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Case4_A1.ec
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pragma Goals:printall.
require import Int Real Distr SmtMap FSet DInterval.
require import BLT_Instance.
require import RandomnessOracle.
theory Case4_A1_Theory.
module Case4_Adv1(A : BLT_Adv, A2:BLT_Adv_Set, RO : ROracle) = {
module TsU_Set_1 = TsU_Set_1(A2)
module BLT_Dummy_Time = BLT_Dummy_Time(Tag_Wrap_1, TsU_Set_1)
module A = A(BLT_Dummy_Time)
var guess : int
proc forge'(pk:pkey) = {
var m : message;
var tg : tag;
var it : int;
var c : chain;
it <@ RO.rndStr();
Tag_Wrap_1.init(pk, witness);
TsU_Set_1.init(initialTime it);
BLT_Dummy_Time.init();
(m,tg, c) <@ A.forge(pk, it);
}
proc forge(pk:pkey) = {
forge'(pk);
guess <$ dinter 1 queryBound;
return (ogetf TsU_1.r.[guess], guess);
}
}.
section Adv1.
op timeX : pkey -> int -> int.
op hash_setX : pkey -> int -> int -> hash_output fset option.
op time_setX : pkey -> int -> int fset.
op message_setX : pkey -> int -> message fset.
op messageX : pkey -> int -> message.
op boolX : pkey -> int -> bool.
op chainX : pkey -> int -> chain.
declare module A <: BLT_Adv{-Tag_Wrap_1, -BLT_Wrap, -TsU, -BLTGame, -BLT_Dummy_Time, -Tag_Wrap, -TsU_1, -RO}.
declare module A2 <: BLT_Adv_Set{-Tag_Wrap_1, -BLT_Wrap, -TsU, -A, -BLTGame, -BLT_Dummy_Time, -Tag_Wrap, -TsU_1, -RO}.
axiom A_ll : forall (O <: BLT_AdvOracle{-A}),
islossless O.sign => islossless O.put => islossless O.get => islossless A(O).forge.
axiom A2_ll : islossless A2.react.
axiom computableAdvs &m pkk skk : (pkk, skk) \in keys =>
Pr [ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m :
toTime BLT_Wrap.qt = timeX pkk RO.t
/\ BLT_Wrap.used = boolX pkk RO.t
/\ fset1 (toTime BLT_Wrap.qt) = time_setX pkk RO.t
/\ (ogetff BLT_Wrap.qs) = message_setX pkk RO.t
/\ BLTGameD.m = messageX pkk RO.t
/\ BLTGameD.c = chainX pkk RO.t
/\ (forall x, TsU.r.[x] = hash_setX pkk RO.t x) ] = 1%r.
lemma a1 &m pkk skk : (pkk, skk) \in keys => Pr [ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m :
toTime BLT_Wrap.qt = timeX pkk RO.t /\ (forall x, x < timeX pkk RO.t => TsU.r.[x] = hash_setX pkk RO.t x) ] = 1%r.
proof. move => h.
have : Pr [ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m :
toTime BLT_Wrap.qt = timeX pkk RO.t
/\ BLT_Wrap.used = boolX pkk RO.t
/\ fset1 (toTime BLT_Wrap.qt) = time_setX pkk RO.t
/\ (ogetff BLT_Wrap.qs) = message_setX pkk RO.t
/\ BLTGameD.m = messageX pkk RO.t
/\ BLTGameD.c = chainX pkk RO.t
/\ (forall x, TsU.r.[x] = hash_setX pkk RO.t x) ] <= Pr [ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m :
toTime BLT_Wrap.qt = timeX pkk RO.t /\ (forall x, x < timeX pkk RO.t => TsU.r.[x] = hash_setX pkk RO.t x) ].
rewrite Pr[mu_sub]. auto. auto.
have : Pr[BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m :
toTime BLT_Wrap.qt = timeX pkk RO.t /\
BLT_Wrap.used = boolX pkk RO.t /\
fset1 (toTime BLT_Wrap.qt) = time_setX pkk RO.t /\
(ogetff BLT_Wrap.qs) = message_setX pkk RO.t /\
BLTGameD.m = messageX pkk RO.t /\
BLTGameD.c = chainX pkk RO.t /\
(forall x, TsU.r.[x] = hash_setX pkk RO.t x) ] = 1%r.
apply (computableAdvs &m pkk skk h).
smt. auto.
rewrite (computableAdvs &m pkk skk h). smt.
qed.
local lemma timeExtractor' :
equiv [ A(BLT_Dummy_Time(Tag_Wrap_1, TsU_Set_1(A2))).forge
~ A(BLT_Wrap(Tag_Wrap, TsU_Set(A2))).forge
: ={rs, glob A, pk, glob A2} /\ TsU.r{2} = TsU_1.r{1} /\ TsU.t{2} = TsU_1.t{1} /\ BLT_Wrap.qs{2} = None
/\ toTime BLT_Wrap.qt{2} = BLT_Dummy_Time.usedTime{1}
/\ BLT_Dummy_Time.used{1} = BLT_Wrap.used{2}
/\ BLT_Wrap.used{2} = Tag_Wrap.usedFlag{2}
/\ BLT_Wrap.used{2} = false
/\ (Tag_Wrap.pk{2}, Tag_Wrap.sk{2}) \in keys
==> toTime BLT_Wrap.qt{2} = BLT_Dummy_Time.usedTime{1}
/\ (forall x, x < BLT_Dummy_Time.usedTime{1} => TsU_1.r.[x]{1} = TsU.r.[x]{2}) ].
