍋 ‹CertReq: M submits the key pair to be certified. The Notes
  event allows KM1 to be lost if M is compromised. Piero remarks
  that the agent mentioned inside the signature is not verified to
  correspond to M. As in CR, each Merchant has fixed key pairs. M
  is only optionally required to send NCA back, so M doesn't do so
  in the model›
  "[| evs3 ∈ set_mr; M = Merchant k; Nonce NM2 ∉ used evs3;
      Key KM1 ∉ used evs3; KM1 ∈ symKeys;
      Gets M {sign (invKey SKi) {Agent X, Nonce NM1, Nonce NCA},
               cert (CA i) EKi onlyEnc (priSK RCA),
               cert (CA i) SKi onlySig (priSK RCA) }
        ∈ set evs3;
      Says M (CA i) {Agent M, Nonce NM1} ∈ set evs3 |]
   ==> Says M (CA i)
            {Crypt KM1 (sign (priSK M) {Agent M, Nonce NM2,
                                          Key (pubSK M), Key (pubEK M)}),
              Crypt EKi (Key KM1)}
         # Notes M {Key KM1, Agent (CA i)}
         # evs3 ∈ set_mr"

| SET_MR4:
         🍋 ‹CertRes: CA issues the certificates for merSK and merEK,
  while checking never to have certified the m even
  separately. NOTE: In Cardholder Registration the
  corresponding rule (6) doesn't use the "sign" primitive. "The
  CertRes shall be signed but not encrypted if the EE is a Merchant
  or Payment Gateway."-- Programmer's Guide, page 191.›
    "[| evs4 ∈ set_mr; M = Merchant k;
        merSK ∉ symKeys; merEK ∉ symKeys;
        Notes (CA i) (Key merSK) ∉ set evs4;
        Notes (CA i) (Key merEK) ∉ set evs4;
        Gets (CA i) {Crypt KM1 (sign (invKey merSK)
                                 {Agent M, Nonce NM2, Key merSK, Key merEK}),
                      Crypt (pubEK (CA i)) (Key KM1) }
          ∈ set evs4 |]
    ==> Says (CA i) M {sign (priSK(CA i)) {Agent M, Nonce NM2, Agent(CA i)},
                        cert M merSK onlySig (priSK (CA i)),
                        cert M merEK onlyEnc (priSK (CA i)),
                        cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)}
          # Notes (CA i) (Key merSK)
          # Notes (CA i) (Key merEK)
          # evs4 ∈ set_mr"


text‹Note possibility proofs are missing.›

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

text‹General facts about message reception›
lemma Gets_imp_Says:
     "[| Gets B X ∈ set evs; evs ∈ set_mr |] ==> ∃A. Says A B X ∈ set evs"
apply (erule rev_mp)
apply (erule set_mr.induct, auto)
done

lemma Gets_imp_knows_Spy:
     "[| Gets B X ∈ set evs; evs ∈ set_mr |] ==> X ∈ knows Spy evs"
by (blast dest!: Gets_imp_Says Says_imp_knows_Spy)


declare Gets_imp_knows_Spy [THEN parts.Inj, dest]

subsubsection‹Proofs on keys›

text‹Spy never sees an agent's private keys! (unless it's bad at start)›
lemma Spy_see_private_Key [simp]:
     "evs ∈ set_mr
      ==> (Key(invKey (publicKey b A)) ∈ parts(knows Spy evs)) = (A ∈ bad)"
apply (erule set_mr.induct)
apply (auto dest!: Gets_imp_knows_Spy [THEN parts.Inj])
done

lemma Spy_analz_private_Key [simp]:
     "evs ∈ set_mr ==>
     (Key(invKey (publicKey b A)) ∈ analz(knows Spy evs)) = (A ∈ bad)"
by auto

declare Spy_see_private_Key [THEN [2] rev_iffD1, dest!]
declare Spy_analz_private_Key [THEN [2] rev_iffD1, dest!]

