chapter {* Analysing WideMouthFrog *} (* *********************************** This file is automatically generated from the AnB file "AnB/WideMouthFrog.AnB". Backend: Open Source Fixedpoint Model Checker version 2009c ************************************ *) theory "WideMouthFrog" imports "../src/ofmc" begin section {* Protocol Model (WideMouthFrog) *} datatype Role = rA | rB | rs datatype Purpose = purposeKAB | purposeNI | purposeTA | purposeTS datatype Agent = honest nat | dishonest nat datatype Nonce = "ni" "nat" | "absTA" "Msg" "nat" | "absTS" "Msg" "nat" | "NI" | "TA" | "TS" and Msg = Nonce "Nonce" | Agent "Agent" | Purpose "Purpose" | pair "Msg*Msg" | scrypt "Msg*Msg" | crypt "Msg*Msg" | inv "Msg" | SID "nat" | Step "nat" | authentication | secrecy (* SymKeys *) | SymKey "Msg" | "absKAB" "Msg" "nat" (* Functions *) | "sk" "Msg" datatype Fact = Iknows Msg | State "Role * (Msg list)" | Secret "Msg * Msg" | Attack "Msg" | Witness "Msg * Msg * Msg * Msg" | Request "Msg * Msg * Msg * Msg * Msg" section {* Inductive Protocol Definition (WideMouthFrog) *} inductive_set WideMouthFrog::"Fact list set" where init_0: "[ Iknows(Nonce((ni Abs_NI)))] : WideMouthFrog" | init_1: "[ Iknows(Agent(dishonest(i)))] : WideMouthFrog" | init_2: "[ State(rA, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), SID(sid)] )] : WideMouthFrog" | init_3: "[ State(rA, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), SID(sid)] )] : WideMouthFrog" | init_4: "[ Iknows(Step(0))] : WideMouthFrog" | init_5: "[ Iknows(SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))))] : WideMouthFrog" | init_6: "[ Iknows(SID(sid))] : WideMouthFrog" | init_7: "[ Iknows(Agent(honest(a)))] : WideMouthFrog" | init_8: "[ State(rB, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), SID(sid)] )] : WideMouthFrog" | init_9: "[ State(rB, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), SID(sid)] )] : WideMouthFrog" | init_10: "[ State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), Agent(dishonest(i)), Agent(dishonest(i)), SID(sid)] )] : WideMouthFrog" | init_11: "[ State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), Agent(honest(a)), SID(sid)] )] : WideMouthFrog" | init_12: "[ State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), Agent(honest(a)), Agent(dishonest(i)), SID(sid)] )] : WideMouthFrog" | init_13: "[ State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), Agent(honest(a)), SID(sid)] )] : WideMouthFrog" | rule_0: "[| t :WideMouthFrog; Iknows(crypt(K, M)) : (set t); Iknows(inv(K)) : (set t)|] ==> ((Iknows(M)) #t) : WideMouthFrog" | rule_1: "[| t :WideMouthFrog; Iknows(crypt(inv(K), M)) : (set t); Iknows(K) : (set t)|] ==> ((Iknows(M)) #t) : WideMouthFrog" | rule_2: "[| t :WideMouthFrog; Iknows(scrypt(K, M)) : (set t); Iknows(K) : (set t)|] ==> ((Iknows(M)) #t) : WideMouthFrog" | rule_3: "[| t :WideMouthFrog; Iknows(pair(M1, M2)) : (set t)|] ==> ((Iknows(M1)) #(Iknows(M2)) #t) : WideMouthFrog" | rule_4: "[| t :WideMouthFrog; Secret(M, Agent(honest(a))) : (set t); Iknows(M) : (set t)|] ==> ((Attack(pair(secrecy, M))) #t) : WideMouthFrog" | rule_5: "[| t :WideMouthFrog; Request(A, B, Purpose(purposeTA), M, SID(sid)) : (set t); ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_TA . M = Nonce((absTA(pair(B, A)) Abs_TA)))|] ==> ((Attack(pair(authentication, pair(A, pair(B, M))))) #t) : WideMouthFrog" | rule_6: "[| t :WideMouthFrog; Request(A, B, Purpose(purposeKAB), M, SID(sid)) : (set t); ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_KAB . M = SymKey((absKAB(pair(B, A)) Abs_KAB)))|] ==> ((Attack(pair(authentication, pair(A, pair(B, M))))) #t) : WideMouthFrog" | rule_7: "[| t :WideMouthFrog; Request(A, B, Purpose(purposeTS), M, SID(sid)) : (set t); ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_TS . M = Nonce((absTS(pair(B, A)) Abs_TS)))|] ==> ((Attack(pair(authentication, pair(A, pair(B, M))))) #t) : WideMouthFrog" | rule_8: "[| t :WideMouthFrog; State(rA, [Agent(A), Step(0), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), SID(sid)] ) : (set t)|] ==> ((Secret(SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)), Agent(B))) #(Witness(Agent(A), Agent(B), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)))) #(State(rA, [Agent(A), Step(1), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), pair(Agent(B), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)))))), SID(sid)] )) #(Iknows(pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), pair(Agent(B), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)))))))) #t) : WideMouthFrog" | rule_9: "[| t :WideMouthFrog; State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SID(sid)] ) : (set t); Iknows(Agent(A)) : (set t); Iknows(scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))) : (set t)|] ==> ((State(rs, [Agent(honest(a)), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SymKey(KAB), Nonce(TA), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB)))), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))), Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))), SID(sid)] )) #(Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))))) #t) : WideMouthFrog" | rule_10: "[| t :WideMouthFrog; State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SID(sid)] ) : (set t); Iknows(Agent(A)) : (set t); Iknows(SymKey(sk(pair(Agent(A), Agent(honest(a)))))) : (set