author | wenzelm |
Sat, 05 Jan 2019 17:24:33 +0100 | |
changeset 69597 | ff784d5a5bfb |
parent 67613 | ce654b0e6d69 |
child 76288 | b82ac7ef65ec |
permissions | -rw-r--r-- |
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(* Title: HOL/Auth/Yahalom.thy |
1985
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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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*) |
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section\<open>The Yahalom Protocol\<close> |
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theory Yahalom imports Public begin |
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text\<open>From page 257 of |
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Burrows, Abadi and Needham (1989). A Logic of Authentication. |
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Proc. Royal Soc. 426 |
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|
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This theory has the prototypical example of a secrecy relation, KeyCryptNonce. |
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\<close> |
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inductive_set yahalom :: "event list set" |
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where |
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(*Initial trace is empty*) |
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Nil: "[] \<in> yahalom" |
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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>evsf \<in> yahalom; X \<in> synth (analz (knows Spy evsf))\<rbrakk> |
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\<Longrightarrow> Says Spy B X # evsf \<in> yahalom" |
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(*A message that has been sent can be received by the |
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intended recipient.*) |
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| Reception: "\<lbrakk>evsr \<in> yahalom; Says A B X \<in> set evsr\<rbrakk> |
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\<Longrightarrow> Gets B X # evsr \<in> yahalom" |
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(*Alice initiates a protocol run*) |
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| YM1: "\<lbrakk>evs1 \<in> yahalom; Nonce NA \<notin> used evs1\<rbrakk> |
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\<Longrightarrow> Says A B \<lbrace>Agent A, Nonce NA\<rbrace> # evs1 \<in> yahalom" |
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(*Bob's response to Alice's message.*) |
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| YM2: "\<lbrakk>evs2 \<in> yahalom; Nonce NB \<notin> used evs2; |
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Gets B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs2\<rbrakk> |
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\<Longrightarrow> Says B Server |
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\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
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# evs2 \<in> yahalom" |
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(*The Server receives Bob's message. He responds by sending a |
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new session key to Alice, with a packet for forwarding to Bob.*) |
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| YM3: "\<lbrakk>evs3 \<in> yahalom; Key KAB \<notin> used evs3; KAB \<in> symKeys; |
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Gets Server |
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\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
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\<in> set evs3\<rbrakk> |
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\<Longrightarrow> Says Server A |
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\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, Nonce NA, Nonce NB\<rbrace>, |
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Crypt (shrK B) \<lbrace>Agent A, Key KAB\<rbrace>\<rbrace> |
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# evs3 \<in> yahalom" |
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| YM4: |
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\<comment> \<open>Alice receives the Server's (?) message, checks her Nonce, and |
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uses the new session key to send Bob his Nonce. The premise |
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\<^term>\<open>A \<noteq> Server\<close> is needed for \<open>Says_Server_not_range\<close>. |
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Alice can check that K is symmetric by its length.\<close> |
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"\<lbrakk>evs4 \<in> yahalom; A \<noteq> Server; K \<in> symKeys; |
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Gets A \<lbrace>Crypt(shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, X\<rbrace> |
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\<in> set evs4; |
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Says A B \<lbrace>Agent A, Nonce NA\<rbrace> \<in> set evs4\<rbrakk> |
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\<Longrightarrow> Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> # evs4 \<in> yahalom" |
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(*This message models possible leaks of session keys. The Nonces |
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identify the protocol run. Quoting Server here ensures they are |
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correct.*) |
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| Oops: "\<lbrakk>evso \<in> yahalom; |
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Says Server A \<lbrace>Crypt (shrK A) |
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\<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, |
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X\<rbrace> \<in> set evso\<rbrakk> |
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\<Longrightarrow> Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso \<in> yahalom" |
