(* Title: HOL/SET_Protocol/Public_SET.thy
Author: Giampaolo Bella
Author: Fabio Massacci
Author: Lawrence C Paulson
*)
section\<open>The Public-Key Theory, Modified for SET\<close>
theory Public_SET
imports Event_SET
begin
subsection\<open>Symmetric and Asymmetric Keys\<close>
text\<open>definitions influenced by the wish to assign asymmetric keys
- since the beginning - only to RCA and CAs, namely we need a partial
function on type Agent\<close>
text\<open>The SET specs mention two signature keys for CAs - we only have one\<close>
consts
publicKey :: "[bool, agent] \<Rightarrow> key"
\<comment> \<open>the boolean is TRUE if a signing key\<close>
abbreviation "pubEK == publicKey False"
abbreviation "pubSK == publicKey True"
(*BEWARE!! priEK, priSK DON'T WORK with inj, range, image, etc.*)
abbreviation "priEK A == invKey (pubEK A)"
abbreviation "priSK A == invKey (pubSK A)"
text\<open>By freeness of agents, no two agents have the same key. Since
\<^term>\<open>True\<noteq>False\<close>, no agent has the same signing and encryption keys.\<close>
specification (publicKey)
injective_publicKey:
"publicKey b A = publicKey c A' \<Longrightarrow> b=c \<and> A=A'"
(*<*)
apply (rule exI [of _ "%b A. 2 * nat_of_agent A + (if b then 1 else 0)"])
apply (auto simp add: inj_on_def inj_nat_of_agent [THEN inj_eq] split: agent.split)
apply (drule_tac f="%x. x mod 2" in arg_cong, simp add: mod_Suc)+
(*or this, but presburger won't abstract out the function applications
apply presburger+
*)
done
(*>*)
axiomatization where
(*No private key equals any public key (essential to ensure that private
keys are private!) *)
privateKey_neq_publicKey [iff]:
"invKey (publicKey b A) \<noteq> publicKey b' A'"
declare privateKey_neq_publicKey [THEN not_sym, iff]
subsection\<open>Initial Knowledge\<close>
text\<open>This information is not necessary. Each protocol distributes any needed
certificates, and anyway our proofs require a formalization of the Spy's
knowledge only. However, the initial knowledge is as follows:
All agents know RCA's public keys;
RCA and CAs know their own respective keys;
RCA (has already certified and therefore) knows all CAs public keys;
Spy knows all keys of all bad agents.\<close>
overloading initState \<equiv> "initState"
begin
primrec initState where
(*<*)
initState_CA:
"initState (CA i) =
(if i=0 then Key ` ({priEK RCA, priSK RCA} \<union>
pubEK ` (range CA) \<union> pubSK ` (range CA))
else {Key (priEK (CA i)), Key (priSK (CA i)),
Key (pubEK (CA i)), Key (pubSK (CA i)),
Key (pubEK RCA), Key (pubSK RCA)})"
| initState_Cardholder:
"initState (Cardholder i) =
{Key (priEK (Cardholder i)), Key (priSK (Cardholder i)),
Key (pubEK (Cardholder i)), Key (pubSK (Cardholder i)),
Key (pubEK RCA), Key (pubSK RCA)}"
| initState_Merchant:
"initState (Merchant i) =
{Key (priEK (Merchant i)), Key (priSK (Merchant i)),
Key (pubEK (Merchant i)), Key (pubSK (Merchant i)),
Key (pubEK RCA), Key (pubSK RCA)}"
| initState_PG:
"initState (PG i) =
{Key (priEK (PG i)), Key (priSK (PG i)),
Key (pubEK (PG i)), Key (pubSK (PG i)),
Key (pubEK RCA), Key (pubSK RCA)}"
(*>*)
| initState_Spy:
"initState Spy = Key ` (invKey ` pubEK ` bad \<union>
invKey ` pubSK ` bad \<union>
range pubEK \<union> range pubSK)"
end
text\<open>Injective mapping from agents to PANs: an agent can have only one card\<close>
consts pan :: "agent \<Rightarrow> nat"
specification (pan)
inj_pan: "inj pan"
\<comment> \<open>No two agents have the same PAN\<close>
(*<*)
apply (rule exI [of _ "nat_of_agent"])
apply (simp add: inj_on_def inj_nat_of_agent [THEN inj_eq])
done
(*>*)
declare inj_pan [THEN inj_eq, iff]
consts
XOR :: "nat*nat \<Rightarrow> nat" \<comment> \<open>no properties are assumed of exclusive-or\<close>
subsection\<open>Signature Primitives\<close>
definition
(* Signature = Message + signed Digest *)
sign :: "[key, msg]\<Rightarrow>msg"
where "sign K X = \<lbrace>X, Crypt K (Hash X) \<rbrace>"
definition
(* Signature Only = signed Digest Only *)
signOnly :: "[key, msg]\<Rightarrow>msg"
where "signOnly K X = Crypt K (Hash X)"
definition
(* Signature for Certificates = Message + signed Message *)
signCert :: "[key, msg]\<Rightarrow>msg"
where "signCert K X = \<lbrace>X, Crypt K X \<rbrace>"
definition
(* Certification Authority's Certificate.
