author | paulson |
Wed, 14 Feb 2001 13:01:02 +0100 | |
changeset 11117 | 55358999077d |
parent 11104 | f2024fed9f0c |
child 11150 | 67387142225e |
permissions | -rw-r--r-- |
1934 | 1 |
(* Title: HOL/Auth/NS_Shared |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1996 University of Cambridge |
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Inductive relation "ns_shared" for Needham-Schroeder Shared-Key protocol. |
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From page 247 of |
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Burrows, Abadi and Needham. A Logic of Authentication. |
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Proc. Royal Soc. 426 (1989) |
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*) |
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theory NS_Shared = Shared: |
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consts ns_shared :: "event list set" |
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ab0a9fbed4c0
Changing "lost" from a parameter of protocol definitions to a constant.
paulson
parents:
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changeset
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inductive "ns_shared" |
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intros |
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(*Initial trace is empty*) |
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Nil: "[] \<in> ns_shared" |
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(*The spy MAY say anything he CAN say. We do not expect him to |
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invent new nonces here, but he can also use NS1. Common to |
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all similar protocols.*) |
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Fake: "\<lbrakk>evs \<in> ns_shared; X \<in> synth (analz (spies evs))\<rbrakk> |
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\<Longrightarrow> Says Spy B X # evs \<in> ns_shared" |
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(*Alice initiates a protocol run, requesting to talk to any B*) |
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NS1: "\<lbrakk>evs1 \<in> ns_shared; Nonce NA \<notin> used evs1\<rbrakk> |
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\<Longrightarrow> Says A Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> # evs1 \<in> ns_shared" |
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(*Server's response to Alice's message. |
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!! It may respond more than once to A's request !! |
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Server doesn't know who the true sender is, hence the A' in |
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the sender field.*) |
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NS2: "\<lbrakk>evs2 \<in> ns_shared; Key KAB \<notin> used evs2; |
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Says A' Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> |
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\<Longrightarrow> Says Server A |
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(Crypt (shrK A) |
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\<lbrace>Nonce NA, Agent B, Key KAB, |
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(Crypt (shrK B) \<lbrace>Key KAB, Agent A\<rbrace>)\<rbrace>) |
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# evs2 \<in> ns_shared" |
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(*We can't assume S=Server. Agent A "remembers" her nonce. |
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Need A \<noteq> Server because we allow messages to self.*) |
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NS3: "\<lbrakk>evs3 \<in> ns_shared; A \<noteq> Server; |
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Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs3; |
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Says A Server \<lbrace>Agent A, Agent B, Nonce NA\<rbrace> \<in> set evs3\<rbrakk> |
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\<Longrightarrow> Says A B X # evs3 \<in> ns_shared" |
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(*Bob's nonce exchange. He does not know who the message came |
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from, but responds to A because she is mentioned inside.*) |
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NS4: "\<lbrakk>evs4 \<in> ns_shared; Nonce NB \<notin> used evs4; |
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Says A' B (Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>) \<in> set evs4\<rbrakk> |
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\<Longrightarrow> Says B A (Crypt K (Nonce NB)) # evs4 \<in> ns_shared" |
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(*Alice responds with Nonce NB if she has seen the key before. |
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Maybe should somehow check Nonce NA again. |
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We do NOT send NB-1 or similar as the Spy cannot spoof such things. |
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Letting the Spy add or subtract 1 lets him send \<forall>nonces. |
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Instead we distinguish the messages by sending the nonce twice.*) |
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NS5: "\<lbrakk>evs5 \<in> ns_shared; |
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Says B' A (Crypt K (Nonce NB)) \<in> set evs5; |
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Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) |
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\<in> set evs5\<rbrakk> |
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\<Longrightarrow> Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) # evs5 \<in> ns_shared" |
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(*This message models possible leaks of session keys. |
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The two Nonces identify the protocol run: the rule insists upon |
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the true senders in order to make them accurate.*) |
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Oops: "\<lbrakk>evso \<in> ns_shared; Says B A (Crypt K (Nonce NB)) \<in> set evso; |
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Says Server A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) |