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definition KeyWithNonce :: "[key, nat, event list] \<Rightarrow> bool" where |
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"KeyWithNonce K NB evs == |
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\<exists>A B na X. |
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Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace> |
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\<in> set evs" |
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declare Says_imp_analz_Spy [dest] |
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declare parts.Body [dest] |
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declare Fake_parts_insert_in_Un [dest] |
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declare analz_into_parts [dest] |
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text\<open>A "possibility property": there are traces that reach the end\<close> |
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lemma "\<lbrakk>A \<noteq> Server; K \<in> symKeys; Key K \<notin> used []\<rbrakk> |
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\<Longrightarrow> \<exists>X NB. \<exists>evs \<in> yahalom. |
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Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs" |
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apply (intro exI bexI) |
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apply (rule_tac [2] yahalom.Nil |
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[THEN yahalom.YM1, THEN yahalom.Reception, |
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THEN yahalom.YM2, THEN yahalom.Reception, |
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THEN yahalom.YM3, THEN yahalom.Reception, |
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THEN yahalom.YM4]) |
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apply (possibility, simp add: used_Cons) |
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done |
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subsection\<open>Regularity Lemmas for Yahalom\<close> |
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lemma Gets_imp_Says: |
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"\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> \<exists>A. Says A B X \<in> set evs" |
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by (erule rev_mp, erule yahalom.induct, auto) |
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text\<open>Must be proved separately for each protocol\<close> |
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lemma Gets_imp_knows_Spy: |
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"\<lbrakk>Gets B X \<in> set evs; evs \<in> yahalom\<rbrakk> \<Longrightarrow> X \<in> knows Spy evs" |
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by (blast dest!: Gets_imp_Says Says_imp_knows_Spy) |
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lemmas Gets_imp_analz_Spy = Gets_imp_knows_Spy [THEN analz.Inj] |
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declare Gets_imp_analz_Spy [dest] |
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text\<open>Lets us treat YM4 using a similar argument as for the Fake case.\<close> |
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lemma YM4_analz_knows_Spy: |
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"\<lbrakk>Gets A \<lbrace>Crypt (shrK A) Y, X\<rbrace> \<in> set evs; evs \<in> yahalom\<rbrakk> |
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\<Longrightarrow> X \<in> analz (knows Spy evs)" |
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by blast |
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lemmas YM4_parts_knows_Spy = |
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YM4_analz_knows_Spy [THEN analz_into_parts] |
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text\<open>For Oops\<close> |
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lemma YM4_Key_parts_knows_Spy: |
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"Says Server A \<lbrace>Crypt (shrK A) \<lbrace>B,K,NA,NB\<rbrace>, X\<rbrace> \<in> set evs |
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\<Longrightarrow> K \<in> parts (knows Spy evs)" |
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by (metis parts.Body parts.Fst parts.Snd Says_imp_knows_Spy parts.Inj) |
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text\<open>Theorems of the form \<^term>\<open>X \<notin> parts (knows Spy evs)\<close> imply |
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that NOBODY sends messages containing X!\<close> |
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text\<open>Spy never sees a good agent's shared key!\<close> |
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lemma Spy_see_shrK [simp]: |
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"evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)" |
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by (erule yahalom.induct, force, |
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drule_tac [6] YM4_parts_knows_Spy, simp_all, blast+) |
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lemma Spy_analz_shrK [simp]: |
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"evs \<in> yahalom \<Longrightarrow> (Key (shrK A) \<in> analz (knows Spy evs)) = (A \<in> bad)" |
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by auto |
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lemma Spy_see_shrK_D [dest!]: |
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"\<lbrakk>Key (shrK A) \<in> parts (knows Spy evs); evs \<in> yahalom\<rbrakk> \<Longrightarrow> A \<in> bad" |
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by (blast dest: Spy_see_shrK) |
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text\<open>Nobody can have used non-existent keys! |
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Needed to apply \<open>analz_insert_Key\<close>\<close> |
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lemma new_keys_not_used [simp]: |
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"\<lbrakk>Key K \<notin> used evs; K \<in> symKeys; evs \<in> yahalom\<rbrakk> |
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\<Longrightarrow> K \<notin> keysFor (parts (spies evs))" |