Contains agent name, a key, a number specifying the key's target use,
a key to sign the entire certificate.
Should prove if signK=priSK RCA and C=CA i,
then Ka=pubEK i or pubSK i depending on T ??
*)
cert :: "[agent, key, msg, key] \<Rightarrow> msg"
where "cert A Ka T signK = signCert signK \<lbrace>Agent A, Key Ka, T\<rbrace>"
definition
(* Cardholder's Certificate.
Contains a PAN, the certified key Ka, the PANSecret PS,
a number specifying the target use for Ka, the signing key signK.
*)
certC :: "[nat, key, nat, msg, key] \<Rightarrow> msg"
where "certC PAN Ka PS T signK =
signCert signK \<lbrace>Hash \<lbrace>Nonce PS, Pan PAN\<rbrace>, Key Ka, T\<rbrace>"
(*cert and certA have no repeated elements, so they could be abbreviations,
but that's tricky and makes proofs slower*)
abbreviation "onlyEnc == Number 0"
abbreviation "onlySig == Number (Suc 0)"
abbreviation "authCode == Number (Suc (Suc 0))"
subsection\<open>Encryption Primitives\<close>
definition EXcrypt :: "[key,key,msg,msg] \<Rightarrow> msg" where
\<comment> \<open>Extra Encryption\<close>
(*K: the symmetric key EK: the public encryption key*)
"EXcrypt K EK M m =
\<lbrace>Crypt K \<lbrace>M, Hash m\<rbrace>, Crypt EK \<lbrace>Key K, m\<rbrace>\<rbrace>"
definition EXHcrypt :: "[key,key,msg,msg] \<Rightarrow> msg" where
\<comment> \<open>Extra Encryption with Hashing\<close>
(*K: the symmetric key EK: the public encryption key*)
"EXHcrypt K EK M m =
\<lbrace>Crypt K \<lbrace>M, Hash m\<rbrace>, Crypt EK \<lbrace>Key K, m, Hash M\<rbrace>\<rbrace>"
definition Enc :: "[key,key,key,msg] \<Rightarrow> msg" where
\<comment> \<open>Simple Encapsulation with SIGNATURE\<close>
(*SK: the sender's signing key
K: the symmetric key
EK: the public encryption key*)
"Enc SK K EK M =
\<lbrace>Crypt K (sign SK M), Crypt EK (Key K)\<rbrace>"
definition EncB :: "[key,key,key,msg,msg] \<Rightarrow> msg" where
\<comment> \<open>Encapsulation with Baggage. Keys as above, and baggage b.\<close>
"EncB SK K EK M b =
\<lbrace>Enc SK K EK \<lbrace>M, Hash b\<rbrace>, b\<rbrace>"
subsection\<open>Basic Properties of pubEK, pubSK, priEK and priSK\<close>
lemma publicKey_eq_iff [iff]:
"(publicKey b A = publicKey b' A') = (b=b' \<and> A=A')"
by (blast dest: injective_publicKey)
lemma privateKey_eq_iff [iff]:
"(invKey (publicKey b A) = invKey (publicKey b' A')) = (b=b' \<and> A=A')"
by auto
lemma not_symKeys_publicKey [iff]: "publicKey b A \<notin> symKeys"
by (simp add: symKeys_def)
lemma not_symKeys_privateKey [iff]: "invKey (publicKey b A) \<notin> symKeys"
by (simp add: symKeys_def)
lemma symKeys_invKey_eq [simp]: "K \<in> symKeys \<Longrightarrow> invKey K = K"
by (simp add: symKeys_def)
lemma symKeys_invKey_iff [simp]: "(invKey K \<in> symKeys) = (K \<in> symKeys)"
by (unfold symKeys_def, auto)
text\<open>Can be slow (or even loop) as a simprule\<close>
lemma symKeys_neq_imp_neq: "(K \<in> symKeys) \<noteq> (K' \<in> symKeys) \<Longrightarrow> K \<noteq> K'"
by blast
text\<open>These alternatives to \<open>symKeys_neq_imp_neq\<close> don't seem any better
in practice.\<close>
lemma publicKey_neq_symKey: "K \<in> symKeys \<Longrightarrow> publicKey b A \<noteq> K"
by blast
lemma symKey_neq_publicKey: "K \<in> symKeys \<Longrightarrow> K \<noteq> publicKey b A"
by blast
