author  wenzelm 
Sat, 13 Feb 2016 12:39:00 +0100  
changeset 62290  658276428cfc 
parent 62286  705d4c4003ea 
child 67399  eab6ce8368fa 
permissions  rwrr 
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theory Needham_Schroeder_Unguided_Attacker_Example 
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imports Needham_Schroeder_Base 
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begin 
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inductive_set ns_public :: "event list set" 
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where 
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(*Initial trace is empty*) 
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Nil: "[] \<in> ns_public" 
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 Fake: "\<lbrakk>evs1 \<in> ns_public; X \<in> synth (analz (spies evs1)) \<rbrakk> 
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\<Longrightarrow> Says Spy A X # evs1 \<in> ns_public" 
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(*Alice initiates a protocol run, sending a nonce to Bob*) 
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 NS1: "\<lbrakk>evs1 \<in> ns_public; Nonce NA \<notin> used evs1\<rbrakk> 
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\<Longrightarrow> Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) 
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# evs1 \<in> ns_public" 
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(*Bob responds to Alice's message with a further nonce*) 
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 NS2: "\<lbrakk>evs2 \<in> ns_public; Nonce NB \<notin> used evs2; 
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Says A' B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk> 
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\<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) 
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# evs2 \<in> ns_public" 
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(*Alice proves her existence by sending NB back to Bob.*) 
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 NS3: "\<lbrakk>evs3 \<in> ns_public; 
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Says A B (Crypt (pubEK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3; 
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Says B' A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk> 
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\<Longrightarrow> Says A B (Crypt (pubEK B) (Nonce NB)) # evs3 \<in> ns_public" 
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declare ListMem_iff[symmetric, code_pred_inline] 
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lemmas [code_pred_intro] = ns_publicp.intros[folded synth'_def] 
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code_pred [skip_proof] ns_publicp unfolding synth'_def by (rule ns_publicp.cases) fastforce+ 
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thm ns_publicp.equation 
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code_pred [generator_cps] ns_publicp . 
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thm ns_publicp.generator_cps_equation 
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lemma "ns_publicp evs ==> \<not> (Says Alice Bob (Crypt (pubEK Bob) (Nonce NB))) : set evs" 
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quickcheck[smart_exhaustive, depth = 5, timeout = 200, expect = counterexample] 
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(*quickcheck[narrowing, size = 6, timeout = 200, verbose, expect = no_counterexample]*) 
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oops 
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lemma 
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"\<lbrakk>ns_publicp evs\<rbrakk> 
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\<Longrightarrow> Says B A (Crypt (pubEK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) : set evs 
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\<Longrightarrow> A \<noteq> Spy \<Longrightarrow> B \<noteq> Spy \<Longrightarrow> A \<noteq> B 
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\<Longrightarrow> Nonce NB \<notin> analz (spies evs)" 
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quickcheck[smart_exhaustive, depth = 6, timeout = 100, expect = no_counterexample] 
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(*quickcheck[narrowing, size = 7, timeout = 200, expect = no_counterexample]*) 
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oops 
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62290  54 
section \<open>Proving the counterexample trace for validation\<close> 
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lemma 
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assumes "A = Alice" "B = Bob" "C = Spy" "NA = 0" "NB = 1" 
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assumes "evs = 
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[Says Alice Spy (Crypt (pubEK Spy) (Nonce 1)), 
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Says Bob Alice (Crypt (pubEK Alice) \<lbrace>Nonce 0, Nonce 1\<rbrace>), 
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Says Spy Bob (Crypt (pubEK Bob) \<lbrace>Nonce 0, Agent Alice\<rbrace>), 

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Says Alice Spy (Crypt (pubEK Spy) \<lbrace>Nonce 0, Agent Alice\<rbrace>)]" (is "_ = [?e3, ?e2, ?e1, ?e0]") 

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shows "A \<noteq> Spy" "B \<noteq> Spy" "evs : ns_public" "Nonce NB : analz (knows Spy evs)" 
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proof  
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from assms show "A \<noteq> Spy" by auto 
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from assms show "B \<noteq> Spy" by auto 
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have "[] : ns_public" by (rule Nil) 
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then have first_step: "[?e0] : ns_public" 
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proof (rule NS1) 
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show "Nonce 0 ~: used []" by eval 
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qed 
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then have "[?e1, ?e0] : ns_public" 
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proof (rule Fake) 
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show "Crypt (pubEK Bob) \<lbrace>Nonce 0, Agent Alice\<rbrace> : synth (analz (knows Spy [?e0]))" 
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by (intro synth.intros(2,3,4,1)) eval+ 
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qed 
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then have "[?e2, ?e1, ?e0] : ns_public" 
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proof (rule NS2) 
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show "Says Spy Bob (Crypt (pubEK Bob) \<lbrace>Nonce 0, Agent Alice\<rbrace>) \<in> set [?e1, ?e0]" by simp 
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show " Nonce 1 ~: used [?e1, ?e0]" by eval 
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qed 
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then show "evs : ns_public" 
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unfolding assms 
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proof (rule NS3) 
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show " Says Alice Spy (Crypt (pubEK Spy) \<lbrace>Nonce 0, Agent Alice\<rbrace>) \<in> set [?e2, ?e1, ?e0]" by simp 
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show "Says Bob Alice (Crypt (pubEK Alice) \<lbrace>Nonce 0, Nonce 1\<rbrace>) : set [?e2, ?e1, ?e0]" by simp 

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qed 
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from assms show "Nonce NB : analz (knows Spy evs)" 
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apply simp 
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apply (rule analz.intros(4)) 
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apply (rule analz.intros(1)) 
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apply (auto simp add: bad_def) 
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done 
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qed 
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end 