(* Title: HOL/Auth/NS_Public_Bad.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1996 University of Cambridge
*)
section\<open>The Needham-Schroeder Public-Key Protocol (Flawed)\<close>
text \<open>Flawed version, vulnerable to Lowe's attack.
From Burrows, Abadi and Needham. A Logic of Authentication.
Proc. Royal Soc. 426 (1989), p. 260\<close>
theory NS_Public_Bad imports Public begin
inductive_set ns_public :: "event list set"
where
Nil: "[] \<in> ns_public"
\<comment> \<open>Initial trace is empty\<close>
| Fake: "\<lbrakk>evsf \<in> ns_public; X \<in> synth (analz (spies evsf))\<rbrakk>
\<Longrightarrow> Says Spy B X # evsf \<in> ns_public"
\<comment> \<open>The spy can say almost anything.\<close>
| NS1: "\<lbrakk>evs1 \<in> ns_public; Nonce NA \<notin> used evs1\<rbrakk>
\<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
# evs1 \<in> ns_public"
\<comment> \<open>Alice initiates a protocol run, sending a nonce to Bob\<close>
| NS2: "\<lbrakk>evs2 \<in> ns_public; Nonce NB \<notin> used evs2;
Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
\<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>)
# evs2 \<in> ns_public"
\<comment> \<open>Bob responds to Alice's message with a further nonce\<close>
| NS3: "\<lbrakk>evs3 \<in> ns_public;
Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk>
\<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 \<in> ns_public"
\<comment> \<open>Alice proves her existence by sending @{term NB} back to Bob.\<close>
declare knows_Spy_partsEs [elim]
declare analz_into_parts [dest]
declare Fake_parts_insert_in_Un [dest]
text \<open>A "possibility property": there are traces that reach the end\<close>
lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubEK 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
subsection \<open>Inductive proofs about @{term ns_public}\<close>
(** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
sends messages containing X! **)
text \<open>Spy never sees another agent's private key! (unless it's bad at start)\<close>
lemma Spy_see_priEK [simp]:
"evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
by (erule ns_public.induct, auto)
lemma Spy_analz_priEK [simp]:
"evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
by auto
subsection \<open>Authenticity properties obtained from {term NS1}\<close>
text \<open>It is impossible to re-use a nonce in both {term NS1} and {term NS2}, provided the nonce
is secret. (Honest users generate fresh nonces.)\<close>
lemma no_nonce_NS1_NS2:
"\<lbrakk>evs \<in> ns_public;
Crypt (pubEK C) \<lbrace>NA', Nonce NA\<rbrace> \<in> parts (spies evs);
Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs)\<rbrakk>
\<Longrightarrow> Nonce NA \<in> analz (spies evs)"
by (induct rule: ns_public.induct) (auto intro: analz_insertI)
text \<open>Unicity for {term NS1}: nonce {term NA} identifies agents {term A} and {term B}\<close>
lemma unique_NA:
assumes NA: "Crypt(pubEK B) \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs)"
"Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs)"
"Nonce NA \<notin> analz (spies evs)"
and evs: "evs \<in> ns_public"
shows "A=A' \<and> B=B'"
using evs NA
by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
text \<open>Secrecy: Spy does not see the nonce sent in msg {term NS1} if {term A} and {term B} are secure
The major premise "Says A B ..." makes it a dest-rule, hence the given assumption order. \<close>
theorem Spy_not_see_NA:
assumes NA: "Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
"A \<notin> bad" "B \<notin> bad"
and evs: "evs \<in> ns_public"
shows "Nonce NA \<notin> analz (spies evs)"
using evs NA
proof (induction rule: ns_public.induct)
case (Fake evsf X B)
then show ?case
by spy_analz
next
case (NS2 evs2 NB A' B NA A)
then show ?case
by simp (metis Says_imp_analz_Spy analz_into_parts parts.simps unique_NA usedI)
next
case (NS3 evs3 A B NA B' NB)
then show ?case
by simp (meson Says_imp_analz_Spy analz_into_parts no_nonce_NS1_NS2)
qed auto
text \<open>Authentication for {term A}: if she receives message 2 and has used {term NA}
to start a run, then {term B} has sent message 2.\<close>
lemma A_trusts_NS2_lemma:
"\<lbrakk>evs \<in> ns_public;
Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts (spies evs);
Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
A \<notin> bad; B \<notin> bad\<rbrakk>
\<Longrightarrow> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
by (induct rule: ns_public.induct) (auto dest: Spy_not_see_NA unique_NA)
theorem A_trusts_NS2:
"\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
Says B' A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
\<Longrightarrow> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
by (blast intro: A_trusts_NS2_lemma)
text \<open>If the encrypted message appears then it originated with Alice in {term NS1}\<close>
lemma B_trusts_NS1:
"\<lbrakk>evs \<in> ns_public;
Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs);
Nonce NA \<notin> analz (spies evs)\<rbrakk>
\<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
by (induct evs rule: ns_public.induct) (use analz_insertI in \<open>auto split: if_split_asm\<close>)
subsection \<open>Authenticity properties obtained from {term NS2}\<close>
text \<open>Unicity for {term NS2}: nonce {term NB} identifies nonce {term NA} and agent {term A}
[proof closely follows that for @{thm [source] unique_NA}]\<close>
lemma unique_NB [dest]:
assumes NB: "Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts(spies evs)"
"Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB\<rbrace> \<in> parts(spies evs)"
"Nonce NB \<notin> analz (spies evs)"
and evs: "evs \<in> ns_public"
shows "A=A' \<and> NA=NA'"
using evs NB
by (induction rule: ns_public.induct) (auto intro!: analz_insertI split: if_split_asm)
text \<open>{term NB} remains secret \emph{provided} Alice never responds with round 3\<close>
theorem Spy_not_see_NB [dest]:
assumes NB: "Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs"
"\<forall>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<notin> set evs"
"A \<notin> bad" "B \<notin> bad"
and evs: "evs \<in> ns_public"
shows "Nonce NB \<notin> analz (spies evs)"
using evs NB evs
proof (induction rule: ns_public.induct)
case Fake
then show ?case by spy_analz
next
case NS2
then show ?case
by (auto intro!: no_nonce_NS1_NS2)
qed auto
text \<open>Authentication for {term B}: if he receives message 3 and has used {term NB}
in message 2, then {term A} has sent message 3 (to somebody) \<close>
lemma B_trusts_NS3_lemma:
"\<lbrakk>evs \<in> ns_public;
Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs);
Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
A \<notin> bad; B \<notin> bad\<rbrakk>
\<Longrightarrow> \<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs"
proof (induction rule: ns_public.induct)
case (NS3 evs3 A B NA B' NB)
then show ?case
by simp (blast intro: no_nonce_NS1_NS2)
qed auto
theorem B_trusts_NS3:
"\<lbrakk>Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
Says A' B (Crypt (pubEK B) (Nonce NB)) \<in> set evs;
A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
\<Longrightarrow> \<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs"
by (blast intro: B_trusts_NS3_lemma)
text \<open>Can we strengthen the secrecy theorem @{thm[source]Spy_not_see_NB}? NO\<close>
lemma "\<lbrakk>evs \<in> ns_public;
Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs;
A \<notin> bad; B \<notin> bad\<rbrakk>
\<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
apply (induction rule: ns_public.induct, simp_all, spy_analz)
(*{term NS1}: by freshness*)
apply blast
(*{term NS2}: by freshness and unicity of {term NB}*)
apply (blast intro: no_nonce_NS1_NS2)
(*{term NS3}: unicity of {term NB} identifies {term A} and {term NA}, but not {term B}*)
apply clarify
apply (frule_tac A' = A in
Says_imp_knows_Spy [THEN parts.Inj, THEN unique_NB], auto)
apply (rename_tac evs3 B' C)
txt\<open>This is the attack!
@{subgoals[display,indent=0,margin=65]}
\<close>
oops
(*
THIS IS THE ATTACK!
Level 8
!!evs. \<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
\<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs \<longrightarrow>
Nonce NB \<notin> analz (spies evs)
1. !!C B' evs3.
\<lbrakk>A \<notin> bad; B \<notin> bad; evs3 \<in> ns_public
Says A C (Crypt (pubEK C) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3;
C \<in> bad;
Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3;
Nonce NB \<notin> analz (spies evs3)\<rbrakk>
\<Longrightarrow> False
*)
end