lemma privateKey_neq_symKey: "K \<in> symKeys \<Longrightarrow> invKey (publicKey b A) \<noteq> K"
by blast
lemma symKey_neq_privateKey: "K \<in> symKeys \<Longrightarrow> K \<noteq> invKey (publicKey b A)"
by blast
lemma analz_symKeys_Decrypt:
"[| Crypt K X \<in> analz H; K \<in> symKeys; Key K \<in> analz H |]
==> X \<in> analz H"
by auto
subsection\<open>"Image" Equations That Hold for Injective Functions\<close>
lemma invKey_image_eq [iff]: "(invKey x \<in> invKey`A) = (x\<in>A)"
by auto
text\<open>holds because invKey is injective\<close>
lemma publicKey_image_eq [iff]:
"(publicKey b A \<in> publicKey c ` AS) = (b=c \<and> A\<in>AS)"
by auto
lemma privateKey_image_eq [iff]:
"(invKey (publicKey b A) \<in> invKey ` publicKey c ` AS) = (b=c \<and> A\<in>AS)"
by auto
lemma privateKey_notin_image_publicKey [iff]:
"invKey (publicKey b A) \<notin> publicKey c ` AS"
by auto
lemma publicKey_notin_image_privateKey [iff]:
"publicKey b A \<notin> invKey ` publicKey c ` AS"
by auto
lemma keysFor_parts_initState [simp]: "keysFor (parts (initState C)) = {}"
apply (simp add: keysFor_def)
apply (induct_tac "C")
apply (auto intro: range_eqI)
done
text\<open>for proving \<open>new_keys_not_used\<close>\<close>
lemma keysFor_parts_insert:
"[| K \<in> keysFor (parts (insert X H)); X \<in> synth (analz H) |]
==> K \<in> keysFor (parts H) | Key (invKey K) \<in> parts H"
by (force dest!:
parts_insert_subset_Un [THEN keysFor_mono, THEN [2] rev_subsetD]
analz_subset_parts [THEN keysFor_mono, THEN [2] rev_subsetD]
intro: analz_into_parts)
lemma Crypt_imp_keysFor [intro]:
"[|K \<in> symKeys; Crypt K X \<in> H|] ==> K \<in> keysFor H"
by (drule Crypt_imp_invKey_keysFor, simp)
text\<open>Agents see their own private keys!\<close>
lemma privateKey_in_initStateCA [iff]:
"Key (invKey (publicKey b A)) \<in> initState A"
by (case_tac "A", auto)
text\<open>Agents see their own public keys!\<close>
lemma publicKey_in_initStateCA [iff]: "Key (publicKey b A) \<in> initState A"
by (case_tac "A", auto)
text\<open>RCA sees CAs' public keys!\<close>
lemma pubK_CA_in_initState_RCA [iff]:
"Key (publicKey b (CA i)) \<in> initState RCA"
by auto
text\<open>Spy knows all public keys\<close>
lemma knows_Spy_pubEK_i [iff]: "Key (publicKey b A) \<in> knows Spy evs"
apply (induct_tac "evs")
apply (simp_all add: imageI knows_Cons split: event.split)
done
declare knows_Spy_pubEK_i [THEN analz.Inj, iff]
(*needed????*)
text\<open>Spy sees private keys of bad agents! [and obviously public keys too]\<close>
lemma knows_Spy_bad_privateKey [intro!]:
"A \<in> bad \<Longrightarrow> Key (invKey (publicKey b A)) \<in> knows Spy evs"
by (rule initState_subset_knows [THEN subsetD], simp)
subsection\<open>Fresh Nonces for Possibility Theorems\<close>
lemma Nonce_notin_initState [iff]: "Nonce N \<notin> parts (initState B)"
by (induct_tac "B", auto)
lemma Nonce_notin_used_empty [simp]: "Nonce N \<notin> used []"
by (simp add: used_Nil)
text\<open>In any trace, there is an upper bound N on the greatest nonce in use.\<close>
lemma Nonce_supply_lemma: "\<exists>N. \<forall>n. N\<le>n \<longrightarrow> Nonce n \<notin> used evs"
apply (induct_tac "evs")
apply (rule_tac x = 0 in exI)
apply (simp_all add: used_Cons split: event.split, safe)
apply (rule msg_Nonce_supply [THEN exE], blast elim!: add_leE)+