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\<in> set evso\<rbrakk> |
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\<Longrightarrow> Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> ns_shared" |
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declare knows_Spy_partsEs [elim] |
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declare analz_subset_parts [THEN subsetD, dest] |
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declare Fake_parts_insert [THEN subsetD, dest] |
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declare image_eq_UN [simp] (*accelerates proofs involving nested images*) |
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(*A "possibility property": there are traces that reach the end*) |
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lemma "A \<noteq> Server \<Longrightarrow> \<exists>N K. \<exists>evs \<in> ns_shared. |
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Says A B (Crypt K \<lbrace>Nonce N, Nonce N\<rbrace>) \<in> set evs" |
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apply (intro exI bexI) |
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apply (rule_tac [2] ns_shared.Nil |
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[THEN ns_shared.NS1, THEN ns_shared.NS2, THEN ns_shared.NS3, |
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THEN ns_shared.NS4, THEN ns_shared.NS5]) |
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apply possibility |
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done |
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(*This version is similar, while instantiating ?K and ?N to epsilon-terms |
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lemma "A \<noteq> Server \<Longrightarrow> \<exists>evs \<in> ns_shared. |
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Says A B (Crypt ?K \<lbrace>Nonce ?N, Nonce ?N\<rbrace>) \<in> set evs" |
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*) |
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(**** Inductive proofs about ns_shared ****) |
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(*Forwarding lemmas, to aid simplification*) |
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(*For reasoning about the encrypted portion of message NS3*) |
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lemma NS3_msg_in_parts_spies: |
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"Says S A (Crypt KA \<lbrace>N, B, K, X\<rbrace>) \<in> set evs \<Longrightarrow> X \<in> parts (spies evs)" |
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by blast |
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(*For reasoning about the Oops message*) |
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lemma Oops_parts_spies: |
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"Says Server A (Crypt (shrK A) \<lbrace>NA, B, K, X\<rbrace>) \<in> set evs |
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\<Longrightarrow> K \<in> parts (spies evs)" |
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by blast |
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(** Theorems of the form X \<notin> parts (spies evs) imply that NOBODY |
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sends messages containing X! **) |
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(*Spy never sees another agent's shared key! (unless it's bad at start)*) |
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lemma Spy_see_shrK [simp]: |
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"evs \<in> ns_shared \<Longrightarrow> (Key (shrK A) \<in> parts (spies evs)) = (A \<in> bad)" |
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apply (erule ns_shared.induct, force, frule_tac [4] NS3_msg_in_parts_spies) |
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apply simp_all |
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apply blast+; |
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done |
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lemma Spy_analz_shrK [simp]: |
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"evs \<in> ns_shared \<Longrightarrow> (Key (shrK A) \<in> analz (spies evs)) = (A \<in> bad)" |
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by auto |
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(*Nobody can have used non-existent keys!*) |
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lemma new_keys_not_used [rule_format, simp]: |
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"evs \<in> ns_shared \<Longrightarrow> Key K \<notin> used evs \<longrightarrow> K \<notin> keysFor (parts (spies evs))" |
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apply (erule ns_shared.induct, force, frule_tac [4] NS3_msg_in_parts_spies) |
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apply simp_all |
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(*Fake, NS2, NS4, NS5*) |
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apply (blast dest!: keysFor_parts_insert)+ |
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done |
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(** Lemmas concerning the form of items passed in messages **) |
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(*Describes the form of K, X and K' when the Server sends this message.*) |
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lemma Says_Server_message_form: |
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"\<lbrakk>Says Server A (Crypt K' \<lbrace>N, Agent B, Key K, X\<rbrace>) \<in> set evs; |
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evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> K \<notin> range shrK \<and> |
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X = (Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>) \<and> |
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K' = shrK A" |
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by (erule rev_mp, erule ns_shared.induct, auto) |
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(*If the encrypted message appears then it originated with the Server*) |
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lemma A_trusts_NS2: |
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"\<lbrakk>Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace> \<in> parts (spies evs); |
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A \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs" |
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apply (erule rev_mp) |