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apply (erule rev_mp) |
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apply (erule yahalom.induct, force, |
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frule_tac [6] YM4_parts_knows_Spy, simp_all) |
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txt\<open>Fake\<close> |
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apply (force dest!: keysFor_parts_insert, auto) |
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done |
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text\<open>Earlier, all protocol proofs declared this theorem. |
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But only a few proofs need it, e.g. Yahalom and Kerberos IV.\<close> |
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lemma new_keys_not_analzd: |
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"\<lbrakk>K \<in> symKeys; evs \<in> yahalom; Key K \<notin> used evs\<rbrakk> |
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\<Longrightarrow> K \<notin> keysFor (analz (knows Spy evs))" |
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by (blast dest: new_keys_not_used intro: keysFor_mono [THEN subsetD]) |
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text\<open>Describes the form of K when the Server sends this message. Useful for |
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Oops as well as main secrecy property.\<close> |
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lemma Says_Server_not_range [simp]: |
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"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> |
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\<in> set evs; evs \<in> yahalom\<rbrakk> |
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\<Longrightarrow> K \<notin> range shrK" |
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by (erule rev_mp, erule yahalom.induct, simp_all) |
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subsection\<open>Secrecy Theorems\<close> |
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(**** |
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The following is to prove theorems of the form |
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Key K \<in> analz (insert (Key KAB) (knows Spy evs)) \<Longrightarrow> |
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Key K \<in> analz (knows Spy evs) |
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A more general formula must be proved inductively. |
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****) |
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text\<open>Session keys are not used to encrypt other session keys\<close> |
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lemma analz_image_freshK [rule_format]: |
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"evs \<in> yahalom \<Longrightarrow> |
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\<forall>K KK. KK \<subseteq> - (range shrK) \<longrightarrow> |
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(Key K \<in> analz (Key`KK \<union> (knows Spy evs))) = |
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(K \<in> KK | Key K \<in> analz (knows Spy evs))" |
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apply (erule yahalom.induct, |
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drule_tac [7] YM4_analz_knows_Spy, analz_freshK, spy_analz, blast) |
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apply (simp only: Says_Server_not_range analz_image_freshK_simps) |
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apply safe |
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done |
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lemma analz_insert_freshK: |
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"\<lbrakk>evs \<in> yahalom; KAB \<notin> range shrK\<rbrakk> \<Longrightarrow> |
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(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = |
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(K = KAB | Key K \<in> analz (knows Spy evs))" |
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by (simp only: analz_image_freshK analz_image_freshK_simps) |
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text\<open>The Key K uniquely identifies the Server's message.\<close> |
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lemma unique_session_keys: |
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"\<lbrakk>Says Server A |
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\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs; |
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Says Server A' |
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\<lbrace>Crypt (shrK A') \<lbrace>Agent B', Key K, na', nb'\<rbrace>, X'\<rbrace> \<in> set evs; |
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evs \<in> yahalom\<rbrakk> |
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\<Longrightarrow> A=A' \<and> B=B' \<and> na=na' \<and> nb=nb'" |
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apply (erule rev_mp, erule rev_mp) |
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apply (erule yahalom.induct, simp_all) |
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txt\<open>YM3, by freshness, and YM4\<close> |
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apply blast+ |
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done |
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text\<open>Crucial secrecy property: Spy does not see the keys sent in msg YM3\<close> |
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lemma secrecy_lemma: |
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"\<lbrakk>A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
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\<Longrightarrow> Says Server A |
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\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, |
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Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
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\<in> set evs \<longrightarrow> |
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Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs \<longrightarrow> |