done
lemma Nonce_supply1: "\<exists>N. Nonce N \<notin> used evs"
by (rule Nonce_supply_lemma [THEN exE], blast)
lemma Nonce_supply: "Nonce (SOME N. Nonce N \<notin> used evs) \<notin> used evs"
apply (rule Nonce_supply_lemma [THEN exE])
apply (rule someI, fast)
done
subsection\<open>Specialized Methods for Possibility Theorems\<close>
ML
\<open>
(*Tactic for possibility theorems*)
fun possibility_tac ctxt =
REPEAT (*omit used_Says so that Nonces start from different traces!*)
(ALLGOALS (simp_tac (ctxt delsimps [@{thm used_Says}, @{thm used_Notes}]))
THEN
REPEAT_FIRST (eq_assume_tac ORELSE'
resolve_tac ctxt [refl, conjI, @{thm Nonce_supply}]))
(*For harder protocols (such as SET_CR!), where we have to set up some
nonces and keys initially*)
fun basic_possibility_tac ctxt =
REPEAT
(ALLGOALS (asm_simp_tac (ctxt setSolver safe_solver))
THEN
REPEAT_FIRST (resolve_tac ctxt [refl, conjI]))
\<close>
method_setup possibility = \<open>
Scan.succeed (SIMPLE_METHOD o possibility_tac)\<close>
"for proving possibility theorems"
method_setup basic_possibility = \<open>
Scan.succeed (SIMPLE_METHOD o basic_possibility_tac)\<close>
"for proving possibility theorems"
subsection\<open>Specialized Rewriting for Theorems About \<^term>\<open>analz\<close> and Image\<close>
lemma insert_Key_singleton: "insert (Key K) H = Key ` {K} \<union> H"
by blast
lemma insert_Key_image:
"insert (Key K) (Key`KK \<union> C) = Key ` (insert K KK) \<union> C"
by blast
text\<open>Needed for \<open>DK_fresh_not_KeyCryptKey\<close>\<close>
lemma publicKey_in_used [iff]: "Key (publicKey b A) \<in> used evs"
by auto
lemma privateKey_in_used [iff]: "Key (invKey (publicKey b A)) \<in> used evs"
by (blast intro!: initState_into_used)
text\<open>Reverse the normal simplification of "image" to build up (not break down)
the set of keys. Based on \<open>analz_image_freshK_ss\<close>, but simpler.\<close>
lemmas analz_image_keys_simps =
simp_thms mem_simps \<comment> \<open>these two allow its use with \<open>only:\<close>\<close>
image_insert [THEN sym] image_Un [THEN sym]
rangeI symKeys_neq_imp_neq
insert_Key_singleton insert_Key_image Un_assoc [THEN sym]
(*General lemmas proved by Larry*)
subsection\<open>Controlled Unfolding of Abbreviations\<close>
text\<open>A set is expanded only if a relation is applied to it\<close>
lemma def_abbrev_simp_relation:
"A = B \<Longrightarrow> (A \<in> X) = (B \<in> X) \<and>
(u = A) = (u = B) \<and>
(A = u) = (B = u)"
by auto
text\<open>A set is expanded only if one of the given functions is applied to it\<close>
lemma def_abbrev_simp_function:
"A = B
\<Longrightarrow> parts (insert A X) = parts (insert B X) \<and>
analz (insert A X) = analz (insert B X) \<and>
keysFor (insert A X) = keysFor (insert B X)"
by auto
subsubsection\<open>Special Simplification Rules for \<^term>\<open>signCert\<close>\<close>
text\<open>Avoids duplicating X and its components!\<close>
lemma parts_insert_signCert:
"parts (insert (signCert K X) H) =
insert \<lbrace>X, Crypt K X\<rbrace> (parts (insert (Crypt K X) H))"
by (simp add: signCert_def insert_commute [of X])
text\<open>Avoids a case split! [X is always available]\<close>
lemma analz_insert_signCert:
"analz (insert (signCert K X) H) =
