(* Title: HOL/Auth/NS_Public ID: $Id: NS_Public.thy,v 1.24 2007/07/11 09:15:16 berghofe Exp $ 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
lemma
∃NB. ∃evs∈ns_public. Says A B (Crypt (pubK B) (Nonce NB)) ∈ set evs
lemma Spy_see_priEK:
evs ∈ ns_public ==> (Key (invKey (pubK A)) ∈ parts (knows Spy evs)) = (A ∈ bad)
lemma Spy_analz_priEK:
evs ∈ ns_public ==> (Key (invKey (pubK A)) ∈ analz (knows Spy evs)) = (A ∈ bad)
lemma no_nonce_NS1_NS2:
[| evs ∈ ns_public;
Crypt (pubK C) {|NA', Nonce NA, Agent D|} ∈ parts (knows Spy evs);
Crypt (pubK B) {|Nonce NA, Agent A|} ∈ parts (knows Spy evs) |]
==> Nonce NA ∈ analz (knows Spy evs)
lemma unique_NA:
[| Crypt (pubK B) {|Nonce NA, Agent A|} ∈ parts (knows Spy evs);
Crypt (pubK B') {|Nonce NA, Agent A'|} ∈ parts (knows Spy evs);
Nonce NA ∉ analz (knows Spy evs); evs ∈ ns_public |]
==> A = A' ∧ B = B'
theorem Spy_not_see_NA:
[| Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) ∈ set evs; A ∉ bad; B ∉ bad;
evs ∈ ns_public |]
==> Nonce NA ∉ analz (knows Spy evs)
lemma A_trusts_NS2_lemma:
[| A ∉ bad; B ∉ bad; evs ∈ ns_public;
Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|} ∈ parts (knows Spy evs);
Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) ∈ set evs |]
==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs
theorem A_trusts_NS2:
[| Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) ∈ set evs;
Says B' A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs;
A ∉ bad; B ∉ bad; evs ∈ ns_public |]
==> Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs
lemma B_trusts_NS1:
[| evs ∈ ns_public;
Crypt (pubK B) {|Nonce NA, Agent A|} ∈ parts (knows Spy evs);
Nonce NA ∉ analz (knows Spy evs) |]
==> Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) ∈ set evs
lemma unique_NB:
[| Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|} ∈ parts (knows Spy evs);
Crypt (pubK A') {|Nonce NA', Nonce NB, Agent B'|} ∈ parts (knows Spy evs);
Nonce NB ∉ analz (knows Spy evs); evs ∈ ns_public |]
==> A = A' ∧ NA = NA' ∧ B = B'
theorem Spy_not_see_NB:
[| Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs; A ∉ bad;
B ∉ bad; evs ∈ ns_public |]
==> Nonce NB ∉ analz (knows Spy evs)
lemma B_trusts_NS3_lemma:
[| A ∉ bad; B ∉ bad; evs ∈ ns_public;
Crypt (pubK B) (Nonce NB) ∈ parts (knows Spy evs);
Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs |]
==> Says A B (Crypt (pubK B) (Nonce NB)) ∈ set evs
theorem B_trusts_NS3:
[| Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs;
Says A' B (Crypt (pubK B) (Nonce NB)) ∈ set evs; A ∉ bad; B ∉ bad;
evs ∈ ns_public |]
==> Says A B (Crypt (pubK B) (Nonce NB)) ∈ set evs
theorem B_trusts_protocol:
[| A ∉ bad; B ∉ bad; evs ∈ ns_public |]
==> Crypt (pubK B) (Nonce NB) ∈ parts (knows Spy evs) -->
Says B A (Crypt (pubK A) {|Nonce NA, Nonce NB, Agent B|}) ∈ set evs -->
Says A B (Crypt (pubK B) {|Nonce NA, Agent A|}) ∈ set evs