t); Iknows(Nonce(TA)) : (set t); Iknows(Agent(B)) : (set t); Iknows(SymKey(KAB)) : (set t)|] ==> ((State(rs, [Agent(honest(a)), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SymKey(KAB), Nonce(TA), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB)))), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))), Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))), SID(sid)] )) #(Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))))) #t) : WideMouthFrog" | rule_11: "[| t :WideMouthFrog; State(rB, [Agent(B), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SID(sid)] ) : (set t); Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB))))) : (set t)|] ==> ((Request(Agent(B), Agent(A), Purpose(purposeKAB), SymKey(KAB), SID(sid))) #(State(rB, [Agent(B), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SymKey(KAB), Nonce(TS), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB)))), SID(sid)] )) #t) : WideMouthFrog" | rule_12: "[| t :WideMouthFrog; State(rB, [Agent(B), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SID(sid)] ) : (set t); Iknows(SymKey(sk(pair(Agent(B), Agent(honest(a)))))) : (set t); Iknows(Nonce(TS)) : (set t); Iknows(Agent(A)) : (set t); Iknows(SymKey(KAB)) : (set t)|] ==> ((Request(Agent(B), Agent(A), Purpose(purposeKAB), SymKey(KAB), SID(sid))) #(State(rB, [Agent(B), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SymKey(KAB), Nonce(TS), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB)))), SID(sid)] )) #t) : WideMouthFrog" section {* Fixed-point Definition (WideMouthFrog) *} definition "WideMouthFrog_fp = {m. ( ? Abs_TA2 Abs_TA1 a21 a22 a23 a24 a25 a26 a27 a28 a29 a30 a31 a32 a33 a34 a17 a18 a19 a20 i19 i20 i21 i22 i23 i24 i18 i12 Abs_TS2 i13 i14 Abs_TS3 i15 Abs_TS4 i16 i17 Abs_NI0 Abs_NI1 i6 a9 Abs_NI2 a10 a11 i7 Abs_KAB2 a12 i8 a13 a14 a15 i9 i10 a16 i11 Abs_KAB3 Abs_TS1 a8 i5 Abs_KAB1 sid0 i4 a5 a6 a7 a3 Abs_TA0 a4 a1 a2 i0 i1 Abs_TS0 i2 a0 i3 Abs_KAB0 . (m = Iknows(Nonce((ni Abs_NI0)))) | (m = Iknows(Agent(dishonest(i0)))) | (m = State(rA, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), SID(sid0)] )) | (m = State(rA, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(honest(a3)), SID(sid0)] )) | (m = Iknows(Step(0))) | (m = Iknows(SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a0))))))) | (m = Iknows(SID(sid0))) | (m = Iknows(Agent(honest(a0)))) | (m = State(rB, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), SID(sid0)] )) | (m = State(rB, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(honest(a3)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(honest(a2)), Agent(honest(a3))))), Agent(dishonest(i1)), Agent(honest(a4)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(0), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(honest(a3)), Agent(honest(a4))))), Agent(honest(a5)), Agent(honest(a6)), SID(sid0)] )) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_TA0)), pair(Agent(honest(a5)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)))))))) | (m = Secret(SymKey((absKAB(pair(Agent(honest(a0)), Agent(dishonest(i0)))) Abs_KAB0)), Agent(dishonest(i1)))) | (m = Witness(Agent(honest(a0)), Agent(dishonest(i0)), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i1)))) Abs_KAB0)))) | (m = State(rA, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i1)))) Abs_TA0)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)), pair(Agent(honest(a5)), scrypt(SymKey(sk(pair(Agent(honest(a6)), Agent(honest(a7))))), pair(Nonce((absTA(pair(Agent(honest(a8)), Agent(dishonest(i3)))) Abs_TA1)), pair(Agent(dishonest(i4)), SymKey((absKAB(pair(Agent(honest(a9)), Agent(dishonest(i5)))) Abs_KAB1)))))), SID(sid0)] )) | (m = Secret(SymKey((absKAB(pair(Agent(honest(a0)), Agent(honest(a1)))) Abs_KAB0)), Agent(honest(a2)))) | (m = Witness(Agent(honest(a0)), Agent(honest(a1)), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB0)))) | (m = State(rA, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(honest(a3)), Nonce((absTA(pair(Agent(honest(a4)), Agent(honest(a5)))) Abs_TA0)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)), pair(Agent(honest(a8)), scrypt(SymKey(sk(pair(Agent(honest(a9)), Agent(honest(a10))))), pair(Nonce((absTA(pair(Agent(honest(a11)), Agent(honest(a12)))) Abs_TA1)), pair(Agent(honest(a13)), SymKey((absKAB(pair(Agent(honest(a14)), Agent(honest(a15)))) Abs_KAB1)))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a3))))), Nonce((ni Abs_NI0)), scrypt(SymKey(sk(pair(Agent(dishonest(i5)), Agent(honest(a4))))), pair(Nonce((ni Abs_NI1)), pair(Agent(dishonest(i6)), SymKey(sk(pair(Agent(dishonest(i7)), Agent(honest(a5)))))))), pair(Agent(dishonest(i8)), scrypt(SymKey(sk(pair(Agent(dishonest(i9)), Agent(honest(a6))))), pair(Nonce((ni Abs_NI2)), pair(Agent(dishonest(i10)), SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a7))))))))), Nonce((absTS(pair(Agent(dishonest(i12)), Agent(dishonest(i13)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i14)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i15)), Agent(dishonest(i16)))) Abs_TS1)), pair(Agent(dishonest(i17)), SymKey(sk(pair(Agent(dishonest(i18)), Agent(honest(a9)))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a5))))), Nonce((ni Abs_NI0)), scrypt(SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a6))))), pair(Nonce((ni