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Key K \<notin> analz (knows Spy evs)" |
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apply (erule yahalom.induct, force, |
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drule_tac [6] YM4_analz_knows_Spy) |
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apply (simp_all add: pushes analz_insert_eq analz_insert_freshK) |
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subgoal \<comment> \<open>Fake\<close> by spy_analz |
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subgoal \<comment> \<open>YM3\<close> by blast |
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subgoal \<comment> \<open>Oops\<close> by (blast dest: unique_session_keys) |
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done |
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text\<open>Final version\<close> |
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lemma Spy_not_see_encrypted_key: |
64364 | 244 |
"\<lbrakk>Says Server A |
61956 | 245 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, |
246 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
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\<in> set evs; |
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Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs; |
64364 | 249 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
250 |
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)" |
|
11251 | 251 |
by (blast dest: secrecy_lemma) |
252 |
||
253 |
||
61830 | 254 |
subsubsection\<open>Security Guarantee for A upon receiving YM3\<close> |
11251 | 255 |
|
61830 | 256 |
text\<open>If the encrypted message appears then it originated with the Server\<close> |
11251 | 257 |
lemma A_trusts_YM3: |
64364 | 258 |
"\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs); |
259 |
A \<notin> bad; evs \<in> yahalom\<rbrakk> |
|
260 |
\<Longrightarrow> Says Server A |
|
61956 | 261 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, |
262 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
|
11251 | 263 |
\<in> set evs" |
264 |
apply (erule rev_mp) |
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apply (erule yahalom.induct, force, |
11251 | 266 |
frule_tac [6] YM4_parts_knows_Spy, simp_all) |
61830 | 267 |
txt\<open>Fake, YM3\<close> |
11251 | 268 |
apply blast+ |
269 |
done |
|
270 |
||
61830 | 271 |
text\<open>The obvious combination of \<open>A_trusts_YM3\<close> with |
272 |
\<open>Spy_not_see_encrypted_key\<close>\<close> |
|
11251 | 273 |
lemma A_gets_good_key: |
64364 | 274 |
"\<lbrakk>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace> \<in> parts (knows Spy evs); |
61956 | 275 |
Notes Spy \<lbrace>na, nb, Key K\<rbrace> \<notin> set evs; |
64364 | 276 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
277 |
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)" |
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by (metis A_trusts_YM3 secrecy_lemma) |
11251 | 279 |
|
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|
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subsubsection\<open>Security Guarantees for B upon receiving YM4\<close> |
11251 | 282 |
|
61830 | 283 |
text\<open>B knows, by the first part of A's message, that the Server distributed |
284 |
the key for A and B. But this part says nothing about nonces.\<close> |
|
11251 | 285 |
lemma B_trusts_YM4_shrK: |
64364 | 286 |
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs); |
287 |
B \<notin> bad; evs \<in> yahalom\<rbrakk> |
|
288 |
\<Longrightarrow> \<exists>NA NB. Says Server A |
|
61956 | 289 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, |
290 |
Nonce NA, Nonce NB\<rbrace>, |
|
291 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
|
11251 | 292 |
\<in> set evs" |
293 |
apply (erule rev_mp) |
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apply (erule yahalom.induct, force, |
11251 | 295 |
frule_tac [6] YM4_parts_knows_Spy, simp_all) |
61830 | 296 |
txt\<open>Fake, YM3\<close> |
11251 | 297 |
apply blast+ |
298 |
done |
|
299 |
||
61830 | 300 |
text\<open>B knows, by the second part of A's message, that the Server |
17411 | 301 |
distributed the key quoting nonce NB. This part says nothing about |
69597 | 302 |
agent names. Secrecy of NB is crucial. Note that \<^term>\<open>Nonce NB |
303 |
\<notin> analz(knows Spy evs)\<close> must be the FIRST antecedent of the |
|
61830 | 304 |
induction formula.\<close> |
17411 | 305 |
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lemma B_trusts_YM4_newK [rule_format]: |
64364 | 307 |
"\<lbrakk>Crypt K (Nonce NB) \<in> parts (knows Spy evs); |
308 |
Nonce NB \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk> |
|
309 |
\<Longrightarrow> \<exists>A B NA. Says Server A |
|
61956 | 310 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, Nonce NB\<rbrace>, |
311 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
|
11251 | 312 |
\<in> set evs" |
313 |
apply (erule rev_mp, erule rev_mp) |
|
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apply (erule yahalom.induct, force, |
11251 | 315 |
frule_tac [6] YM4_parts_knows_Spy) |
64364 | 316 |
apply (analz_mono_contra, simp_all) |
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subgoal \<comment> \<open>Fake\<close> by blast |
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subgoal \<comment> \<open>YM3\<close> by blast |
61830 | 319 |
txt\<open>YM4. A is uncompromised because NB is secure |
320 |
A's certificate guarantees the existence of the Server message\<close> |
|
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apply (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad |
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dest: Says_imp_spies |
11251 | 323 |
parts.Inj [THEN parts.Fst, THEN A_trusts_YM3]) |
324 |
done |
|
325 |
||
326 |
||
61830 | 327 |
subsubsection\<open>Towards proving secrecy of Nonce NB\<close> |
11251 | 328 |
|
61830 | 329 |
text\<open>Lemmas about the predicate KeyWithNonce\<close> |
11251 | 330 |
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lemma KeyWithNonceI: |
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"Says Server A |
61956 | 333 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB\<rbrace>, X\<rbrace> |