(*This is to state that the signed keys received in step 4
  are into parts - rather than installing sign_def each time.
  Needed in Spy_see_priSK_RCA, Spy_see_priEK and in Spy_see_priSK
 Goal "[|Gets C {Crypt KM1
  (sign K {Agent M, Nonce NM2, Key merSK, Key merEK}), X}
  ∈ set evs; evs ∈ set_mr |]
  ==> Key merSK ∈ parts (knows Spy evs) ∧
  Key merEK ∈ parts (knows Spy evs)"
 by (fast_tac (claset() addss (simpset())) 1);
 qed "signed_keys_in_parts";
???*)

text‹Proofs on certificates -
  they hold, as in CR, because RCA's keys are secure›

lemma Crypt_valid_pubEK:
     "[| Crypt (priSK RCA) {Agent (CA i), Key EKi, onlyEnc}
           ∈ parts (knows Spy evs);
         evs ∈ set_mr |] ==> EKi = pubEK (CA i)"
apply (erule rev_mp)
apply (erule set_mr.induct, auto)
done

lemma certificate_valid_pubEK:
    "[| cert (CA i) EKi onlyEnc (priSK RCA) ∈ parts (knows Spy evs);
        evs ∈ set_mr |]
     ==> EKi = pubEK (CA i)"
apply (unfold cert_def signCert_def)
apply (blast dest!: Crypt_valid_pubEK)
done

lemma Crypt_valid_pubSK:
     "[| Crypt (priSK RCA) {Agent (CA i), Key SKi, onlySig}
           ∈ parts (knows Spy evs);
         evs ∈ set_mr |] ==> SKi = pubSK (CA i)"
apply (erule rev_mp)
apply (erule set_mr.induct, auto)
done

lemma certificate_valid_pubSK:
    "[| cert (CA i) SKi onlySig (priSK RCA) ∈ parts (knows Spy evs);
        evs ∈ set_mr |] ==> SKi = pubSK (CA i)"
apply (unfold cert_def signCert_def)
apply (blast dest!: Crypt_valid_pubSK)
done

lemma Gets_certificate_valid:
     "[| Gets A { X, cert (CA i) EKi onlyEnc (priSK RCA),
                      cert (CA i) SKi onlySig (priSK RCA)} ∈ set evs;
         evs ∈ set_mr |]
      ==> EKi = pubEK (CA i) ∧ SKi = pubSK (CA i)"
by (blast dest: certificate_valid_pubEK certificate_valid_pubSK)


text‹Nobody can have used non-existent keys!›
lemma new_keys_not_used [rule_format,simp]:
     "evs ∈ set_mr
      ==> Key K ∉ used evs ⟶ K ∈ symKeys ⟶
          K ∉ keysFor (parts (knows Spy evs))"
apply (erule set_mr.induct, simp_all)
apply (force dest!: usedI keysFor_parts_insert)  🍋 ‹Fake›
apply force  🍋 ‹Message 2›
apply (blast dest: Gets_certificate_valid)  🍋 ‹Message 3›
apply force  🍋 ‹Message 4›
done


subsubsection‹New Versions: As Above, but Generalized with the Kk Argument›

lemma gen_new_keys_not_used [rule_format]:
     "evs ∈ set_mr
      ==> Key K ∉ used evs ⟶ K ∈ symKeys ⟶
          K ∉ keysFor (parts (Key`KK ∪ knows Spy evs))"
by auto

lemma gen_new_keys_not_analzd:
     "[|Key K ∉ used evs; K ∈ symKeys; evs ∈ set_mr |]
      ==> K ∉ keysFor (analz (Key`KK ∪ knows Spy evs))"
by (blast intro: keysFor_mono [THEN [2] rev_subsetD]
          dest: gen_new_keys_not_used)

lemma analz_Key_image_insert_eq:
     "[|Key K ∉ used evs; K ∈ symKeys; evs ∈ set_mr |]
      ==> analz (Key ` (insert K KK) ∪ knows Spy evs) =
          insert (Key K) (analz (Key ` KK ∪ knows Spy evs))"
by (simp add: gen_new_keys_not_analzd)


lemma Crypt_parts_imp_used:
     "[|Crypt K X ∈ parts (knows Spy evs);
        K ∈ symKeys; evs ∈ set_mr |] ==> Key K ∈ used evs"
apply (rule ccontr)
apply (force dest: new_keys_not_used Crypt_imp_invKey_keysFor)
done

lemma Crypt_analz_imp_used:
     "[|Crypt K X ∈ analz (knows Spy evs);
        K ∈ symKeys; evs ∈ set_mr |] ==> Key K ∈ used evs"
by (blast intro: Crypt_parts_imp_used)

text‹Rewriting rule for private encryption keys. Analogous rewriting rules
 for other keys aren't needed.›

lemma parts_image_priEK:
     "[|Key (priEK (CA i)) ∈ parts (Key`KK ∪ (knows Spy evs));
        evs ∈ set_mr|] ==> priEK (CA i) ∈ KK | CA i ∈ bad"
by auto