Abs_NI1)), pair(Agent(honest(a7)), SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a8)))))))), pair(Agent(dishonest(i5)), scrypt(SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a9))))), pair(Nonce((ni Abs_NI2)), pair(Agent(honest(a10)), SymKey(sk(pair(Agent(dishonest(i7)), Agent(honest(a11))))))))), Nonce((absTS(pair(Agent(honest(a12)), Agent(dishonest(i8)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a13)), Agent(honest(a14))))), pair(Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i9)))) Abs_TS1)), pair(Agent(dishonest(i10)), SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a16)))))))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(honest(a0)), Agent(honest(a1))))), pair(Nonce((absTS(pair(Agent(honest(a2)), Agent(dishonest(i0)))) Abs_TS0)), pair(Agent(dishonest(i1)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))))))))) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a0))))), pair(Nonce((absTS(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_TS0)), pair(Agent(dishonest(i3)), SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a1)))))))))) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(honest(a0)), Agent(honest(a1))))), pair(Nonce((absTA(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_TA0)), pair(Agent(honest(a4)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(honest(a6)))) Abs_KAB0))))))) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(honest(a0)), Agent(honest(a1))))), pair(Nonce((absTA(pair(Agent(honest(a2)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i2)))) Abs_KAB0))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_TA0)), pair(Agent(honest(a5)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(dishonest(i0)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a0)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)), pair(Agent(dishonest(i2)), SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a0))))))))) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a3))))), Nonce((absTS(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i7)), Agent(honest(a4))))), pair(Nonce((absTS(pair(Agent(dishonest(i8)), Agent(dishonest(i9)))) Abs_TS1)), pair(Agent(dishonest(i10)), SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a5)))))))), pair(Agent(dishonest(i12)), scrypt(SymKey(sk(pair(Agent(dishonest(i13)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(dishonest(i14)), Agent(dishonest(i15)))) Abs_TS2)), pair(Agent(dishonest(i16)), SymKey(sk(pair(Agent(dishonest(i17)), Agent(honest(a7))))))))), Nonce((absTS(pair(Agent(dishonest(i18)), Agent(dishonest(i19)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i20)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i21)), Agent(dishonest(i22)))) Abs_TS4)), pair(Agent(dishonest(i23)), SymKey(sk(pair(Agent(dishonest(i24)), Agent(honest(a9)))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(honest(a2)), Agent(honest(a3))))), Agent(dishonest(i1)), Agent(honest(a4)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a5))))), Nonce((absTS(pair(Agent(honest(a6)), Agent(dishonest(i3)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a7)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(honest(a9)), Agent(dishonest(i4)))) Abs_TS1)), pair(Agent(dishonest(i5)), SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a10)))))))), pair(Agent(honest(a11)), scrypt(SymKey(sk(pair(Agent(honest(a12)), Agent(honest(a13))))), pair(Nonce((absTS(pair(Agent(honest(a14)), Agent(dishonest(i7)))) Abs_TS2)), pair(Agent(dishonest(i8)), SymKey(sk(pair(Agent(dishonest(i9)), Agent(honest(a15))))))))), Nonce((absTS(pair(Agent(dishonest(i10)), Agent(honest(a16)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a17))))), pair(Nonce((absTS(pair(Agent(dishonest(i12)), Agent(honest(a18)))) Abs_TS4)), pair(Agent(honest(a19)), SymKey(sk(pair(Agent(dishonest(i13)), Agent(honest(a20)))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(honest(a2)), Agent(honest(a3))))), Agent(dishonest(i1)), Agent(honest(a4)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((absTA(pair(Agent(honest(a6)), Agent(dishonest(i3)))) Abs_TA0)), scrypt(SymKey(sk(pair(Agent(honest(a7)), Agent(honest(a8))))), pair(Nonce((absTA(pair(Agent(honest(a9)), Agent(dishonest(i4)))) Abs_TA1)), pair(Agent(dishonest(i5)), SymKey((absKAB(pair(Agent(honest(a10)), Agent(dishonest(i6)))) Abs_KAB1))))), pair(Agent(honest(a11)), scrypt(SymKey(sk(pair(Agent(honest(a12)), Agent(honest(a13))))), pair(Nonce((absTA(pair(Agent(honest(a14)), Agent(dishonest(i7)))) Abs_TA2)), pair(Agent(dishonest(i8)), SymKey((absKAB(pair(Agent(honest(a15)), Agent(dishonest(i9)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(dishonest(i10)), Agent(honest(a16)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a17))))), pair(Nonce((absTS(pair(Agent(dishonest(i12)), Agent(honest(a18)))) Abs_TS1)), pair(Agent(honest(a19)), SymKey((absKAB(pair(Agent(honest(a20)), Agent(dishonest(i13)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(honest(a3)), Agent(honest(a4))))), Agent(honest(a5)), Agent(honest(a6)), SymKey((absKAB(pair(Agent(honest(a7)), Agent(honest(a8)))) Abs_KAB0)), Nonce((absTA(pair(Agent(honest(a9)), Agent(honest(a10)))) Abs_TA0)), scrypt(SymKey(sk(pair(Agent(honest(a11)), Agent(honest(a12))))), pair(Nonce((absTA(pair(Agent(honest(a13)), Agent(honest(a14)))) Abs_TA1)), pair(Agent(honest(a15)), SymKey((absKAB(pair(Agent(honest(a16)), Agent(honest(a17)))) Abs_KAB1))))), pair(Agent(honest(a18)), scrypt(SymKey(sk(pair(Agent(honest(a19)), Agent(honest(a20))))), pair(Nonce((absTA(pair(Agent(honest(a21)), Agent(honest(a22)))) Abs_TA2)), pair(Agent(honest(a23)), SymKey((absKAB(pair(Agent(honest(a24)), Agent(honest(a25)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(honest(a26)), Agent(honest(a27)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a28)), Agent(honest(a29))))), pair(Nonce((absTS(pair(Agent(honest(a30)), Agent(honest(a31)))) Abs_TS1)), pair(Agent(honest(a32)), SymKey((absKAB(pair(Agent(honest(a33)), Agent(honest(a34)))) Abs_KAB3))))), SID(sid0)] )) | (m = Request(Agent(honest(a0)), Agent(dishonest(i0)), Purpose(purposeKAB), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a1))))), SID(sid0))) | (m = State(rB, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a3))))), Nonce((absTS(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a5)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(honest(a7)), Agent(dishonest(i3)))) Abs_TS1)), pair(Agent(dishonest(i4)), SymKey(sk(pair(Agent(dishonest(i5)), Agent(honest(a8)))))))), SID(sid0)] )) | (m = Request(Agent(honest(a0)), Agent(dishonest(i0)), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i1)))) Abs_KAB0)), SID(sid0))) | (m = State(rB, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i1)))) Abs_KAB0)), Nonce((absTA(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_TA0)), scrypt(SymKey(sk(pair(Agent(honest(a5)), Agent(honest(a6))))), pair(Nonce((absTA(pair(Agent(honest(a7)), Agent(dishonest(i3)))) Abs_TA1)), pair(Agent(dishonest(i4)), SymKey((absKAB(pair(Agent(honest(a8)), Agent(dishonest(i5)))) Abs_KAB1))))), SID(sid0)] )) | (m = Request(Agent(honest(a0)), Agent(honest(a1)), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_KAB0)), SID(sid0))) | (m = State(rB, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(honest(a3)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(honest(a5)))) Abs_KAB0)), Nonce((absTA(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_TA0)), scrypt(SymKey(sk(pair(Agent(honest(a8)), Agent(honest(a9))))), pair(Nonce((absTA(pair(Agent(honest(a10)), Agent(honest(a11)))) Abs_TA1)), pair(Agent(honest(a12)), SymKey((absKAB(pair(Agent(honest(a13)), Agent(honest(a14)))) Abs_KAB1))))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(honest(a0)), Agent(honest(a1))))), pair(Nonce((absTS(pair(Agent(honest(a2)), Agent(honest(a3)))) Abs_TS0)), pair(Agent(honest(a4)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(honest(a6)))) Abs_KAB0))))))) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a0))))), pair(Nonce((absTS(pair(Agent(dishonest(i1)), Agent(honest(a1)))) Abs_TS0)), pair(Agent(honest(a2)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i2)))) Abs_KAB0))))))) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a0))))), pair(Nonce((absTS(pair(Agent(dishonest(i1)), Agent(honest(a1)))) Abs_TS0)), pair(Agent(honest(a2)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a3)))))))))) | (m = Iknows(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_TA0)), pair(Agent(honest(a5)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(dishonest(i0)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a0)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)), pair(Agent(dishonest(i2)), SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a0))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i0)))) Abs_KAB0))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey((absKAB(pair(Agent(honest(a2)), Agent(dishonest(i1)))) Abs_KAB0)))))) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((absTS(pair(Agent(dishonest(i3)), Agent(honest(a6)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a7))))), pair(Nonce((absTS(pair(Agent(dishonest(i5)), Agent(honest(a8)))) Abs_TS1)), pair(Agent(honest(a9)), SymKey((absKAB(pair(Agent(honest(a10)), Agent(dishonest(i6)))) Abs_KAB1))))), pair(Agent(dishonest(i7)), scrypt(SymKey(sk(pair(Agent(dishonest(i8)), Agent(honest(a11))))), pair(Nonce((absTS(pair(Agent(dishonest(i9)), Agent(honest(a12)))) Abs_TS2)), pair(Agent(honest(a13)), SymKey((absKAB(pair(Agent(honest(a14)), Agent(dishonest(i10)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i11)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(honest(a16)), Agent(honest(a17))))), pair(Nonce((absTS(pair(Agent(honest(a18)), Agent(dishonest(i12)))) Abs_TS4)), pair(Agent(dishonest(i13)), SymKey((absKAB(pair(Agent(honest(a19)), Agent(dishonest(i14)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a5))))), Nonce((absTS(pair(Agent(dishonest(i3)), Agent(honest(a6)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a7))))), pair(Nonce((absTS(pair(Agent(dishonest(i5)), Agent(honest(a8)))) Abs_TS1)), pair(Agent(honest(a9)), SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a10)))))))), pair(Agent(dishonest(i7)), scrypt(SymKey(sk(pair(Agent(dishonest(i8)), Agent(honest(a11))))), pair(Nonce((absTS(pair(Agent(dishonest(i9)), Agent(honest(a12)))) Abs_TS2)), pair(Agent(honest(a13)), SymKey(sk(pair(Agent(dishonest(i10)), Agent(honest(a14))))))))), Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i11)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(honest(a16)), Agent(honest(a17))))), pair(Nonce((absTS(pair(Agent(honest(a18)), Agent(dishonest(i12)))) Abs_TS4)), pair(Agent(dishonest(i13)), SymKey(sk(pair(Agent(dishonest(i14)), Agent(honest(a19)))