64364 | 334 |
\<in> set evs \<Longrightarrow> KeyWithNonce K NB evs" |
11251 | 335 |
by (unfold KeyWithNonce_def, blast) |
336 |
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lemma KeyWithNonce_Says [simp]: |
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"KeyWithNonce K NB (Says S A X # evs) = |
67613 | 339 |
(Server = S \<and> |
61956 | 340 |
(\<exists>B n X'. X = \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, n, Nonce NB\<rbrace>, X'\<rbrace>) |
11251 | 341 |
| KeyWithNonce K NB evs)" |
342 |
by (simp add: KeyWithNonce_def, blast) |
|
343 |
||
344 |
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345 |
lemma KeyWithNonce_Notes [simp]: |
11251 | 346 |
"KeyWithNonce K NB (Notes A X # evs) = KeyWithNonce K NB evs" |
347 |
by (simp add: KeyWithNonce_def) |
|
348 |
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lemma KeyWithNonce_Gets [simp]: |
11251 | 350 |
"KeyWithNonce K NB (Gets A X # evs) = KeyWithNonce K NB evs" |
351 |
by (simp add: KeyWithNonce_def) |
|
352 |
||
61830 | 353 |
text\<open>A fresh key cannot be associated with any nonce |
354 |
(with respect to a given trace).\<close> |
|
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lemma fresh_not_KeyWithNonce: |
67613 | 356 |
"Key K \<notin> used evs \<Longrightarrow> \<not> KeyWithNonce K NB evs" |
11251 | 357 |
by (unfold KeyWithNonce_def, blast) |
358 |
||
61830 | 359 |
text\<open>The Server message associates K with NB' and therefore not with any |
360 |
other nonce NB.\<close> |
|
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lemma Says_Server_KeyWithNonce: |
64364 | 362 |
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, Nonce NB'\<rbrace>, X\<rbrace> |
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\<in> set evs; |
64364 | 364 |
NB \<noteq> NB'; evs \<in> yahalom\<rbrakk> |
67613 | 365 |
\<Longrightarrow> \<not> KeyWithNonce K NB evs" |
11251 | 366 |
by (unfold KeyWithNonce_def, blast dest: unique_session_keys) |
367 |
||
368 |
||
61830 | 369 |
text\<open>The only nonces that can be found with the help of session keys are |
11251 | 370 |
those distributed as nonce NB by the Server. The form of the theorem |
61830 | 371 |
recalls \<open>analz_image_freshK\<close>, but it is much more complicated.\<close> |
11251 | 372 |
|
373 |
||
61830 | 374 |
text\<open>As with \<open>analz_image_freshK\<close>, we take some pains to express the |
375 |
property as a logical equivalence so that the simplifier can apply it.\<close> |
|
11251 | 376 |
lemma Nonce_secrecy_lemma: |
67613 | 377 |
"P \<longrightarrow> (X \<in> analz (G \<union> H)) \<longrightarrow> (X \<in> analz H) \<Longrightarrow> |
378 |
P \<longrightarrow> (X \<in> analz (G \<union> H)) = (X \<in> analz H)" |
|
11251 | 379 |
by (blast intro: analz_mono [THEN subsetD]) |
380 |
||
381 |
lemma Nonce_secrecy: |
|
64364 | 382 |
"evs \<in> yahalom \<Longrightarrow> |
67613 | 383 |
(\<forall>KK. KK \<subseteq> - (range shrK) \<longrightarrow> |
384 |
(\<forall>K \<in> KK. K \<in> symKeys \<longrightarrow> \<not> KeyWithNonce K NB evs) \<longrightarrow> |
|
385 |
(Nonce NB \<in> analz (Key`KK \<union> (knows Spy evs))) = |
|
11251 | 386 |
(Nonce NB \<in> analz (knows Spy evs)))" |
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387 |
apply (erule yahalom.induct, |
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388 |
frule_tac [7] YM4_analz_knows_Spy) |
11251 | 389 |
apply (safe del: allI impI intro!: Nonce_secrecy_lemma [THEN impI, THEN allI]) |
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390 |
apply (simp_all del: image_insert image_Un |
11251 | 391 |
add: analz_image_freshK_simps split_ifs |
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392 |
all_conj_distrib ball_conj_distrib |
11251 | 393 |
analz_image_freshK fresh_not_KeyWithNonce |
394 |
imp_disj_not1 (*Moves NBa\<noteq>NB to the front*) |
|
395 |
Says_Server_KeyWithNonce) |
|
69597 | 396 |
txt\<open>For Oops, simplification proves \<^prop>\<open>NBa\<noteq>NB\<close>. By |
397 |
\<^term>\<open>Says_Server_KeyWithNonce\<close>, we get \<^prop>\<open>\<not> KeyWithNonce K NB |
|
398 |
evs\<close>; then simplification can apply the induction hypothesis with |
|
399 |
\<^term>\<open>KK = {K}\<close>.\<close> |
|
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400 |
subgoal \<comment> \<open>Fake\<close> by spy_analz |
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401 |
subgoal \<comment> \<open>YM2\<close> by blast |
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402 |
subgoal \<comment> \<open>YM3\<close> by blast |
69597 | 403 |
subgoal \<comment> \<open>YM4: If \<^prop>\<open>A \<in> bad\<close> then \<^term>\<open>NBa\<close> is known, therefore \<^prop>\<open>NBa \<noteq> NB\<close>.\<close> |
64364 | 404 |
by (metis A_trusts_YM3 Gets_imp_analz_Spy Gets_imp_knows_Spy KeyWithNonce_def |
405 |
Spy_analz_shrK analz.Fst analz.Snd analz_shrK_Decrypt parts.Fst parts.Inj) |
|
11251 | 406 |
done |
407 |
||
408 |
||
61830 | 409 |
text\<open>Version required below: if NB can be decrypted using a session key then |
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it was distributed with that key. The more general form above is required |
61830 | 411 |
for the induction to carry through.\<close> |
11251 | 412 |
lemma single_Nonce_secrecy: |
64364 | 413 |
"\<lbrakk>Says Server A |
61956 | 414 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key KAB, na, Nonce NB'\<rbrace>, X\<rbrace> |
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415 |
\<in> set evs; |
64364 | 416 |
NB \<noteq> NB'; KAB \<notin> range shrK; evs \<in> yahalom\<rbrakk> |
417 |
\<Longrightarrow> (Nonce NB \<in> analz (insert (Key KAB) (knows Spy evs))) = |
|
11251 | 418 |
(Nonce NB \<in> analz (knows Spy evs))" |
419 |
by (simp_all del: image_insert image_Un imp_disjL |
|
420 |
add: analz_image_freshK_simps split_ifs |
|
13507 | 421 |
Nonce_secrecy Says_Server_KeyWithNonce) |
11251 | 422 |
|
423 |
||
61830 | 424 |
subsubsection\<open>The Nonce NB uniquely identifies B's message.\<close> |
11251 | 425 |
|
426 |
lemma unique_NB: |
|
64364 | 427 |
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs); |
61956 | 428 |
Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace> \<in> parts (knows Spy evs); |
64364 | 429 |