text‹trivial proof because (priEK (CA i)) never appears even in (parts evs)›
lemma analz_image_priEK:
     "evs ∈ set_mr ==>
          (Key (priEK (CA i)) ∈ analz (Key`KK ∪ (knows Spy evs))) =
          (priEK (CA i) ∈ KK | CA i ∈ bad)"
by (blast dest!: parts_image_priEK intro: analz_mono [THEN [2] rev_subsetD])


subsection‹Secrecy of Session Keys›

text‹This holds because if (priEK (CA i)) appears in any traffic then it must
  be known to the Spy, by ‹Spy_see_private_Key›\›
lemma merK_neq_priEK:
     "[|Key merK ∉ analz (knows Spy evs);
        Key merK ∈ parts (knows Spy evs);
        evs ∈ set_mr|] ==> merK ≠ priEK C"
by blast

text‹Lemma for message 4: either merK is compromised (when we don't care)
  or else merK hasn't been used to encrypt K.›
lemma msg4_priEK_disj:
     "[|Gets B {Crypt KM1
                       (sign K {Agent M, Nonce NM2, Key merSK, Key merEK}),
                 Y} ∈ set evs;
        evs ∈ set_mr|]
  ==> (Key merSK ∈ analz (knows Spy evs) | merSK ∉ range(λC. priEK C))
   ∧ (Key merEK ∈ analz (knows Spy evs) | merEK ∉ range(λC. priEK C))"
apply (unfold sign_def)
apply (blast dest: merK_neq_priEK)
done


lemma Key_analz_image_Key_lemma:
     "P ⟶ (Key K ∈ analz (Key`KK ∪ H)) ⟶ (K∈KK | Key K ∈ analz H)
      ==>
      P ⟶ (Key K ∈ analz (Key`KK ∪ H)) = (K∈KK | Key K ∈ analz H)"
by (blast intro: analz_mono [THEN [2] rev_subsetD])

lemma symKey_compromise:
     "evs ∈ set_mr ==>
      (∀SK KK. SK ∈ symKeys ⟶ (∀K ∈ KK. K ∉ range(λC. priEK C)) ⟶
               (Key SK ∈ analz (Key`KK ∪ (knows Spy evs))) =
               (SK ∈ KK | Key SK ∈ analz (knows Spy evs)))"
apply (erule set_mr.induct)
apply (safe del: impI intro!: Key_analz_image_Key_lemma [THEN impI])
apply (drule_tac [7] msg4_priEK_disj)
apply (frule_tac [6] Gets_certificate_valid)
apply (safe del: impI)
apply (simp_all del: image_insert image_Un imp_disjL
         add: analz_image_keys_simps abbrev_simps analz_knows_absorb
              analz_knows_absorb2 analz_Key_image_insert_eq notin_image_iff
              Spy_analz_private_Key analz_image_priEK)
  🍋 ‹5 seconds on a 1.6GHz machine›
apply spy_analz  🍋 ‹Fake›
apply auto  🍋 ‹Message 3›
done

lemma symKey_secrecy [rule_format]:
     "[|CA i ∉ bad; K ∈ symKeys; evs ∈ set_mr|]
      ==> ∀X m. Says (Merchant m) (CA i) X ∈ set evs ⟶
                Key K ∈ parts{X} ⟶
                Merchant m ∉ bad ⟶
                Key K ∉ analz (knows Spy evs)"
apply (erule set_mr.induct)
apply (drule_tac [7] msg4_priEK_disj)
apply (frule_tac [6] Gets_certificate_valid)
apply (safe del: impI)
apply (simp_all del: image_insert image_Un imp_disjL
         add: analz_image_keys_simps abbrev_simps analz_knows_absorb
              analz_knows_absorb2 analz_Key_image_insert_eq
              symKey_compromise notin_image_iff Spy_analz_private_Key
              analz_image_priEK)
apply spy_analz  🍋 ‹Fake›
apply force  🍋 ‹Message 1›
apply (auto intro: analz_into_parts [THEN usedI] in_parts_Says_imp_used)  🍋 ‹Message 3›
done

subsection‹Unicity›

lemma msg4_Says_imp_Notes:
 "[|Says (CA i) M {sign (priSK (CA i)) {Agent M, Nonce NM2, Agent (CA i)},
                    cert M merSK onlySig (priSK (CA i)),
                    cert M merEK onlyEnc (priSK (CA i)),
                    cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)} ∈ set evs;
    evs ∈ set_mr |]
  ==> Notes (CA i) (Key merSK) ∈ set evs
   ∧ Notes (CA i) (Key merEK) ∈ set evs"
apply (erule rev_mp)
apply (erule set_mr.induct)
apply (simp_all (no_asm_simp))
done

