Theory NS_Public

theory NS_Public
imports Public
(*  Title:      HOL/Auth/NS_Public.thy
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).
*)


header{*Verifying the Needham-Schroeder-Lowe Public-Key Protocol*}

theory NS_Public imports Public begin

inductive_set ns_public :: "event list set"
where
(*Initial trace is empty*)
Nil: "[] ∈ 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: "[|evsf ∈ ns_public; X ∈ synth (analz (spies evsf))|]
==> Says Spy B X # evsf ∈ ns_public"


(*Alice initiates a protocol run, sending a nonce to Bob*)
| NS1: "[|evs1 ∈ ns_public; Nonce NA ∉ used evs1|]
==> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
# evs1 ∈ ns_public"


(*Bob responds to Alice's message with a further nonce*)
| NS2: "[|evs2 ∈ ns_public; Nonce NB ∉ used evs2;
Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs2|]
==> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
# evs2 ∈ ns_public"


(*Alice proves her existence by sending NB back to Bob.*)
| NS3: "[|evs3 ∈ ns_public;
Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs3;
Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
∈ set evs3|]
==> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 ∈ ns_public"


declare knows_Spy_partsEs [elim]
declare knows_Spy_partsEs [elim]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]
declare image_eq_UN [simp] (*accelerates proofs involving nested images*)

(*A "possibility property": there are traces that reach the end*)
lemma "∃NB. ∃evs ∈ ns_public. Says A B (Crypt (pubEK B) (Nonce NB)) ∈ 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], possibility)
done

(** Theorems of the form X ∉ 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_priEK [simp]:
"evs ∈ ns_public ==> (Key (priEK A) ∈ parts (spies evs)) = (A ∈ bad)"
by (erule ns_public.induct, auto)

lemma Spy_analz_priEK [simp]:
"evs ∈ ns_public ==> (Key (priEK A) ∈ analz (spies evs)) = (A ∈ bad)"
by auto

subsection{*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 ∈ ns_public
==> Crypt (pubEK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> ∈ parts (spies evs) -->
Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> ∈ parts (spies evs) -->
Nonce NA ∈ 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:
"[|Crypt(pubEK B) \<lbrace>Nonce NA, Agent A \<rbrace> ∈ parts(spies evs);
Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> ∈ parts(spies evs);
Nonce NA ∉ analz (spies evs); evs ∈ ns_public|]
==> A=A' ∧ 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:
"[|Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Nonce NA ∉ analz (spies evs)"

apply (erule rev_mp)
apply (erule ns_public.induct, simp_all, 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]:
"[|A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> ∈ parts (spies evs) -->
Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs -->
Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs"

apply (erule ns_public.induct, simp_all)
(*Fake, NS1*)
apply (blast dest: Spy_not_see_NA)+
done

theorem A_trusts_NS2:
"[|Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs;
Says B' A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ 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 ∈ ns_public
==> Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> ∈ parts (spies evs) -->
Nonce NA ∉ analz (spies evs) -->
Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs"

apply (erule ns_public.induct, simp_all)
(*Fake*)
apply (blast intro!: analz_insertI)
done


subsection{*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]:
"[|Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> ∈ parts(spies evs);
Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> ∈ parts(spies evs);
Nonce NB ∉ analz (spies evs); evs ∈ ns_public|]
==> A=A' ∧ NA=NA' ∧ 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]:
"[|Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Nonce NB ∉ analz (spies evs)"

apply (erule rev_mp)
apply (erule ns_public.induct, simp_all, 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]:
"[|A ∉ bad; B ∉ bad; evs ∈ ns_public|] ==>
Crypt (pubEK B) (Nonce NB) ∈ parts (spies evs) -->
Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs -->
Says A B (Crypt (pubEK B) (Nonce NB)) ∈ set evs"

by (erule ns_public.induct, auto)

theorem B_trusts_NS3:
"[|Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs;
Says A' B (Crypt (pubEK B) (Nonce NB)) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Says A B (Crypt (pubEK B) (Nonce NB)) ∈ set evs"

by (blast intro: B_trusts_NS3_lemma)

subsection{*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:
"[|A ∉ bad; B ∉ bad; evs ∈ ns_public|] ==>
Crypt (pubEK B) (Nonce NB) ∈ parts (spies evs) -->
Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) ∈ set evs -->
Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) ∈ set evs"

by (erule ns_public.induct, auto)

end