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(honest(a3)), Agent(honest(a4))))), Agent(honest(a5)), Agent(honest(a6)), SymKey((absKAB(pair(Agent(honest(a7)), Agent(honest(a8)))) Abs_KAB0)), Nonce((absTS(pair(Agent(honest(a9)), Agent(honest(a10)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a11)), Agent(honest(a12))))), pair(Nonce((absTS(pair(Agent(honest(a13)), Agent(honest(a14)))) Abs_TS1)), pair(Agent(honest(a15)), SymKey((absKAB(pair(Agent(honest(a16)), Agent(honest(a17)))) Abs_KAB1))))), pair(Agent(honest(a18)), scrypt(SymKey(sk(pair(Agent(honest(a19)), Agent(honest(a20))))), pair(Nonce((absTS(pair(Agent(honest(a21)), Agent(honest(a22)))) Abs_TS2)), pair(Agent(honest(a23)), SymKey((absKAB(pair(Agent(honest(a24)), Agent(honest(a25)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(honest(a26)), Agent(honest(a27)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(honest(a28)), Agent(honest(a29))))), pair(Nonce((absTS(pair(Agent(honest(a30)), Agent(honest(a31)))) Abs_TS4)), pair(Agent(honest(a32)), SymKey((absKAB(pair(Agent(honest(a33)), Agent(honest(a34)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey(sk(pair(Agent(dishonest(i2)), Agent(honest(a5))))), Nonce((absTS(pair(Agent(dishonest(i3)), Agent(dishonest(i4)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i5)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(dishonest(i6)), Agent(dishonest(i7)))) Abs_TS1)), pair(Agent(honest(a7)), SymKey(sk(pair(Agent(dishonest(i8)), Agent(honest(a8)))))))), pair(Agent(dishonest(i9)), scrypt(SymKey(sk(pair(Agent(dishonest(i10)), Agent(honest(a9))))), pair(Nonce((absTS(pair(Agent(dishonest(i11)), Agent(dishonest(i12)))) Abs_TS2)), pair(Agent(honest(a10)), SymKey(sk(pair(Agent(dishonest(i13)), Agent(honest(a11))))))))), Nonce((absTS(pair(Agent(honest(a12)), Agent(dishonest(i14)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(honest(a13)), Agent(honest(a14))))), pair(Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i15)))) Abs_TS4)), pair(Agent(dishonest(i16)), SymKey(sk(pair(Agent(dishonest(i17)), Agent(honest(a16)))))))), SID(sid0)] )) | (m = State(rB, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(honest(a3)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(honest(a5)))) Abs_KAB0)), Nonce((absTS(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a8)), Agent(honest(a9))))), pair(Nonce((absTS(pair(Agent(honest(a10)), Agent(honest(a11)))) Abs_TS1)), pair(Agent(honest(a12)), SymKey((absKAB(pair(Agent(honest(a13)), Agent(honest(a14)))) Abs_KAB1))))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(honest(a0)), Agent(honest(a1))))), pair(Nonce((absTS(pair(Agent(honest(a2)), Agent(dishonest(i0)))) Abs_TS0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i2)))) Abs_KAB0))))))) | (m = Iknows(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)))) | (m = Iknows(SymKey((absKAB(pair(Agent(honest(a0)), Agent(dishonest(i0)))) Abs_KAB0)))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_TA0)), pair(Agent(honest(a5)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(dishonest(i0)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a0)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)), pair(Agent(dishonest(i2)), SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a0))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i0)))) Abs_KAB0))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey((absKAB(pair(Agent(honest(a2)), Agent(dishonest(i1)))) Abs_KAB0)))))) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(honest(a2)), Agent(honest(a3))))), Agent(dishonest(i1)), Agent(honest(a4)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((absTS(pair(Agent(honest(a6)), Agent(dishonest(i3)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a7)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(honest(a9)), Agent(dishonest(i4)))) Abs_TS1)), pair(Agent(dishonest(i5)), SymKey((absKAB(pair(Agent(honest(a10)), Agent(dishonest(i6)))) Abs_KAB1))))), pair(Agent(honest(a11)), scrypt(SymKey(sk(pair(Agent(honest(a12)), Agent(honest(a13))))), pair(Nonce((absTS(pair(Agent(honest(a14)), Agent(dishonest(i7)))) Abs_TS2)), pair(Agent(dishonest(i8)), SymKey((absKAB(pair(Agent(honest(a15)), Agent(dishonest(i9)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(dishonest(i10)), Agent(honest(a16)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a17))))), pair(Nonce((absTS(pair(Agent(dishonest(i12)), Agent(honest(a18)))) Abs_TS4)), pair(Agent(honest(a19)), SymKey((absKAB(pair(Agent(honest(a20)), Agent(dishonest(i13)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i4)))) Abs_KAB0)), Nonce((absTS(pair(Agent(dishonest(i5)), Agent(honest(a4)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a5))))), pair(Nonce((absTS(pair(Agent(dishonest(i7)), Agent(honest(a6)))) Abs_TS1)), pair(Agent(dishonest(i8)), SymKey((absKAB(pair(Agent(honest(a7)), Agent(dishonest(i9)))) Abs_KAB1))))), pair(Agent(dishonest(i10)), scrypt(SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i12)), Agent(honest(a9)))) Abs_TS2)), pair(Agent(dishonest(i13)), SymKey((absKAB(pair(Agent(honest(a10)), Agent(dishonest(i14)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(dishonest(i15)), Agent(dishonest(i16)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i17)), Agent(honest(a11))))), pair(Nonce((absTS(pair(Agent(dishonest(i18)), Agent(dishonest(i19)))) Abs_TS4)