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apply (erule ns_shared.induct, force, drule_tac [4] NS3_msg_in_parts_spies) |
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apply auto |
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done |
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lemma cert_A_form: |
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"\<lbrakk>Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace> \<in> parts (spies evs); |
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A \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> K \<notin> range shrK \<and> X = (Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>)" |
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by (blast dest!: A_trusts_NS2 Says_Server_message_form) |
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(*EITHER describes the form of X when the following message is sent, |
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OR reduces it to the Fake case. |
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Use Says_Server_message_form if applicable.*) |
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lemma Says_S_message_form: |
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"\<lbrakk>Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs; |
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evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> (K \<notin> range shrK \<and> X = (Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>)) |
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\<or> X \<in> analz (spies evs)" |
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apply (frule Says_imp_knows_Spy) |
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(*mystery: why is this frule needed?*) |
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apply (blast dest: cert_A_form analz.Inj) |
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done |
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(*Alternative version also provable |
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lemma Says_S_message_form2: |
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"\<lbrakk>Says S A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs; |
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evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> Says Server A (Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace>) \<in> set evs |
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\<or> X \<in> analz (spies evs)" |
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apply (case_tac "A \<in> bad") |
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apply (force dest!: Says_imp_knows_Spy [THEN analz.Inj]); |
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by (blast dest!: A_trusts_NS2 Says_Server_message_form) |
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*) |
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(**** |
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SESSION KEY COMPROMISE THEOREM. To prove theorems of the form |
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Key K \<in> analz (insert (Key KAB) (spies evs)) \<Longrightarrow> |
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Key K \<in> analz (spies evs) |
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A more general formula must be proved inductively. |
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****) |
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1934 | 199 |
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11104 | 200 |
(*NOT useful in this form, but it says that session keys are not used |
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to encrypt messages containing other keys, in the actual protocol. |
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We require that agents should behave like this subsequently also.*) |
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lemma "\<lbrakk>evs \<in> ns_shared; Kab \<notin> range shrK\<rbrakk> \<Longrightarrow> |
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(Crypt KAB X) \<in> parts (spies evs) \<and> |
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Key K \<in> parts {X} \<longrightarrow> Key K \<in> parts (spies evs)" |
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apply (erule ns_shared.induct, force, frule_tac [4] NS3_msg_in_parts_spies) |
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apply simp_all |
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(*Fake*) |
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apply (blast dest: parts_insert_subset_Un) |
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(*Base, NS4 and NS5*) |
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apply auto |
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done |
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(** Session keys are not used to encrypt other session keys **) |
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(*The equality makes the induction hypothesis easier to apply*) |
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lemma analz_image_freshK [rule_format]: |
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"evs \<in> ns_shared \<Longrightarrow> |
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\<forall>K KK. KK \<subseteq> - (range shrK) \<longrightarrow> |
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(Key K \<in> analz (Key`KK \<union> (spies evs))) = |
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(K \<in> KK \<or> Key K \<in> analz (spies evs))" |
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apply (erule ns_shared.induct, force) |
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apply (frule_tac [7] Says_Server_message_form) |
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apply (erule_tac [4] Says_S_message_form [THEN disjE]) |
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11117 | 227 |
apply analz_freshK |
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apply spy_analz |
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11104 | 229 |
done |
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lemma analz_insert_freshK: |
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"\<lbrakk>evs \<in> ns_shared; KAB \<notin> range shrK\<rbrakk> \<Longrightarrow> |
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Key K \<in> analz (insert (Key KAB) (spies evs)) = |
