--- a/src/HOL/Auth/NS_Public_Bad.thy Fri Oct 14 10:30:37 2022 +0100
+++ b/src/HOL/Auth/NS_Public_Bad.thy Fri Oct 14 14:57:28 2022 +0100
@@ -1,78 +1,68 @@
(* Title: HOL/Auth/NS_Public_Bad.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.
-Flawed version, vulnerable to Lowe's attack.
-
-From page 260 of
- Burrows, Abadi and Needham. A Logic of Authentication.
- Proc. Royal Soc. 426 (1989)
*)
section\<open>Verifying the Needham-Schroeder Public-Key Protocol\<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
- (*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.*)
+ 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"
-
- (*Alice initiates a protocol run, sending a nonce to Bob*)
+ \<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"
-
- (*Bob responds to Alice's message with a further nonce*)
+ \<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"
-
- (*Alice proves her existence by sending NB back to Bob.*)
+ \<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]
-(*A "possibility property": there are traces that reach the end*)
+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
+ 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 ****)
+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! **)
-(*Spy never sees another agent's private key! (unless it's bad at start)*)
+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)
+ "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
+ "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
+ by auto
-(*** Authenticity properties obtained from NS2 ***)
+subsection \<open>Authenticity properties obtained from {term NS1}\<close>
-(*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
- is secret. (Honest users generate fresh nonces.)*)
+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 [rule_format]:
"evs \<in> ns_public
\<Longrightarrow> Crypt (pubEK C) \<lbrace>NA', Nonce NA\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -81,32 +71,42 @@
by (induct rule: ns_public.induct) (auto intro: analz_insertI)
-(*Unicity for NS1: nonce NA identifies agents A and B*)
+text \<open>Unicity for {term NS1}: nonce {term NA} identifies agents {term A} and {term B}\<close>
lemma unique_NA:
- "\<lbrakk>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); 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, auto intro: analz_insertI)
-done
+ 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)
-(*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. *)
+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:
- "\<lbrakk>Says A B (Crypt(pubEK 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, spy_analz)
-apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
-done
+ 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
-(*Authentication for A: if she receives message 2 and has used NA
- to start a run, then B has sent message 2.*)
+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 [rule_format]:
"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
\<Longrightarrow> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -119,10 +119,10 @@
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)
+ by (blast intro: A_trusts_NS2_lemma)
-(*If the encrypted message appears then it originated with Alice in NS1*)
+text \<open>If the encrypted message appears then it originated with Alice in {term NS1}\<close>
lemma B_trusts_NS1 [rule_format]:
"evs \<in> ns_public
\<Longrightarrow> Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
@@ -131,65 +131,72 @@
by (induction evs rule: ns_public.induct) (use analz_insertI in auto)
-
-(*** Authenticity properties obtained from NS2 ***)
+subsection \<open>Authenticity properties obtained from {term NS2}\<close>
-(*Unicity for NS2: nonce NB identifies nonce NA and agent A
- [proof closely follows that for unique_NA] *)
+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]:
- "\<lbrakk>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); evs \<in> ns_public\<rbrakk>
- \<Longrightarrow> A=A' \<and> NA=NA'"
-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
+ 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)
-(*NB remains secret PROVIDED Alice never responds with round 3*)
+text \<open>{term NB} remains secret \emph{provided} Alice never responds with round 3\<close>
theorem Spy_not_see_NB [dest]:
- "\<lbrakk>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; evs \<in> ns_public\<rbrakk>
- \<Longrightarrow> Nonce NB \<notin> 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) (*speeds up the next step*)
-apply (blast intro: no_nonce_NS1_NS2)+
-done
+ 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
-(*Authentication for B: if he receives message 3 and has used NB
- in message 2, then A has sent message 3--to somebody....*)
-
+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 [rule_format]:
- "\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> ns_public\<rbrakk>
- \<Longrightarrow> Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
- Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs \<longrightarrow>
- (\<exists>C. Says A C (Crypt (pubEK C) (Nonce NB)) \<in> set evs)"
-apply (erule ns_public.induct, auto)
-by (blast intro: no_nonce_NS1_NS2)+
+ "\<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)
+ by (blast intro: B_trusts_NS3_lemma)
-(*Can we strengthen the secrecy theorem Spy_not_see_NB? NO*)
-lemma "\<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)"
-apply (erule ns_public.induct, simp_all, spy_analz)
-(*NS1: by freshness*)
+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
-(*NS2: by freshness and unicity of NB*)
+(*{term NS2}: by freshness and unicity of {term NB}*)
apply (blast intro: no_nonce_NS1_NS2)
-(*NS3: unicity of NB identifies A and NA, but not B*)
+(*{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)