), pair(Agent(dishonest(i20)), SymKey((absKAB(pair(Agent(honest(a12)), Agent(dishonest(i21)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey(sk(pair(Agent(dishonest(i4)), Agent(honest(a3))))), Nonce((absTS(pair(Agent(dishonest(i5)), Agent(honest(a4)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a5))))), pair(Nonce((absTS(pair(Agent(dishonest(i7)), Agent(honest(a6)))) Abs_TS1)), pair(Agent(dishonest(i8)), SymKey(sk(pair(Agent(dishonest(i9)), Agent(honest(a7)))))))), pair(Agent(dishonest(i10)), scrypt(SymKey(sk(pair(Agent(dishonest(i11)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i12)), Agent(honest(a9)))) Abs_TS2)), pair(Agent(dishonest(i13)), SymKey(sk(pair(Agent(dishonest(i14)), Agent(honest(a10))))))))), Nonce((absTS(pair(Agent(dishonest(i15)), Agent(dishonest(i16)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i17)), Agent(honest(a11))))), pair(Nonce((absTS(pair(Agent(dishonest(i18)), Agent(dishonest(i19)))) Abs_TS4)), pair(Agent(dishonest(i20)), SymKey(sk(pair(Agent(dishonest(i21)), Agent(honest(a12)))))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i4)))) Abs_KAB0)), Nonce((absTS(pair(Agent(dishonest(i5)), Agent(dishonest(i6)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i7)), Agent(honest(a4))))), pair(Nonce((absTS(pair(Agent(dishonest(i8)), Agent(dishonest(i9)))) Abs_TS1)), pair(Agent(dishonest(i10)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i11)))) Abs_KAB1))))), pair(Agent(dishonest(i12)), scrypt(SymKey(sk(pair(Agent(dishonest(i13)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(dishonest(i14)), Agent(dishonest(i15)))) Abs_TS2)), pair(Agent(dishonest(i16)), SymKey((absKAB(pair(Agent(honest(a7)), Agent(dishonest(i17)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(dishonest(i18)), Agent(dishonest(i19)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(dishonest(i20)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i21)), Agent(dishonest(i22)))) Abs_TS4)), pair(Agent(dishonest(i23)), SymKey((absKAB(pair(Agent(honest(a9)), Agent(dishonest(i24)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1))))), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))), Agent(dishonest(i2)), Agent(dishonest(i3)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i4)))) Abs_KAB0)), Nonce((ni Abs_NI0)), scrypt(SymKey(sk(pair(Agent(dishonest(i5)), Agent(honest(a4))))), pair(Nonce((ni Abs_NI1)), pair(Agent(dishonest(i6)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i7)))) Abs_KAB1))))), pair(Agent(dishonest(i8)), scrypt(SymKey(sk(pair(Agent(dishonest(i9)), Agent(honest(a6))))), pair(Nonce((ni Abs_NI2)), pair(Agent(dishonest(i10)), SymKey((absKAB(pair(Agent(honest(a7)), Agent(dishonest(i11)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(dishonest(i12)), Agent(dishonest(i13)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i14)), Agent(honest(a8))))), pair(Nonce((absTS(pair(Agent(dishonest(i15)), Agent(dishonest(i16)))) Abs_TS1)), pair(Agent(dishonest(i17)), SymKey((absKAB(pair(Agent(honest(a9)), Agent(dishonest(i18)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((absTS(pair(Agent(dishonest(i3)), Agent(dishonest(i4)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(dishonest(i5)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(dishonest(i6)), Agent(dishonest(i7)))) Abs_TS1)), pair(Agent(honest(a7)), SymKey((absKAB(pair(Agent(honest(a8)), Agent(dishonest(i8)))) Abs_KAB1))))), pair(Agent(dishonest(i9)), scrypt(SymKey(sk(pair(Agent(dishonest(i10)), Agent(honest(a9))))), pair(Nonce((absTS(pair(Agent(dishonest(i11)), Agent(dishonest(i12)))) Abs_TS2)), pair(Agent(honest(a10)), SymKey((absKAB(pair(Agent(honest(a11)), Agent(dishonest(i13)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(honest(a12)), Agent(dishonest(i14)))) Abs_TS3)), scrypt(SymKey(sk(pair(Agent(honest(a13)), Agent(honest(a14))))), pair(Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i15)))) Abs_TS4)), pair(Agent(dishonest(i16)), SymKey((absKAB(pair(Agent(honest(a16)), Agent(dishonest(i17)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rs, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a3))))), Agent(honest(a4)), Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a5)), Agent(dishonest(i2)))) Abs_KAB0)), Nonce((ni Abs_NI0)), scrypt(SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a6))))), pair(Nonce((ni Abs_NI1)), pair(Agent(honest(a7)), SymKey((absKAB(pair(Agent(honest(a8)), Agent(dishonest(i4)))) Abs_KAB1))))), pair(Agent(dishonest(i5)), scrypt(SymKey(sk(pair(Agent(dishonest(i6)), Agent(honest(a9))))), pair(Nonce((ni Abs_NI2)), pair(Agent(honest(a10)), SymKey((absKAB(pair(Agent(honest(a11)), Agent(dishonest(i7)))) Abs_KAB2)))))), Nonce((absTS(pair(Agent(honest(a12)), Agent(dishonest(i8)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a13)), Agent(honest(a14))))), pair(Nonce((absTS(pair(Agent(honest(a15)), Agent(dishonest(i9)))) Abs_TS1)), pair(Agent(dishonest(i10)), SymKey((absKAB(pair(Agent(honest(a16)), Agent(dishonest(i11)))) Abs_KAB3))))), SID(sid0)] )) | (m = State(rB, [Agent(honest(a0)), Step(1), SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), Agent(dishonest(i0)), SymKey((absKAB(pair(Agent(honest(a3)), Agent(dishonest(i1)))) Abs_KAB0)), Nonce((absTS(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_TS0)), scrypt(SymKey(sk(pair(Agent(honest(a5)), Agent(honest(a6))))), pair(Nonce((absTS(pair(Agent(honest(a7)), Agent(dishonest(i3)))) Abs_TS1)), pair(Agent(dishonest(i4)), SymKey((absKAB(pair(Agent(honest(a8)), Agent(dishonest(i5)))) Abs_KAB1))))), SID(sid0)] )) | (m = Iknows(scrypt(SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a0))))), pair(Nonce((absTS(pair(Agent(dishonest(i1)), Agent(dishonest(i2)))) Abs_TS0)), pair(Agent(dishonest(i3)), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i4)))) Abs_KAB0))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(honest(a4)))) Abs_TA0)), pair(Agent(honest(a5)), SymKey((absKAB(pair(Agent(honest(a6)), Agent(honest(a7)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(honest(a0)), scrypt(SymKey(sk(pair(Agent(honest(a1)), Agent(honest(a2))))), pair(Nonce((absTA(pair(Agent(honest(a3)), Agent(dishonest(i0)))) Abs_TA0)), pair(Agent(dishonest(i1)), SymKey((absKAB(pair(Agent(honest(a4)), Agent(dishonest(i2)))) Abs_KAB0)))))))) | (m = Iknows(pair(Agent(dishonest(i0)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a0)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)), pair(Agent(dishonest(i2)), SymKey(sk(pair(Agent(dishonest(i3)), Agent(honest(a0))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey(sk(pair(Agent(dishonest(i0)), Agent(honest(a1)))))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey(sk(pair(Agent(dishonest(i1)), Agent(honest(a2))))))))) | (m = Iknows(pair(Agent(honest(a0)), SymKey((absKAB(pair(Agent(honest(a1)), Agent(dishonest(i0)))) Abs_KAB0))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(honest(a0)))) Abs_TS0)), pair(Agent(honest(a1)), SymKey((absKAB(pair(Agent(honest(a2)), Agent(dishonest(i1)))) Abs_KAB0)))))) | (m = Iknows(pair(Agent(dishonest(i0)), SymKey((absKAB(pair(Agent(honest(a0)), Agent(dishonest(i1)))) Abs_KAB0))))) | (m = Iknows(pair(Nonce((absTS(pair(Agent(dishonest(i0)), Agent(dishonest(i1)))) Abs_TS0)), pair(Agent(dishonest(i2)), SymKey((absKAB(pair(Agent(honest(a0)), Agent(dishonest(i3)))) Abs_KAB0)))))) )}" section {* Checking Fixed-point (WideMouthFrog) *} lemma fp_attack_free: "~ (Attack m : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_0: "Iknows(Nonce((ni Abs_NI))) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_1: "Iknows(Agent(dishonest(i))) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_2: "State(rA, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_3: "State(rA, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_4: "Iknows(Step(0)) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_5: "Iknows(SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a)))))) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_6: "Iknows(SID(sid)) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_7: "Iknows(Agent(honest(a))) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_8: "State(rB, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_9: "State(rB, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_10: "State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), Agent(dishonest(i)), Agent(dishonest(i)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_11: "State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(dishonest(i)), Agent(honest(a)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_12: "State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), SymKey(sk(pair(Agent(dishonest(i)), Agent(honest(a))))), Agent(honest(a)), Agent(dishonest(i)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma init_13: "State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), SymKey(sk(pair(Agent(honest(a)), Agent(honest(a))))), Agent(honest(a)), Agent(honest(a)), SID(sid)] ) : WideMouthFrog_fp" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_0: "[| Iknows(crypt(K, M)) : WideMouthFrog_fp; Iknows(inv(K)) : WideMouthFrog_fp|] ==> (Iknows(M) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_1: "[| Iknows(crypt(inv(K), M)) : WideMouthFrog_fp; Iknows(K) : WideMouthFrog_fp|] ==> (Iknows(M) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_2: "[| Iknows(scrypt(K, M)) : WideMouthFrog_fp; Iknows(K) : WideMouthFrog_fp|] ==> (Iknows(M) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_3: "[| Iknows(pair(M1, M2)) : WideMouthFrog_fp|] ==> (Iknows(M1) : WideMouthFrog_fp) & (Iknows(M2) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_4: "[| Secret(M, Agent(honest(a))) : WideMouthFrog_fp; Iknows(M) : WideMouthFrog_fp|] ==> (Attack(pair(secrecy, M)) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_5: "[| Request(A, B, Purpose(purposeTA), M, SID(sid)) : WideMouthFrog_fp; ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_TA . M = Nonce((absTA(pair(B, A)) Abs_TA)))|] ==> (Attack(pair(authentication, pair(A, pair(B, M)))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_6: "[| Request(A, B, Purpose(purposeKAB), M, SID(sid)) : WideMouthFrog_fp; ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_KAB . M = SymKey((absKAB(pair(B, A)) Abs_KAB)))|] ==> (Attack(pair(authentication, pair(A, pair(B, M)))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_7: "[| Request(A, B, Purpose(purposeTS), M, SID(sid)) : WideMouthFrog_fp; ~ ( ? i . B = Agent(dishonest(i))); ~ ( ? B A Abs_TS . M = Nonce((absTS(pair(B, A)) Abs_TS)))|] ==> (Attack(pair(authentication, pair(A, pair(B, M)))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_8: "[| State(rA, [Agent(A), Step(0), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), SID(sid)] ) : WideMouthFrog_fp|] ==> (Secret(SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)), Agent(B)) : WideMouthFrog_fp) & (Witness(Agent(A), Agent(B), Purpose(purposeKAB), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB))) : WideMouthFrog_fp) & (State(rA, [Agent(A), Step(1), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), pair(Agent(B), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB)))))), SID(sid)] ) : WideMouthFrog_fp) & (Iknows(pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce((absTA(pair(Agent(A), Agent(B))) Abs_TA)), pair(Agent(B), SymKey((absKAB(pair(Agent(A), Agent(B))) Abs_KAB))))))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_9: "[| State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SID(sid)] ) : WideMouthFrog_fp; Iknows(Agent(A)) : WideMouthFrog_fp; Iknows(scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))) : WideMouthFrog_fp|] ==> (State(rs, [Agent(honest(a)), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SymKey(KAB), Nonce(TA), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB)))), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))), Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))), SID(sid)] ) : WideMouthFrog_fp) & (Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB))))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_10: "[| State(rs, [Agent(honest(a)), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SID(sid)] ) : WideMouthFrog_fp; Iknows(Agent(A)) : WideMouthFrog_fp; Iknows(SymKey(sk(pair(Agent(A), Agent(honest(a)))))) : WideMouthFrog_fp; Iknows(Nonce(TA)) : WideMouthFrog_fp; Iknows(Agent(B)) : WideMouthFrog_fp; Iknows(SymKey(KAB)) : WideMouthFrog_fp|] ==> (State(rs, [Agent(honest(a)), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), SymKey(sk(pair(Agent(A), Agent(honest(a))))), Agent(B), Agent(A), SymKey(KAB), Nonce(TA), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB)))), pair(Agent(A), scrypt(SymKey(sk(pair(Agent(A), Agent(honest(a))))), pair(Nonce(TA), pair(Agent(B), SymKey(KAB))))), Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB)))), SID(sid)] ) : WideMouthFrog_fp) & (Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce((absTS(pair(Agent(B), Agent(A))) Abs_TS)), pair(Agent(A), SymKey(KAB))))) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_11: "[| State(rB, [Agent(B), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SID(sid)] ) : WideMouthFrog_fp; Iknows(scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB))))) : WideMouthFrog_fp|] ==> (Request(Agent(B), Agent(A), Purpose(purposeKAB), SymKey(KAB), SID(sid)) : WideMouthFrog_fp) & (State(rB, [Agent(B), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SymKey(KAB), Nonce(TS), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB)))), SID(sid)] ) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ lemma rule_12: "[| State(rB, [Agent(B), Step(0), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SID(sid)] ) : WideMouthFrog_fp; Iknows(SymKey(sk(pair(Agent(B), Agent(honest(a)))))) : WideMouthFrog_fp; Iknows(Nonce(TS)) : WideMouthFrog_fp; Iknows(Agent(A)) : WideMouthFrog_fp; Iknows(SymKey(KAB)) : WideMouthFrog_fp|] ==> (Request(Agent(B), Agent(A), Purpose(purposeKAB), SymKey(KAB), SID(sid)) : WideMouthFrog_fp) & (State(rB, [Agent(B), Step(1), SymKey(sk(pair(Agent(B), Agent(honest(a))))), Agent(A), SymKey(KAB), Nonce(TS), scrypt(SymKey(sk(pair(Agent(B), Agent(honest(a))))), pair(Nonce(TS), pair(Agent(A), SymKey(KAB)))), SID(sid)] ) : WideMouthFrog_fp)" by(simp only: WideMouthFrog_fp_def, simp only: set2pred, simp, auto?)+ section {* Security Proof(s) (WideMouthFrog) *} lemma over_approx: "t : WideMouthFrog ==> (set t) <= WideMouthFrog_fp" apply(rule WideMouthFrog.induct, simp_all) apply(propagate_fp, cut_tac init_0, (assumption | simp)+) apply(propagate_fp, cut_tac init_1, (assumption | simp)+) apply(propagate_fp, cut_tac init_2, (assumption | simp)+) apply(propagate_fp, cut_tac init_3, (assumption | simp)+) apply(propagate_fp, cut_tac init_4, (assumption | simp)+) apply(propagate_fp, cut_tac init_5, (assumption | simp)+) apply(propagate_fp, cut_tac init_6, (assumption | simp)+) apply(propagate_fp, cut_tac init_7, (assumption | simp)+) apply(propagate_fp, cut_tac init_8, (assumption | simp)+) apply(propagate_fp, cut_tac init_9, (assumption | simp)+) apply(propagate_fp, cut_tac init_10, (assumption | simp)+) apply(propagate_fp, cut_tac init_11, (assumption | simp)+) apply(propagate_fp, cut_tac init_12, (assumption | simp)+) apply(propagate_fp, cut_tac init_13, (assumption | simp)+) apply(propagate_fp, cut_tac rule_0, (assumption | simp)+) apply(propagate_fp, cut_tac rule_1, (assumption | simp)+) apply(propagate_fp, cut_tac rule_2, (assumption | simp)+) apply(propagate_fp, cut_tac rule_3, (assumption | simp)+) apply(propagate_fp, cut_tac rule_4, (assumption | simp)+) apply(propagate_fp, cut_tac rule_5, (assumption | simp)+) apply(propagate_fp, cut_tac rule_6, (assumption | simp)+) apply(propagate_fp, cut_tac rule_7, (assumption | simp)+) apply(propagate_fp, cut_tac rule_8, (assumption | simp)+) apply(propagate_fp, cut_tac rule_9, (assumption | simp)+) apply(propagate_fp, cut_tac rule_10, (assumption | simp)+) apply(propagate_fp, cut_tac rule_11, (assumption | simp)+) apply(propagate_fp, cut_tac rule_12, (assumption | simp)+) done end (* theory *)