wenzelm@37936: (*  Title:      HOL/Auth/NS_Public.thy
paulson@2318:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
paulson@2318:     Copyright   1996  University of Cambridge
paulson@2318: 
paulson@2318: Inductive relation "ns_public" for the Needham-Schroeder Public-Key protocol.
paulson@2538: Version incorporating Lowe's fix (inclusion of B's identity in round 2).
paulson@2318: *)
paulson@2318: 
wenzelm@58889: section{*Verifying the Needham-Schroeder-Lowe Public-Key Protocol*}
paulson@13956: 
haftmann@16417: theory NS_Public imports Public begin
paulson@2318: 
berghofe@23746: inductive_set ns_public :: "event list set"
berghofe@23746:   where 
paulson@2318:          (*Initial trace is empty*)
paulson@11104:    Nil:  "[] \<in> ns_public"
paulson@2318: 
paulson@2318:          (*The spy MAY say anything he CAN say.  We do not expect him to
paulson@2318:            invent new nonces here, but he can also use NS1.  Common to
paulson@2318:            all similar protocols.*)
berghofe@23746:  | Fake: "\<lbrakk>evsf \<in> ns_public;  X \<in> synth (analz (spies evsf))\<rbrakk>
paulson@11366:           \<Longrightarrow> Says Spy B X  # evsf \<in> ns_public"
paulson@2318: 
paulson@2318:          (*Alice initiates a protocol run, sending a nonce to Bob*)
berghofe@23746:  | NS1:  "\<lbrakk>evs1 \<in> ns_public;  Nonce NA \<notin> used evs1\<rbrakk>
paulson@13922:           \<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>)
paulson@11104:                  # evs1  \<in>  ns_public"
paulson@2318: 
paulson@2318:          (*Bob responds to Alice's message with a further nonce*)
berghofe@23746:  | NS2:  "\<lbrakk>evs2 \<in> ns_public;  Nonce NB \<notin> used evs2;
paulson@13922:            Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk>
paulson@13922:           \<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
paulson@11104:                 # evs2  \<in>  ns_public"
paulson@2318: 
paulson@2318:          (*Alice proves her existence by sending NB back to Bob.*)
berghofe@23746:  | NS3:  "\<lbrakk>evs3 \<in> ns_public;
paulson@13922:            Says A  B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3;
paulson@13922:            Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>)
paulson@11104:               \<in> set evs3\<rbrakk>
paulson@13922:           \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 \<in> ns_public"
paulson@11104: 
paulson@11104: declare knows_Spy_partsEs [elim]
paulson@14200: declare knows_Spy_partsEs [elim]
paulson@14200: declare analz_into_parts [dest]
haftmann@56681: declare Fake_parts_insert_in_Un [dest]
paulson@11104: 
paulson@11104: (*A "possibility property": there are traces that reach the end*)
paulson@13922: lemma "\<exists>NB. \<exists>evs \<in> ns_public. Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
paulson@11104: apply (intro exI bexI)
paulson@11104: apply (rule_tac [2] ns_public.Nil [THEN ns_public.NS1, THEN ns_public.NS2, 
paulson@14207:                                    THEN ns_public.NS3], possibility)
paulson@13926: done
paulson@11104: 
paulson@11104: (** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY
paulson@11104:     sends messages containing X! **)
paulson@11104: 
paulson@11104: (*Spy never sees another agent's private key! (unless it's bad at start)*)
paulson@13922: lemma Spy_see_priEK [simp]: 
paulson@13922:       "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> parts (spies evs)) = (A \<in> bad)"
paulson@11104: by (erule ns_public.induct, auto)
paulson@11104: 
paulson@13922: lemma Spy_analz_priEK [simp]: 
paulson@13922:       "evs \<in> ns_public \<Longrightarrow> (Key (priEK A) \<in> analz (spies evs)) = (A \<in> bad)"
paulson@11104: by auto
paulson@11104: 
paulson@14200: subsection{*Authenticity properties obtained from NS2*}