evs \<in> yahalom; B \<notin> bad; B' \<notin> bad\<rbrakk> |
67613 | 430 |
\<Longrightarrow> NA' = NA \<and> A' = A \<and> B' = B" |
11251 | 431 |
apply (erule rev_mp, erule rev_mp) |
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432 |
apply (erule yahalom.induct, force, |
11251 | 433 |
frule_tac [6] YM4_parts_knows_Spy, simp_all) |
61830 | 434 |
txt\<open>Fake, and YM2 by freshness\<close> |
11251 | 435 |
apply blast+ |
436 |
done |
|
437 |
||
438 |
||
61830 | 439 |
text\<open>Variant useful for proving secrecy of NB. Because nb is assumed to be |
440 |
secret, we no longer must assume B, B' not bad.\<close> |
|
11251 | 441 |
lemma Says_unique_NB: |
64364 | 442 |
"\<lbrakk>Says C S \<lbrace>X, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> |
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\<in> set evs; |
61956 | 444 |
Gets S' \<lbrace>X', Crypt (shrK B') \<lbrace>Agent A', Nonce NA', nb\<rbrace>\<rbrace> |
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\<in> set evs; |
64364 | 446 |
nb \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk> |
67613 | 447 |
\<Longrightarrow> NA' = NA \<and> A' = A \<and> B' = B" |
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by (blast dest!: Gets_imp_Says Crypt_Spy_analz_bad |
11251 | 449 |
dest: Says_imp_spies unique_NB parts.Inj analz.Inj) |
450 |
||
451 |
||
61830 | 452 |
subsubsection\<open>A nonce value is never used both as NA and as NB\<close> |
11251 | 453 |
|
454 |
lemma no_nonce_YM1_YM2: |
|
64364 | 455 |
"\<lbrakk>Crypt (shrK B') \<lbrace>Agent A', Nonce NB, nb'\<rbrace> \<in> parts(knows Spy evs); |
456 |
Nonce NB \<notin> analz (knows Spy evs); evs \<in> yahalom\<rbrakk> |
|
457 |
\<Longrightarrow> Crypt (shrK B) \<lbrace>Agent A, na, Nonce NB\<rbrace> \<notin> parts(knows Spy evs)" |
|
11251 | 458 |
apply (erule rev_mp, erule rev_mp) |
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|
459 |
apply (erule yahalom.induct, force, |
11251 | 460 |
frule_tac [6] YM4_parts_knows_Spy) |
461 |
apply (analz_mono_contra, simp_all) |
|
61830 | 462 |
txt\<open>Fake, YM2\<close> |
11251 | 463 |
apply blast+ |
464 |
done |
|
465 |
||
61830 | 466 |
text\<open>The Server sends YM3 only in response to YM2.\<close> |
11251 | 467 |
lemma Says_Server_imp_YM2: |
64364 | 468 |
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, k, na, nb\<rbrace>, X\<rbrace> \<in> set evs; |
469 |
evs \<in> yahalom\<rbrakk> |
|
470 |
\<Longrightarrow> Gets Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, na, nb\<rbrace>\<rbrace> |
|
11251 | 471 |
\<in> set evs" |
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472 |
by (erule rev_mp, erule yahalom.induct, auto) |
11251 | 473 |
|
61830 | 474 |
text\<open>A vital theorem for B, that nonce NB remains secure from the Spy.\<close> |
64364 | 475 |
theorem Spy_not_see_NB : |
476 |
"\<lbrakk>Says B Server |
|
61956 | 477 |
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
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478 |
\<in> set evs; |
61956 | 479 |
(\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs); |
64364 | 480 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
481 |
\<Longrightarrow> Nonce NB \<notin> analz (knows Spy evs)" |
|
11251 | 482 |
apply (erule rev_mp, erule rev_mp) |
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483 |
apply (erule yahalom.induct, force, |
11251 | 484 |
frule_tac [6] YM4_analz_knows_Spy) |
485 |
apply (simp_all add: split_ifs pushes new_keys_not_analzd analz_insert_eq |
|
486 |
analz_insert_freshK) |
|
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487 |
subgoal \<comment> \<open>Fake\<close> by spy_analz |
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488 |
subgoal \<comment> \<open>YM1: NB=NA is impossible anyway, but NA is secret because it is fresh!\<close> by blast |
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489 |
subgoal \<comment> \<open>YM2\<close> by blast |
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|
490 |
subgoal \<comment> \<open>YM3: because no NB can also be an NA\<close> |
64364 | 491 |
by (blast dest!: no_nonce_YM1_YM2 dest: Gets_imp_Says Says_unique_NB) |
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3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
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|
492 |
subgoal \<comment> \<open>YM4: key K is visible to Spy, contradicting session key secrecy theorem\<close> |
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
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parents:
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|
493 |
\<comment> \<open>Case analysis on whether Aa is bad; |
69597 | 494 |
use \<open>Says_unique_NB\<close> to identify message components: \<^term>\<open>Aa=A\<close>, \<^term>\<open>Ba=B\<close>\<close> |
64364 | 495 |
apply clarify |
496 |
apply (blast dest!: Says_unique_NB analz_shrK_Decrypt |
|
497 |
parts.Inj [THEN parts.Fst, THEN A_trusts_YM3] |
|
498 |
dest: Gets_imp_Says Says_imp_spies Says_Server_imp_YM2 |
|
499 |
Spy_not_see_encrypted_key) |
|
500 |
done |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67226
diff
changeset
|
501 |
subgoal \<comment> \<open>Oops case: if the nonce is betrayed now, show that the Oops event is |
64364 | 502 |
covered by the quantified Oops assumption.\<close> |
503 |
apply clarsimp |
|
504 |
apply (metis Says_Server_imp_YM2 Gets_imp_Says Says_Server_not_range Says_unique_NB no_nonce_YM1_YM2 parts.Snd single_Nonce_secrecy spies_partsEs(1)) |
|
505 |
done |
|
11251 | 506 |
done |
507 |
||
61830 | 508 |
text\<open>B's session key guarantee from YM4. The two certificates contribute to a |
11251 | 509 |
single conclusion about the Server's message. Note that the "Notes Spy" |
61830 | 510 |
assumption must quantify over \<open>\<forall>\<close> POSSIBLE keys instead of our particular K. |
11251 | 511 |
If this run is broken and the spy substitutes a certificate containing an |
61830 | 512 |
old key, B has no means of telling.\<close> |
11251 | 513 |
lemma B_trusts_YM4: |
64364 | 514 |
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, |
61956 | 515 |
Crypt K (Nonce NB)\<rbrace> \<in> set evs; |
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f20fbb141673
Conversion of all main protocols from "Shared" to "Public".