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(K = KAB \<or> Key K \<in> analz (spies evs))" |
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by (simp only: analz_image_freshK analz_image_freshK_simps) |
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(** The session key K uniquely identifies the message **) |
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1934 | 240 |
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11104 | 241 |
(*In messages of this form, the session key uniquely identifies the rest*) |
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lemma unique_session_keys: |
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"\<lbrakk>Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs; |
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Says Server A' (Crypt (shrK A') \<lbrace>NA', Agent B', Key K, X'\<rbrace>) \<in> set evs; |
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evs \<in> ns_shared\<rbrakk> \<Longrightarrow> A=A' \<and> NA=NA' \<and> B=B' \<and> X = X'" |
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apply (erule rev_mp, erule rev_mp, erule ns_shared.induct) |
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apply simp_all |
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apply blast+ |
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done |
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(** Crucial secrecy property: Spy does not see the keys sent in msg NS2 **) |
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lemma secrecy_lemma [rule_format]: |
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"\<lbrakk>Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, |
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Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) |
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\<in> set evs; |
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A \<notin> bad; B \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> (\<forall>NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs) \<longrightarrow> |
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Key K \<notin> analz (spies evs)" |
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apply (erule rev_mp) |
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apply (erule ns_shared.induct, force) |
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apply (frule_tac [7] Says_Server_message_form) |
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apply (frule_tac [4] Says_S_message_form) |
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apply (erule_tac [5] disjE) |
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apply (simp_all add: analz_insert_eq analz_insert_freshK pushes split_ifs) |
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apply spy_analz (*Fake*) |
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apply blast (*NS2*) |
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(*NS3, Server sub-case*) (**LEVEL 6 **) |
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apply clarify |
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apply (frule Says_imp_knows_Spy [THEN parts.Inj, THEN A_trusts_NS2]) |
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apply (blast dest: Says_imp_knows_Spy [THEN analz.Inj, THEN Crypt_Spy_analz_bad]) |
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apply assumption |
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apply (blast dest: unique_session_keys)+ (*also proves Oops*) |
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done |
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277 |
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(*Final version: Server's message in the most abstract form*) |
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lemma Spy_not_see_encrypted_key: |
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"\<lbrakk>Says Server A (Crypt K' \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs; |
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\<forall>NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs; |
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A \<notin> bad; B \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> Key K \<notin> analz (spies evs)" |
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apply (frule Says_Server_message_form, assumption) |
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apply (auto dest: Says_Server_message_form secrecy_lemma) |
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done |
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287 |
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288 |
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(**** Guarantees available at various stages of protocol ***) |
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1934 | 290 |
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11104 | 291 |
(*If the encrypted message appears then it originated with the Server*) |
292 |
lemma B_trusts_NS3: |
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293 |
"\<lbrakk>Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<in> parts (spies evs); |
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294 |
B \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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\<Longrightarrow> \<exists>NA. Says Server A |
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(Crypt (shrK A) \<lbrace>NA, Agent B, Key K, |
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Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) |
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\<in> set evs" |
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299 |
apply (erule rev_mp) |
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300 |
apply (erule ns_shared.induct, force, drule_tac [4] NS3_msg_in_parts_spies) |
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301 |
apply auto |
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302 |
done |
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303 |
||
304 |
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305 |
lemma A_trusts_NS4_lemma [rule_format]: |
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306 |
"evs \<in> ns_shared \<Longrightarrow> |
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307 |
Key K \<notin> analz (spies evs) \<longrightarrow> |
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308 |
Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs \<longrightarrow> |
