--- a/src/HOL/Auth/NS_Public.thy Tue Feb 13 01:32:54 2001 +0100
+++ b/src/HOL/Auth/NS_Public.thy Tue Feb 13 13:16:27 2001 +0100
@@ -7,39 +7,199 @@
Version incorporating Lowe's fix (inclusion of B's identity in round 2).
*)
-NS_Public = Public +
+theory NS_Public = Public:
-consts ns_public :: event list set
+consts ns_public :: "event list set"
inductive ns_public
- intrs
+ intros
(*Initial trace is empty*)
- Nil "[]: ns_public"
+ Nil: "[] \<in> ns_public"
(*The spy MAY say anything he CAN say. We do not expect him to
invent new nonces here, but he can also use NS1. Common to
all similar protocols.*)
- Fake "[| evs: ns_public; X: synth (analz (spies evs)) |]
- ==> Says Spy B X # evs : ns_public"
+ Fake: "\<lbrakk>evs \<in> ns_public; X \<in> synth (analz (spies evs))\<rbrakk>
+ \<Longrightarrow> Says Spy B X # evs \<in> ns_public"
(*Alice initiates a protocol run, sending a nonce to Bob*)
- NS1 "[| evs1: ns_public; Nonce NA ~: used evs1 |]
- ==> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|})
- # evs1 : ns_public"
+ NS1: "\<lbrakk>evs1 \<in> ns_public; Nonce NA \<notin> used evs1\<rbrakk>
+ \<Longrightarrow> Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
+ # evs1 \<in> ns_public"
(*Bob responds to Alice's message with a further nonce*)
- NS2 "[| evs2: ns_public; Nonce NB ~: used evs2;
- Says A' B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set evs2 |]
- ==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|})
- # evs2 : ns_public"
+ NS2: "\<lbrakk>evs2 \<in> ns_public; Nonce NB \<notin> used evs2;
+ Says A' B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
+ \<Longrightarrow> Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
+ # evs2 \<in> ns_public"
(*Alice proves her existence by sending NB back to Bob.*)
- NS3 "[| evs3: ns_public;
- Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) : set evs3;
- Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|})
- : set evs3 |]
- ==> Says A B (Crypt (pubK B) (Nonce NB)) # evs3 : ns_public"
+ NS3: "\<lbrakk>evs3 \<in> ns_public;
+ Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
+ Says B' A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
+ \<in> set evs3\<rbrakk>
+ \<Longrightarrow> Says A B (Crypt (pubK B) (Nonce NB)) # evs3 \<in> ns_public"
+
+ (*No Oops message. Should there be one?*)
+
+declare knows_Spy_partsEs [elim]
+declare analz_subset_parts [THEN subsetD, dest]
+declare Fake_parts_insert [THEN subsetD, dest]
+declare image_eq_UN [simp] (*accelerates proofs involving nested images*)
+
+(*A "possibility property": there are traces that reach the end*)
+lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
+apply (intro exI bexI)
+apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2,
+ THEN ns_public.NS3])
+by possibility
+
+
+(**** Inductive proofs about ns_public ****)
+
+(** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
+ sends messages containing X! **)
+
+(*Spy never sees another agent's private key! (unless it's bad at start)*)
+lemma Spy_see_priK [simp]:
+ "evs \<in> ns_public \<Longrightarrow> (Key (priK A) \<in> parts (spies evs)) = (A \<in> bad)"
+by (erule ns_public.induct, auto)
+
+lemma Spy_analz_priK [simp]:
+ "evs \<in> ns_public \<Longrightarrow> (Key (priK A) \<in> analz (spies evs)) = (A \<in> bad)"
+by auto
+
+
+(*** Authenticity properties obtained from NS2 ***)
+
+
+(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
+ is secret. (Honest users generate fresh nonces.)*)
+lemma no_nonce_NS1_NS2 [rule_format]:
+ "evs \<in> ns_public
+ \<Longrightarrow> Crypt (pubK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+ Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+ Nonce NA \<in> analz (spies evs)"
+apply (erule ns_public.induct, simp_all)
+apply (blast intro: analz_insertI)+
+done
+
+(*Unicity for NS1: nonce NA identifies agents A and B*)
+lemma unique_NA:
+ "\<lbrakk>Crypt(pubK B) \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs);
+ Crypt(pubK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs);