paulson@11104: 
paulson@11104: 
paulson@11104: (*It is impossible to re-use a nonce in both NS1 and NS2, provided the nonce
paulson@11104:   is secret.  (Honest users generate fresh nonces.)*)
paulson@11104: lemma no_nonce_NS1_NS2 [rule_format]: 
paulson@11104:       "evs \<in> ns_public 
paulson@13922:        \<Longrightarrow> Crypt (pubEK C) \<lbrace>NA', Nonce NA, Agent D\<rbrace> \<in> parts (spies evs) \<longrightarrow>
paulson@13922:            Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>  
paulson@11104:            Nonce NA \<in> analz (spies evs)"
paulson@11104: apply (erule ns_public.induct, simp_all)
paulson@11104: apply (blast intro: analz_insertI)+
paulson@11104: done
paulson@11104: 
paulson@11104: (*Unicity for NS1: nonce NA identifies agents A and B*)
paulson@11104: lemma unique_NA: 
paulson@13922:      "\<lbrakk>Crypt(pubEK B)  \<lbrace>Nonce NA, Agent A \<rbrace> \<in> parts(spies evs);  
paulson@13922:        Crypt(pubEK B') \<lbrace>Nonce NA, Agent A'\<rbrace> \<in> parts(spies evs);  
paulson@11104:        Nonce NA \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
paulson@11104:       \<Longrightarrow> A=A' \<and> B=B'"
paulson@11104: apply (erule rev_mp, erule rev_mp, erule rev_mp)   
paulson@11104: apply (erule ns_public.induct, simp_all)
paulson@11104: (*Fake, NS1*)
paulson@11104: apply (blast intro: analz_insertI)+
paulson@11104: done
paulson@11104: 
paulson@11104: 
paulson@11104: (*Secrecy: Spy does not see the nonce sent in msg NS1 if A and B are secure
paulson@11104:   The major premise "Says A B ..." makes it a dest-rule, so we use
paulson@11104:   (erule rev_mp) rather than rule_format. *)
paulson@11104: theorem Spy_not_see_NA: 
paulson@13922:       "\<lbrakk>Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;
paulson@11104:         A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
paulson@11104:        \<Longrightarrow> Nonce NA \<notin> analz (spies evs)"
paulson@11104: apply (erule rev_mp)   
paulson@13507: apply (erule ns_public.induct, simp_all, spy_analz)
paulson@11104: apply (blast dest: unique_NA intro: no_nonce_NS1_NS2)+
paulson@11104: done
paulson@11104: 
paulson@2318: 
paulson@11104: (*Authentication for A: if she receives message 2 and has used NA
paulson@11104:   to start a run, then B has sent message 2.*)
paulson@11104: lemma A_trusts_NS2_lemma [rule_format]: 
paulson@11104:    "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
paulson@13922:     \<Longrightarrow> Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts (spies evs) \<longrightarrow>
wenzelm@32960:         Says A B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs \<longrightarrow>
wenzelm@32960:         Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
paulson@11104: apply (erule ns_public.induct, simp_all)
paulson@11104: (*Fake, NS1*)
paulson@11104: apply (blast dest: Spy_not_see_NA)+
paulson@11104: done
paulson@11104: 
paulson@11104: theorem A_trusts_NS2: 
paulson@13922:      "\<lbrakk>Says A  B (Crypt(pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs;   
paulson@13922:        Says B' A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
paulson@11104:        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                     
paulson@13922:       \<Longrightarrow> Says B A (Crypt(pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs"
paulson@11104: by (blast intro: A_trusts_NS2_lemma)
paulson@11104: 
paulson@11104: 
paulson@11104: (*If the encrypted message appears then it originated with Alice in NS1*)
paulson@11104: lemma B_trusts_NS1 [rule_format]:
paulson@11104:      "evs \<in> ns_public                                         
paulson@13922:       \<Longrightarrow> Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace> \<in> parts (spies evs) \<longrightarrow>