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309 |
Crypt K (Nonce NB) \<in> parts (spies evs) \<longrightarrow> |
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310 |
Says B A (Crypt K (Nonce NB)) \<in> set evs" |
|
311 |
apply (erule ns_shared.induct, force, drule_tac [4] NS3_msg_in_parts_spies) |
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312 |
apply analz_mono_contra |
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313 |
apply simp_all |
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314 |
apply blast (*Fake*) |
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315 |
(*NS2: contradiction from the assumptions |
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316 |
Key K \<notin> used evs2 and Crypt K (Nonce NB) \<in> parts (spies evs2) *) |
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317 |
apply (force dest!: Crypt_imp_keysFor) |
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318 |
apply blast (*NS3*) |
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(*NS4*) |
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320 |
apply clarify; |
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321 |
apply (frule Says_imp_knows_Spy [THEN analz.Inj]) |
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322 |
apply (blast dest: Says_imp_knows_Spy [THEN analz.Inj] Crypt_Spy_analz_bad |
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323 |
B_trusts_NS3 unique_session_keys) |
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324 |
done |
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325 |
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326 |
(*This version no longer assumes that K is secure*) |
|
327 |
lemma A_trusts_NS4: |
|
328 |
"\<lbrakk>Crypt K (Nonce NB) \<in> parts (spies evs); |
|
329 |
Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace> \<in> parts (spies evs); |
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330 |
\<forall>NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs; |
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331 |
A \<notin> bad; B \<notin> bad; evs \<in> ns_shared\<rbrakk> |
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332 |
\<Longrightarrow> Says B A (Crypt K (Nonce NB)) \<in> set evs" |
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333 |
by (blast intro: A_trusts_NS4_lemma |
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334 |
dest: A_trusts_NS2 Spy_not_see_encrypted_key) |
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335 |
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336 |
(*If the session key has been used in NS4 then somebody has forwarded |
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337 |
component X in some instance of NS4. Perhaps an interesting property, |
|
338 |
but not needed (after all) for the proofs below.*) |
|
339 |
theorem NS4_implies_NS3 [rule_format]: |
|
340 |
"evs \<in> ns_shared \<Longrightarrow> |
|
341 |
Key K \<notin> analz (spies evs) \<longrightarrow> |
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342 |
Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace>) \<in> set evs \<longrightarrow> |
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343 |
Crypt K (Nonce NB) \<in> parts (spies evs) \<longrightarrow> |
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344 |
(\<exists>A'. Says A' B X \<in> set evs)" |
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345 |
apply (erule ns_shared.induct, force, drule_tac [4] NS3_msg_in_parts_spies) |
|
346 |
apply analz_mono_contra |
|
347 |
apply (simp_all add: ex_disj_distrib) |
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348 |
apply blast (*Fake*) |
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349 |
apply (blast dest!: new_keys_not_used Crypt_imp_keysFor) (*NS2*) |
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350 |
apply blast (*NS3*) |
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351 |
(*NS4*) |
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352 |
apply (case_tac "Ba \<in> bad") |
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353 |
apply (blast dest: Says_imp_knows_Spy [THEN analz.Inj] Crypt_Spy_analz_bad); |
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354 |
apply (frule Says_imp_knows_Spy [THEN parts.Inj, THEN B_trusts_NS3], |
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355 |
assumption+) |
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356 |
apply (blast dest: unique_session_keys) |
|
357 |
done |
|
358 |
||
359 |
||
360 |
lemma B_trusts_NS5_lemma [rule_format]: |
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361 |
"\<lbrakk>B \<notin> bad; evs \<in> ns_shared\<rbrakk> \<Longrightarrow> |
|
362 |
Key K \<notin> analz (spies evs) \<longrightarrow> |
|
363 |
Says Server A |
|
364 |
(Crypt (shrK A) \<lbrace>NA, Agent B, Key K, |
|
365 |
Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs \<longrightarrow> |
|
366 |
Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace> \<in> parts (spies evs) \<longrightarrow> |
|
367 |
Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) \<in> set evs" |
|
368 |
apply (erule ns_shared.induct, force, drule_tac [4] NS3_msg_in_parts_spies) |
|
369 |
apply analz_mono_contra |
|
370 |
apply simp_all |
|
371 |
apply blast (*Fake*) |
|
372 |
apply (blast dest!: new_keys_not_used Crypt_imp_keysFor) (*NS2*) |
|
373 |
apply (blast dest!: cert_A_form) (*NS3*) |
|
374 |
(**LEVEL 5**) |
|
375 |
(*NS5*) |
|
376 |
apply clarify |
|
377 |
apply (case_tac "Aa \<in> bad") |
|
378 |
apply (blast dest: Says_imp_knows_Spy [THEN analz.Inj] Crypt_Spy_analz_bad); |
|
379 |
apply (blast dest: A_trusts_NS2 unique_session_keys) |
|
380 |
done |
|
381 |
||
382 |
||
383 |
(*Very strong Oops condition reveals protocol's weakness*) |
|
384 |
lemma B_trusts_NS5: |
|
385 |
"\<lbrakk>Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace> \<in> parts (spies evs); |
|
386 |
Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace> \<in> parts (spies evs); |
|
387 |
\<forall>NA NB. Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs; |
|
388 |
A \<notin> bad; B \<notin> bad; evs \<in> ns_shared\<rbrakk> |
|
389 |
\<Longrightarrow> Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) \<in> set evs" |
|
390 |
apply (drule B_trusts_NS3, clarify+) |
|
391 |
apply (blast intro: B_trusts_NS5_lemma |
|
392 |
dest: dest: Spy_not_see_encrypted_key) |
|
393 |
(*surprisingly delicate proof due to quantifier interactions*) |
|
394 |
done |
|
1934 | 395 |
|
396 |
end |