+ Nonce NA \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> A=A' \<and> B=B'"
+apply (erule rev_mp, erule rev_mp, erule rev_mp)
+apply (erule ns_public.induct, simp_all)
+(*Fake, NS1*)
+apply (blast intro: analz_insertI)+
+done
+
+
+(*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure
+ The major premise "Says A B ..." makes it a dest-rule, so we use
+ (erule rev_mp) rather than rule_format. *)
+theorem Spy_not_see_NA:
+ "\<lbrakk>Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> Nonce NA \<notin> analz (spies evs)"
+apply (erule rev_mp)
+apply (erule ns_public.induct, simp_all)
+apply spy_analz
+apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
+done
+
- (**Oops message??**)
+(*Authentication for A: if she receives message 2 and has used NA
+ to start a run, then B has sent message 2.*)
+lemma A_trusts_NS2_lemma [rule_format]:
+ "\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+ Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs \<longrightarrow>
+ Says B A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
+apply (erule ns_public.induct, simp_all)
+(*Fake, NS1*)
+apply (blast dest: Spy_not_see_NA)+
+done
+
+theorem A_trusts_NS2:
+ "\<lbrakk>Says A B (Crypt(pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
+ Says B' A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> Says B A (Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
+by (blast intro: A_trusts_NS2_lemma)
+
+
+(*If the encrypted message appears then it originated with Alice in NS1*)
+lemma B_trusts_NS1 [rule_format]:
+ "evs \<in> ns_public
+ \<Longrightarrow> Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
+ Nonce NA \<notin> analz (spies evs) \<longrightarrow>
+ Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
+apply (erule ns_public.induct, simp_all)
+(*Fake*)
+apply (blast intro!: analz_insertI)
+done
+
+
+
+(*** Authenticity properties obtained from NS2 ***)
+
+(*Unicity for NS2: nonce NB identifies nonce NA and agents A, B
+ [unicity of B makes Lowe's fix work]
+ [proof closely follows that for unique_NA] *)
+
+lemma unique_NB [dest]:
+ "\<lbrakk>Crypt(pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts(spies evs);
+ Crypt(pubK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> \<in> parts(spies evs);
+ Nonce NB \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> A=A' \<and> NA=NA' \<and> B=B'"
+apply (erule rev_mp, erule rev_mp, erule rev_mp)
+apply (erule ns_public.induct, simp_all)
+(*Fake, NS2*)
+apply (blast intro: analz_insertI)+
+done
+
+
+(*Secrecy: Spy does not see the nonce sent in msg NS2 if A and B are secure*)
+theorem Spy_not_see_NB [dest]:
+ "\<lbrakk>Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
+apply (erule rev_mp)
+apply (erule ns_public.induct, simp_all)
+apply spy_analz
+apply (blast intro: no_nonce_NS1_NS2)+
+done
+
+
+(*Authentication for B: if he receives message 3 and has used NB
+ in message 2, then A has sent message 3.*)
+lemma B_trusts_NS3_lemma [rule_format]:
+ "\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow>
+ Crypt (pubK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
+ Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
+ Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
+by (erule ns_public.induct, auto)
+
+theorem B_trusts_NS3:
+ "\<lbrakk>Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
+ Says A' B (Crypt (pubK B) (Nonce NB)) \<in> set evs;
+ A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
+ \<Longrightarrow> Says A B (Crypt (pubK B) (Nonce NB)) \<in> set evs"
+by (blast intro: B_trusts_NS3_lemma)
+
+(*** Overall guarantee for B ***)
+
+
+(*If NS3 has been sent and the nonce NB agrees with the nonce B joined with
+ NA, then A initiated the run using NA.*)
+theorem B_trusts_protocol:
+ "\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk> \<Longrightarrow>
+ Crypt (pubK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
+ Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
+ Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
+by (erule ns_public.induct, auto)
end