# Theory Guard_NS_Public

```(*  Title:      HOL/Auth/Guard/Guard_NS_Public.thy
Author:     Frederic Blanqui, University of Cambridge Computer Laboratory

Incorporating Lowe's fix (inclusion of B's identity in round 2).
*)

section‹Needham-Schroeder-Lowe Public-Key Protocol›

theory Guard_NS_Public imports Guard_Public begin

subsection‹messages used in the protocol›

abbreviation (input)
ns1 :: "agent => agent => nat => event" where
"ns1 A B NA == Says A B (Crypt (pubK B) ⦃Nonce NA, Agent A⦄)"

abbreviation (input)
ns1' :: "agent => agent => agent => nat => event" where
"ns1' A' A B NA == Says A' B (Crypt (pubK B) ⦃Nonce NA, Agent A⦄)"

abbreviation (input)
ns2 :: "agent => agent => nat => nat => event" where
"ns2 B A NA NB == Says B A (Crypt (pubK A) ⦃Nonce NA, Nonce NB, Agent B⦄)"

abbreviation (input)
ns2' :: "agent => agent => agent => nat => nat => event" where
"ns2' B' B A NA NB == Says B' A (Crypt (pubK A) ⦃Nonce NA, Nonce NB, Agent B⦄)"

abbreviation (input)
ns3 :: "agent => agent => nat => event" where
"ns3 A B NB == Says A B (Crypt (pubK B) (Nonce NB))"

subsection‹definition of the protocol›

inductive_set nsp :: "event list set"
where

Nil: "[] ∈ nsp"

| Fake: "⟦evs ∈ nsp; X ∈ synth (analz (spies evs))⟧ ⟹ Says Spy B X # evs ∈ nsp"

| NS1: "⟦evs1 ∈ nsp; Nonce NA ∉ used evs1⟧ ⟹ ns1 A B NA # evs1 ∈ nsp"

| NS2: "⟦evs2 ∈ nsp; Nonce NB ∉ used evs2; ns1' A' A B NA ∈ set evs2⟧ ⟹
ns2 B A NA NB # evs2 ∈ nsp"

| NS3: "⋀A B B' NA NB evs3. ⟦evs3 ∈ nsp; ns1 A B NA ∈ set evs3; ns2' B' B A NA NB ∈ set evs3⟧ ⟹
ns3 A B NB # evs3 ∈ nsp"

subsection‹declarations for tactics›

declare knows_Spy_partsEs [elim]
declare Fake_parts_insert [THEN subsetD, dest]
declare initState.simps [simp del]

subsection‹general properties of nsp›

lemma nsp_has_no_Gets: "evs ∈ nsp ⟹ ∀A X. Gets A X ∉ set evs"
by (erule nsp.induct, auto)

lemma nsp_is_Gets_correct [iff]: "Gets_correct nsp"
by (auto simp: Gets_correct_def dest: nsp_has_no_Gets)

lemma nsp_is_one_step [iff]: "one_step nsp"
unfolding one_step_def by (clarify, ind_cases "ev#evs ∈ nsp" for ev evs, auto)

lemma nsp_has_only_Says' [rule_format]: "evs ∈ nsp ⟹
ev ∈ set evs ⟶ (∃A B X. ev=Says A B X)"
by (erule nsp.induct, auto)

lemma nsp_has_only_Says [iff]: "has_only_Says nsp"
by (auto simp: has_only_Says_def dest: nsp_has_only_Says')

lemma nsp_is_regular [iff]: "regular nsp"
apply (simp only: regular_def, clarify)
by (erule nsp.induct, auto simp: initState.simps knows.simps)

subsection‹nonce are used only once›

lemma NA_is_uniq [rule_format]: "evs ∈ nsp ⟹
Crypt (pubK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs)
⟶ Crypt (pubK B') ⦃Nonce NA, Agent A'⦄ ∈ parts (spies evs)
⟶ Nonce NA ∉ analz (spies evs) ⟶ A=A' ∧ B=B'"
apply (erule nsp.induct, simp_all)
by (blast intro: analz_insertI)+

lemma no_Nonce_NS1_NS2 [rule_format]: "evs ∈ nsp ⟹
Crypt (pubK B') ⦃Nonce NA', Nonce NA, Agent A'⦄ ∈ parts (spies evs)
⟶ Crypt (pubK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs)
⟶ Nonce NA ∈ analz (spies evs)"
apply (erule nsp.induct, simp_all)
by (blast intro: analz_insertI)+

lemma no_Nonce_NS1_NS2' [rule_format]:
"⟦Crypt (pubK B') ⦃Nonce NA', Nonce NA, Agent A'⦄ ∈ parts (spies evs);
Crypt (pubK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs); evs ∈ nsp⟧