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parents:
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diff
changeset
|
516 |
Says B Server |
61956 | 517 |
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
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Conversion of all main protocols from "Shared" to "Public".
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parents:
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diff
changeset
|
518 |
\<in> set evs; |
61956 | 519 |
\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs; |
64364 | 520 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
521 |
\<Longrightarrow> Says Server A |
|
61956 | 522 |
\<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, |
523 |
Nonce NA, Nonce NB\<rbrace>, |
|
524 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>\<rbrace> |
|
11251 | 525 |
\<in> set evs" |
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f20fbb141673
Conversion of all main protocols from "Shared" to "Public".
paulson
parents:
14200
diff
changeset
|
526 |
by (blast dest: Spy_not_see_NB Says_unique_NB |
11251 | 527 |
Says_Server_imp_YM2 B_trusts_YM4_newK) |
528 |
||
529 |
||
530 |
||
61830 | 531 |
text\<open>The obvious combination of \<open>B_trusts_YM4\<close> with |
532 |
\<open>Spy_not_see_encrypted_key\<close>\<close> |
|
11251 | 533 |
lemma B_gets_good_key: |
64364 | 534 |
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, |
61956 | 535 |
Crypt K (Nonce NB)\<rbrace> \<in> set evs; |
14207
f20fbb141673
Conversion of all main protocols from "Shared" to "Public".
paulson
parents:
14200
diff
changeset
|
536 |
Says B Server |
61956 | 537 |
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
14207
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Conversion of all main protocols from "Shared" to "Public".
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parents:
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diff
changeset
|
538 |
\<in> set evs; |
61956 | 539 |
\<forall>k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs; |
64364 | 540 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
541 |
\<Longrightarrow> Key K \<notin> analz (knows Spy evs)" |
|
32367
a508148f7c25
Removal of redundant settings of unification trace and search bounds.
paulson
parents:
23746
diff
changeset
|
542 |
by (metis B_trusts_YM4 Spy_not_see_encrypted_key) |
11251 | 543 |
|
544 |
||
61830 | 545 |
subsection\<open>Authenticating B to A\<close> |
11251 | 546 |
|
61830 | 547 |
text\<open>The encryption in message YM2 tells us it cannot be faked.\<close> |
11251 | 548 |
lemma B_Said_YM2 [rule_format]: |
64364 | 549 |
"\<lbrakk>Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace> \<in> parts (knows Spy evs); |
550 |
evs \<in> yahalom\<rbrakk> |
|
67613 | 551 |
\<Longrightarrow> B \<notin> bad \<longrightarrow> |
61956 | 552 |
Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> |
11251 | 553 |
\<in> set evs" |
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Conversion of all main protocols from "Shared" to "Public".
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parents:
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diff
changeset
|
554 |
apply (erule rev_mp, erule yahalom.induct, force, |
11251 | 555 |
frule_tac [6] YM4_parts_knows_Spy, simp_all) |
61830 | 556 |
txt\<open>Fake\<close> |
11251 | 557 |
apply blast |
558 |
done |
|
559 |
||
61830 | 560 |
text\<open>If the server sends YM3 then B sent YM2\<close> |
11251 | 561 |
lemma YM3_auth_B_to_A_lemma: |
64364 | 562 |
"\<lbrakk>Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace> |
563 |
\<in> set evs; evs \<in> yahalom\<rbrakk> |
|
67613 | 564 |
\<Longrightarrow> B \<notin> bad \<longrightarrow> |
61956 | 565 |
Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> |
11251 | 566 |
\<in> set evs" |
567 |
apply (erule rev_mp, erule yahalom.induct, simp_all) |
|
61830 | 568 |
txt\<open>YM3, YM4\<close> |
11251 | 569 |
apply (blast dest!: B_Said_YM2)+ |
570 |
done |
|
571 |
||
61830 | 572 |
text\<open>If A receives YM3 then B has used nonce NA (and therefore is alive)\<close> |
64364 | 573 |
theorem YM3_auth_B_to_A: |
574 |
"\<lbrakk>Gets A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA, nb\<rbrace>, X\<rbrace> |
|
14207
f20fbb141673
Conversion of all main protocols from "Shared" to "Public".
