(* Title: HOL/Auth/NS_Public
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
Version incorporating Lowe's fix (inclusion of B's identity in round 2).
*)
theory NS_Public = Public:
consts ns_public :: "event list set"
inductive ns_public
intros
(*Initial trace is empty*)
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: "\<lbrakk>evsf \<in> ns_public; X \<in> synth (analz (spies evsf))\<rbrakk>
\<Longrightarrow> Says Spy B X # evsf \<in> ns_public"
(*Alice initiates a protocol run, sending a nonce to Bob*)
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: "\<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: "\<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"
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
(*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