⟹ Nonce NA ∈ analz (spies evs)"
by (rule no_Nonce_NS1_NS2, auto)

lemma NB_is_uniq [rule_format]: "evs ∈ nsp ⟹
Crypt (pubK A) ⦃Nonce NA, Nonce NB, Agent B⦄ ∈ parts (spies evs)
⟶ Crypt (pubK A') ⦃Nonce NA', Nonce NB, Agent B'⦄ ∈ parts (spies evs)
⟶ Nonce NB ∉ analz (spies evs) ⟶ A=A' ∧ B=B' ∧ NA=NA'"
apply (erule nsp.induct, simp_all)
by (blast intro: analz_insertI)+

subsection‹guardedness of NA›

lemma ns1_imp_Guard [rule_format]: "⟦evs ∈ nsp; A ∉ bad; B ∉ bad⟧ ⟹
ns1 A B NA ∈ set evs ⟶ Guard NA {priK A,priK B} (spies evs)"
apply (erule nsp.induct)
(* Nil *)
apply simp_all
(* Fake *)
apply safe
apply (erule in_synth_Guard, erule Guard_analz, simp)
(* NS1 *)
apply blast
apply blast
apply blast
apply (drule Nonce_neq, simp+, rule No_Nonce, simp)
(* NS2 *)
apply (frule_tac A=A in Nonce_neq, simp+)
apply (case_tac "NAa=NA")
apply (drule Guard_Nonce_analz, simp+)
apply (drule Says_imp_knows_Spy)+
apply (drule_tac B=B and A'=Aa in NA_is_uniq, auto)
(* NS3 *)
apply (case_tac "NB=NA", clarify)
apply (drule Guard_Nonce_analz, simp+)
apply (drule Says_imp_knows_Spy)+
by (drule no_Nonce_NS1_NS2, auto)

subsection‹guardedness of NB›

lemma ns2_imp_Guard [rule_format]: "⟦evs ∈ nsp; A ∉ bad; B ∉ bad⟧ ⟹
ns2 B A NA NB ∈ set evs ⟶ Guard NB {priK A,priK B} (spies evs)"
apply (erule nsp.induct)
(* Nil *)
apply simp_all
(* Fake *)
apply safe
apply (erule in_synth_Guard, erule Guard_analz, simp)
(* NS1 *)
apply (frule Nonce_neq, simp+, blast, rule No_Nonce, simp)
(* NS2 *)
apply blast
apply blast
apply blast
apply (frule_tac A=B and n=NB in Nonce_neq, simp+)
apply (case_tac "NAa=NB")
apply (drule Guard_Nonce_analz, simp+)
apply (drule Says_imp_knows_Spy)+
apply (drule no_Nonce_NS1_NS2, auto)
(* NS3 *)
apply (case_tac "NBa=NB", clarify)
apply (drule Guard_Nonce_analz, simp+)
apply (drule Says_imp_knows_Spy)+
apply (drule_tac A=Aa and A'=A in NB_is_uniq)
apply auto[1]
done

subsection‹Agents' Authentication›

lemma B_trusts_NS1: "⟦evs ∈ nsp; A ∉ bad; B ∉ bad⟧ ⟹
Crypt (pubK B) ⦃Nonce NA, Agent A⦄ ∈ parts (spies evs)
⟶ Nonce NA ∉ analz (spies evs) ⟶ ns1 A B NA ∈ set evs"
apply (erule nsp.induct, simp_all)
by (blast intro: analz_insertI)+

lemma A_trusts_NS2: "⟦evs ∈ nsp; A ∉ bad; B ∉ bad⟧ ⟹ ns1 A B NA ∈ set evs
⟶ Crypt (pubK A) ⦃Nonce NA, Nonce NB, Agent B⦄ ∈ parts (spies evs)
⟶ ns2 B A NA NB ∈ set evs"
apply (erule nsp.induct, simp_all, safe)
apply (frule_tac B=B in ns1_imp_Guard, simp+)
apply (drule Guard_Nonce_analz, simp+, blast)
apply (frule_tac B=B in ns1_imp_Guard, simp+)
apply (drule Guard_Nonce_analz, simp+, blast)
apply (frule_tac B=B in ns1_imp_Guard, simp+)
by (drule Guard_Nonce_analz, simp+, blast+)

lemma B_trusts_NS3: "⟦evs ∈ nsp; A ∉ bad; B ∉ bad⟧ ⟹ ns2 B A NA NB ∈ set evs
⟶ Crypt (pubK B) (Nonce NB) ∈ parts (spies evs) ⟶ ns3 A B NB ∈ set evs"
apply (erule nsp.induct, simp_all, safe)
apply (frule_tac B=B in ns2_imp_Guard, simp+)
apply (drule Guard_Nonce_analz, simp+, blast)
apply (frule_tac B=B in ns2_imp_Guard, simp+)
apply (drule Guard_Nonce_analz, simp+, blast)
apply (frule_tac B=B in ns2_imp_Guard, simp+)
apply (drule Guard_Nonce_analz, simp+, blast, blast)
apply (frule_tac B=B in ns2_imp_Guard, simp+)
by (drule Guard_Nonce_analz, auto dest: Says_imp_knows_Spy NB_is_uniq)

end
```