text‹Unicity of merSK wrt a given CA:
  merSK uniquely identifies the other components, including merEK›
lemma merSK_unicity:
 "[|Says (CA i) M {sign (priSK(CA i)) {Agent M, Nonce NM2, Agent (CA i)},
                    cert M merSK onlySig (priSK (CA i)),
                    cert M merEK onlyEnc (priSK (CA i)),
                    cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)} ∈ set evs;
    Says (CA i) M' {sign (priSK(CA i)) {Agent M', Nonce NM2', Agent (CA i)},
                    cert M' merSK onlySig (priSK (CA i)),
                    cert M' merEK' onlyEnc (priSK (CA i)),
                    cert (CA i) (pubSK(CA i)) onlySig (priSK RCA)} ∈ set evs;
    evs ∈ set_mr |] ==> M=M' ∧ NM2=NM2' ∧ merEK=merEK'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule set_mr.induct)
apply (simp_all (no_asm_simp))
apply (blast dest!: msg4_Says_imp_Notes)
done

text‹Unicity of merEK wrt a given CA:
  merEK uniquely identifies the other components, including merSK›
lemma merEK_unicity:
 "[|Says (CA i) M {sign (priSK(CA i)) {Agent M, Nonce NM2, Agent (CA i)},
                    cert M merSK onlySig (priSK (CA i)),
                    cert M merEK onlyEnc (priSK (CA i)),
                    cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)} ∈ set evs;
    Says (CA i) M' {sign (priSK(CA i)) {Agent M', Nonce NM2', Agent (CA i)},
                     cert M' merSK' onlySig (priSK (CA i)),
                     cert M' merEK onlyEnc (priSK (CA i)),
                     cert (CA i) (pubSK(CA i)) onlySig (priSK RCA)} ∈ set evs;
    evs ∈ set_mr |]
  ==> M=M' ∧ NM2=NM2' ∧ merSK=merSK'"
apply (erule rev_mp)
apply (erule rev_mp)
apply (erule set_mr.induct)
apply (simp_all (no_asm_simp))
apply (blast dest!: msg4_Says_imp_Notes)
done


text‹-No interest on secrecy of nonces: they appear to be used
  only for freshness.
  -No interest on secrecy of merSK or merEK, as in CR.
  -There's no equivalent of the PAN›


subsection‹Primary Goals of Merchant Registration›

subsubsection‹The merchant's certificates really were created by the CA,
 provided the CA is uncompromised›

text‹The assumption 🍋‹CA i ≠ RCA› is required: step 2 uses
  certificates of the same form.›
lemma certificate_merSK_valid_lemma [intro]:
     "[|Crypt (priSK (CA i)) {Agent M, Key merSK, onlySig}
          ∈ parts (knows Spy evs);
        CA i ∉ bad; CA i ≠ RCA; evs ∈ set_mr|]
 ==> ∃X Y Z. Says (CA i) M
                  {X, cert M merSK onlySig (priSK (CA i)), Y, Z} ∈ set evs"
apply (erule rev_mp)
apply (erule set_mr.induct)
apply (simp_all (no_asm_simp))
apply auto
done

lemma certificate_merSK_valid:
     "[| cert M merSK onlySig (priSK (CA i)) ∈ parts (knows Spy evs);
         CA i ∉ bad; CA i ≠ RCA; evs ∈ set_mr|]
 ==> ∃X Y Z. Says (CA i) M
                  {X, cert M merSK onlySig (priSK (CA i)), Y, Z} ∈ set evs"
by auto

lemma certificate_merEK_valid_lemma [intro]:
     "[|Crypt (priSK (CA i)) {Agent M, Key merEK, onlyEnc}
          ∈ parts (knows Spy evs);
        CA i ∉ bad; CA i ≠ RCA; evs ∈ set_mr|]
 ==> ∃X Y Z. Says (CA i) M
                  {X, Y, cert M merEK onlyEnc (priSK (CA i)), Z} ∈ set evs"
apply (erule rev_mp)
apply (erule set_mr.induct)
apply (simp_all (no_asm_simp))
apply auto
done

lemma certificate_merEK_valid:
     "[| cert M merEK onlyEnc (priSK (CA i)) ∈ parts (knows Spy evs);
         CA i ∉ bad; CA i ≠ RCA; evs ∈ set_mr|]
 ==> ∃X Y Z. Says (CA i) M
                  {X, Y, cert M merEK onlyEnc (priSK (CA i)), Z} ∈ set evs"
by auto

text‹The two certificates - for merSK and for merEK - cannot be proved to
  have originated together›

end