header{*Verifying the Needham-Schroeder Public-Key Protocol*}
theory NS_Public_Bad imports Public begin
inductive_set ns_public :: "event list set"
where
Nil: "[] ∈ ns_public"
| Fake: "[|evsf ∈ ns_public; X ∈ synth (analz (spies evsf))|]
==> Says Spy B X # evsf ∈ ns_public"
| NS1: "[|evs1 ∈ ns_public; Nonce NA ∉ used evs1|]
==> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
# evs1 ∈ ns_public"
| 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\<rbrace>)
# evs2 ∈ ns_public"
| 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\<rbrace>) ∈ set evs3|]
==> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 ∈ ns_public"
declare knows_Spy_partsEs [elim]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]
declare image_eq_UN [simp]
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])
by possibility
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
lemma no_nonce_NS1_NS2 [rule_format]:
"evs ∈ ns_public
==> Crypt (pubEK C) \<lbrace>NA', Nonce NA\<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
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)
apply (blast intro!: analz_insertI)+
done
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
lemma A_trusts_NS2_lemma [rule_format]:
"[|A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<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\<rbrace>) ∈ set evs"
apply (erule ns_public.induct)
apply (auto dest: Spy_not_see_NA unique_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\<rbrace>) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) ∈ set evs"
by (blast intro: A_trusts_NS2_lemma)
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)
apply (blast intro!: analz_insertI)
done
lemma unique_NB [dest]:
"[|Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> ∈ parts(spies evs);
Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB\<rbrace> ∈ parts(spies evs);
Nonce NB ∉ analz (spies evs); evs ∈ ns_public|]
==> A=A' ∧ NA=NA'"
apply (erule rev_mp, erule rev_mp, erule rev_mp)
apply (erule ns_public.induct, simp_all)
apply (blast intro!: analz_insertI)+
done
theorem Spy_not_see_NB [dest]:
"[|Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) ∈ set evs;
∀C. Says A C (Crypt (pubEK C) (Nonce NB)) ∉ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Nonce NB ∉ analz (spies evs)"
apply (erule rev_mp, erule rev_mp)
apply (erule ns_public.induct, simp_all, spy_analz)
apply (simp_all add: all_conj_distrib)
apply (blast intro: no_nonce_NS1_NS2)+
done
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\<rbrace>) ∈ set evs -->
(∃C. Says A C (Crypt (pubEK C) (Nonce NB)) ∈ set evs)"
apply (erule ns_public.induct, auto)
by (blast intro: no_nonce_NS1_NS2)+
theorem B_trusts_NS3:
"[|Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) ∈ set evs;
Says A' B (Crypt (pubEK B) (Nonce NB)) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> ∃C. Says A C (Crypt (pubEK C) (Nonce NB)) ∈ set evs"
by (blast intro: B_trusts_NS3_lemma)
lemma "[|A ∉ bad; B ∉ bad; evs ∈ ns_public|]
==> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) ∈ set evs
--> Nonce NB ∉ analz (spies evs)"
apply (erule ns_public.induct, simp_all, spy_analz)
apply blast
apply (blast intro: no_nonce_NS1_NS2)
apply clarify
apply (frule_tac A' = A in
Says_imp_knows_Spy [THEN parts.Inj, THEN unique_NB], auto)
apply (rename_tac C B' evs3)
txt{*This is the attack!
@{subgoals[display,indent=0,margin=65]}
*}
oops
end