paulson
parents:
14200
diff
changeset
|
575 |
\<in> set evs; |
64364 | 576 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
577 |
\<Longrightarrow> Says B Server \<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, nb\<rbrace>\<rbrace> |
|
11251 | 578 |
\<in> set evs" |
32367
a508148f7c25
Removal of redundant settings of unification trace and search bounds.
paulson
parents:
23746
diff
changeset
|
579 |
by (metis A_trusts_YM3 Gets_imp_analz_Spy YM3_auth_B_to_A_lemma analz.Fst |
a508148f7c25
Removal of redundant settings of unification trace and search bounds.
paulson
parents:
23746
diff
changeset
|
580 |
not_parts_not_analz) |
11251 | 581 |
|
582 |
||
61830 | 583 |
subsection\<open>Authenticating A to B using the certificate |
69597 | 584 |
\<^term>\<open>Crypt K (Nonce NB)\<close>\<close> |
11251 | 585 |
|
61830 | 586 |
text\<open>Assuming the session key is secure, if both certificates are present then |
11251 | 587 |
A has said NB. We can't be sure about the rest of A's message, but only |
61830 | 588 |
NB matters for freshness.\<close> |
64364 | 589 |
theorem A_Said_YM3_lemma [rule_format]: |
11251 | 590 |
"evs \<in> yahalom |
67613 | 591 |
\<Longrightarrow> Key K \<notin> analz (knows Spy evs) \<longrightarrow> |
592 |
Crypt K (Nonce NB) \<in> parts (knows Spy evs) \<longrightarrow> |
|
593 |
Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs) \<longrightarrow> |
|
594 |
B \<notin> bad \<longrightarrow> |
|
61956 | 595 |
(\<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs)" |
14207
f20fbb141673
Conversion of all main protocols from "Shared" to "Public".
paulson
parents:
14200
diff
changeset
|
596 |
apply (erule yahalom.induct, force, |
11251 | 597 |
frule_tac [6] YM4_parts_knows_Spy) |
598 |
apply (analz_mono_contra, simp_all) |
|
67443
3abf6a722518
standardized towards new-style formal comments: isabelle update_comments;
wenzelm
parents:
67226
diff
changeset
|
599 |
subgoal \<comment> \<open>Fake\<close> by blast |
69597 | 600 |
subgoal \<comment> \<open>YM3 because the message \<^term>\<open>Crypt K (Nonce NB)\<close> could not exist\<close> |
64364 | 601 |
by (force dest!: Crypt_imp_keysFor) |
69597 | 602 |
subgoal \<comment> \<open>YM4: was \<^term>\<open>Crypt K (Nonce NB)\<close> the very last message? If not, use the induction hypothesis, |
64364 | 603 |
otherwise by unicity of session keys\<close> |
604 |
by (blast dest!: Gets_imp_Says A_trusts_YM3 B_trusts_YM4_shrK Crypt_Spy_analz_bad |
|
11251 | 605 |
dest: Says_imp_knows_Spy [THEN parts.Inj] unique_session_keys) |
606 |
done |
|
607 |
||
61830 | 608 |
text\<open>If B receives YM4 then A has used nonce NB (and therefore is alive). |
11251 | 609 |
Moreover, A associates K with NB (thus is talking about the same run). |
61830 | 610 |
Other premises guarantee secrecy of K.\<close> |
64364 | 611 |
theorem YM4_imp_A_Said_YM3 [rule_format]: |
612 |
"\<lbrakk>Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace>, |
|
61956 | 613 |
Crypt K (Nonce NB)\<rbrace> \<in> set evs; |
11251 | 614 |
Says B Server |
61956 | 615 |
\<lbrace>Agent B, Crypt (shrK B) \<lbrace>Agent A, Nonce NA, Nonce NB\<rbrace>\<rbrace> |
11251 | 616 |
\<in> set evs; |
61956 | 617 |
(\<forall>NA k. Notes Spy \<lbrace>Nonce NA, Nonce NB, k\<rbrace> \<notin> set evs); |
64364 | 618 |
A \<notin> bad; B \<notin> bad; evs \<in> yahalom\<rbrakk> |
619 |
\<Longrightarrow> \<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs" |
|
32367
a508148f7c25
Removal of redundant settings of unification trace and search bounds.
paulson
parents:
23746
diff
changeset
|
620 |
by (metis A_Said_YM3_lemma B_gets_good_key Gets_imp_analz_Spy YM4_parts_knows_Spy analz.Fst not_parts_not_analz) |
64364 | 621 |
|
1985
84cf16192e03
Tidied many proofs, using AddIffs to let equivalences take
paulson
parents:
diff
changeset
|
622 |
end |