wenzelm@32960:           Nonce NA \<notin> analz (spies evs) \<longrightarrow>
wenzelm@32960:           Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
paulson@11104: apply (erule ns_public.induct, simp_all)
paulson@11104: (*Fake*)
paulson@11104: apply (blast intro!: analz_insertI)
paulson@11104: done
paulson@11104: 
paulson@11104: 
paulson@14200: subsection{*Authenticity properties obtained from NS2*}
paulson@11104: 
paulson@11104: (*Unicity for NS2: nonce NB identifies nonce NA and agents A, B 
paulson@11104:   [unicity of B makes Lowe's fix work]
paulson@11104:   [proof closely follows that for unique_NA] *)
paulson@11104: 
paulson@11104: lemma unique_NB [dest]: 
paulson@13922:      "\<lbrakk>Crypt(pubEK A)  \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace> \<in> parts(spies evs);
paulson@13922:        Crypt(pubEK A') \<lbrace>Nonce NA', Nonce NB, Agent B'\<rbrace> \<in> parts(spies evs);
paulson@11104:        Nonce NB \<notin> analz (spies evs); evs \<in> ns_public\<rbrakk>
paulson@11104:       \<Longrightarrow> A=A' \<and> NA=NA' \<and> B=B'"
paulson@11104: apply (erule rev_mp, erule rev_mp, erule rev_mp)   
paulson@11104: apply (erule ns_public.induct, simp_all)
paulson@11104: (*Fake, NS2*)
paulson@11104: apply (blast intro: analz_insertI)+
paulson@11104: done
paulson@11104: 
paulson@11104: 
paulson@11104: (*Secrecy: Spy does not see the nonce sent in msg NS2 if A and B are secure*)
paulson@11104: theorem Spy_not_see_NB [dest]:
paulson@13922:      "\<lbrakk>Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
paulson@11104:        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>
paulson@11104:       \<Longrightarrow> Nonce NB \<notin> analz (spies evs)"
paulson@11104: apply (erule rev_mp)
paulson@13507: apply (erule ns_public.induct, simp_all, spy_analz)
paulson@11104: apply (blast intro: no_nonce_NS1_NS2)+
paulson@11104: done
paulson@11104: 
paulson@11104: 
paulson@11104: (*Authentication for B: if he receives message 3 and has used NB
paulson@11104:   in message 2, then A has sent message 3.*)
paulson@11104: lemma B_trusts_NS3_lemma [rule_format]:
paulson@11104:      "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk> \<Longrightarrow>
paulson@13922:       Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
paulson@13922:       Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
paulson@13922:       Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
paulson@11104: by (erule ns_public.induct, auto)
paulson@11104: 
paulson@11104: theorem B_trusts_NS3:
paulson@13922:      "\<lbrakk>Says B A  (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs;
paulson@13922:        Says A' B (Crypt (pubEK B) (Nonce NB)) \<in> set evs;             
paulson@11104:        A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk>                    
paulson@13922:       \<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) \<in> set evs"
paulson@11104: by (blast intro: B_trusts_NS3_lemma)
paulson@11104: 
paulson@14200: subsection{*Overall guarantee for B*}
paulson@11104: 
paulson@11104: (*If NS3 has been sent and the nonce NB agrees with the nonce B joined with
paulson@11104:   NA, then A initiated the run using NA.*)
paulson@11104: theorem B_trusts_protocol:
paulson@11104:      "\<lbrakk>A \<notin> bad;  B \<notin> bad;  evs \<in> ns_public\<rbrakk> \<Longrightarrow>
paulson@13922:       Crypt (pubEK B) (Nonce NB) \<in> parts (spies evs) \<longrightarrow>
paulson@13922:       Says B A  (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB, Agent B\<rbrace>) \<in> set evs \<longrightarrow>
paulson@13922:       Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"
paulson@11104: by (erule ns_public.induct, auto)
paulson@2318: 
paulson@2318: end