insert \<lbrace>X, Crypt K X\<rbrace> (insert (Crypt K X) (analz (insert X H)))"
by (simp add: signCert_def insert_commute [of X])
lemma keysFor_insert_signCert: "keysFor (insert (signCert K X) H) = keysFor H"
by (simp add: signCert_def)
text\<open>Controlled rewrite rules for \<^term>\<open>signCert\<close>, just the definitions
of the others. Encryption primitives are just expanded, despite their huge
redundancy!\<close>
lemmas abbrev_simps [simp] =
parts_insert_signCert analz_insert_signCert keysFor_insert_signCert
sign_def [THEN def_abbrev_simp_relation]
sign_def [THEN def_abbrev_simp_function]
signCert_def [THEN def_abbrev_simp_relation]
signCert_def [THEN def_abbrev_simp_function]
certC_def [THEN def_abbrev_simp_relation]
certC_def [THEN def_abbrev_simp_function]
cert_def [THEN def_abbrev_simp_relation]
cert_def [THEN def_abbrev_simp_function]
EXcrypt_def [THEN def_abbrev_simp_relation]
EXcrypt_def [THEN def_abbrev_simp_function]
EXHcrypt_def [THEN def_abbrev_simp_relation]
EXHcrypt_def [THEN def_abbrev_simp_function]
Enc_def [THEN def_abbrev_simp_relation]
Enc_def [THEN def_abbrev_simp_function]
EncB_def [THEN def_abbrev_simp_relation]
EncB_def [THEN def_abbrev_simp_function]
subsubsection\<open>Elimination Rules for Controlled Rewriting\<close>
lemma Enc_partsE:
"!!R. [|Enc SK K EK M \<in> parts H;
[|Crypt K (sign SK M) \<in> parts H;
Crypt EK (Key K) \<in> parts H|] ==> R|]
==> R"
by (unfold Enc_def, blast)
lemma EncB_partsE:
"!!R. [|EncB SK K EK M b \<in> parts H;
[|Crypt K (sign SK \<lbrace>M, Hash b\<rbrace>) \<in> parts H;
Crypt EK (Key K) \<in> parts H;
b \<in> parts H|] ==> R|]
==> R"
by (unfold EncB_def Enc_def, blast)
lemma EXcrypt_partsE:
"!!R. [|EXcrypt K EK M m \<in> parts H;
[|Crypt K \<lbrace>M, Hash m\<rbrace> \<in> parts H;
Crypt EK \<lbrace>Key K, m\<rbrace> \<in> parts H|] ==> R|]
==> R"
by (unfold EXcrypt_def, blast)
subsection\<open>Lemmas to Simplify Expressions Involving \<^term>\<open>analz\<close>\<close>
lemma analz_knows_absorb:
"Key K \<in> analz (knows Spy evs)
==> analz (Key ` (insert K H) \<union> knows Spy evs) =
analz (Key ` H \<union> knows Spy evs)"
by (simp add: analz_insert_eq Un_upper2 [THEN analz_mono, THEN subsetD])
lemma analz_knows_absorb2:
"Key K \<in> analz (knows Spy evs)
==> analz (Key ` (insert X (insert K H)) \<union> knows Spy evs) =
analz (Key ` (insert X H) \<union> knows Spy evs)"
apply (subst insert_commute)
apply (erule analz_knows_absorb)
done
lemma analz_insert_subset_eq:
"[|X \<in> analz (knows Spy evs); knows Spy evs \<subseteq> H|]
==> analz (insert X H) = analz H"
apply (rule analz_insert_eq)
apply (blast intro: analz_mono [THEN [2] rev_subsetD])
done
lemmas analz_insert_simps =
analz_insert_subset_eq Un_upper2
subset_insertI [THEN [2] subset_trans]
subsection\<open>Freshness Lemmas\<close>
lemma in_parts_Says_imp_used:
"[|Key K \<in> parts {X}; Says A B X \<in> set evs|] ==> Key K \<in> used evs"
by (blast intro: parts_trans dest!: Says_imp_knows_Spy [THEN parts.Inj])
text\<open>A useful rewrite rule with \<^term>\<open>analz_image_keys_simps\<close>\<close>
lemma Crypt_notin_image_Key: "Crypt K X \<notin> Key ` KK"
by auto
lemma fresh_notin_analz_knows_Spy:
"Key K \<notin> used evs \<Longrightarrow> Key K \<notin> analz (knows Spy evs)"
by (auto dest: analz_into_parts)
end