proof. proc*.
call (_: BLT_Wrap.used, ={glob A2} /\ (toTime BLT_Wrap.qt{2} = BLT_Dummy_Time.usedTime{1}) /\ (forall x, x < BLT_Dummy_Time.usedTime{1} => TsU_1.r.[x]{1} = TsU.r.[x]{2})
/\ (BLT_Dummy_Time.used{1} => BLT_Dummy_Time.usedTime{1} <= TsU_1.t{1} )
/\ (BLT_Wrap.used => toTime BLT_Wrap.qt <= TsU.t){2}
/\ BLT_Dummy_Time.used{1} = BLT_Wrap.used{2} /\ TsU_1.t{1} = TsU.t{2} /\ TsU.r{2} = TsU_1.r{1}
/\ (Tag_Wrap.pk{2}, Tag_Wrap.sk{2}) \in keys /\ Tag_Wrap.usedFlag{2} = BLT_Wrap.used{2} , (toTime BLT_Wrap.qt{2} = BLT_Dummy_Time.usedTime{1}) /\ (forall x, x < BLT_Dummy_Time.usedTime{1} => TsU_1.r.[x]{1} = TsU.r.[x]{2}) /\ (BLT_Dummy_Time.used{1} => BLT_Dummy_Time.usedTime{1} <= TsU_1.t{1} ) /\ (BLT_Wrap.used => toTime BLT_Wrap.qt <= TsU.t){2} /\ BLT_Dummy_Time.used{1} = BLT_Wrap.used{2} /\ Tag_Wrap.usedFlag{2} = BLT_Wrap.used{2}
/\ (Tag_Wrap.pk{2}, Tag_Wrap.sk{2}) \in keys /\ BLT_Dummy_Time.used{1} = BLT_Wrap.used{2}).
apply A_ll.
proc. wp. if. auto.
inline*.
wp. call (_: true ==> true). proc*. call {1} A2_ll. call {2} A2_ll. skip. auto.
wp. skip. progress. smt (toTimeProp). smt. smt. smt. smt. smt. skip. smt.
move => &2 used. proc.
inline*. if. wp. call (_:true). apply A2_ll. wp. skip. smt.
wp. skip. smt.
move => &1. proc.
inline*. if. wp. call (_:true). apply A2_ll. wp. skip. smt.
wp. skip. smt.
proc. inline*. wp.
call (_:true). wp.
skip. progress.
move => &2 used. proc.
inline*. wp. call (_:true). apply A2_ll. wp. skip. progress. smt. smt.
move => &1. proc.
inline*. wp. call (_:true). apply A2_ll. wp. skip. progress. smt. smt.
proc. inline*. wp.
skip. progress.
move => &2 used. proc.
inline*. wp. skip. progress.
move => &1. proc.
inline*. wp. skip. smt.
skip. progress. smt.
smt.
qed.
local lemma timeExtractor pkk skk : (pkk, skk) \in keys => equiv [ Case4_Adv1(A, A2, RO).forge' ~ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main
: ={glob A, glob A2, glob RO} /\ pk{1} = pk{2} /\ (pkk, skk) = (pk{2}, sk{2}) ==> toTime BLT_Wrap.qt{2} = BLT_Dummy_Time.usedTime{1}
/\ (forall x, x < BLT_Dummy_Time.usedTime{1} => TsU_1.r.[x]{1} = TsU.r.[x]{2}) /\ ={RO.t} ].
proof. move => pksk. proc. inline*.
wp. call timeExtractor'. wp. wp. skip. progress.
qed.
local lemma a2 &m pkk skk : (pkk, skk) \in keys =>
Pr [ Case4_Adv1(A,A2, RO).forge'(pkk) @ &m : BLT_Dummy_Time.usedTime = timeX pkk RO.t /\
(forall x, x < timeX pkk RO.t => TsU_1.r.[x] = hash_setX pkk RO.t x) ]
= Pr [ BLTGameD(TsU_Set(A2),Tag_Wrap, A).main(pkk, skk) @ &m : toTime BLT_Wrap.qt = timeX pkk RO.t /\ (forall x, x < timeX pkk RO.t => TsU.r.[x] = hash_setX pkk RO.t x) ].
proof. move => pr.
byequiv (_ : ={pk, glob RO, glob A, glob A2, glob Tag_Wrap, glob TsU, glob BLT_Wrap}
/\ (pk{2}, sk{2}) = (pkk, skk) ==> BLT_Dummy_Time.usedTime{1} = toTime BLT_Wrap.qt{2} /\ (forall x, x < toTime BLT_Wrap.qt{2} => TsU_1.r{1}.[x] = TsU.r{2}.[x]) /\ ={RO.t}).
conseq (timeExtractor pkk skk pr). progress.
progress. progress. progress. smt. smt. qed.
local lemma a222 &m pkk skk: (pkk, skk) \in keys => Pr [ Case4_Adv1(A,A2, RO).forge'(pkk) @ &m : BLT_Dummy_Time.usedTime = timeX pkk RO.t /\ (forall x, x < timeX pkk RO.t => TsU_1.r.[x] = hash_setX pkk RO.t x) ] = 1%r.
proof.
move => pr. rewrite (a2 &m pkk skk). auto. apply (a1 &m pkk skk pr). qed.
lemma a1_premise pkk skk : phoare [ Case4_Adv1(A,A2, RO).forge' : (pkk, skk) \in keys /\ pk = pkk ==> BLT_Dummy_Time.usedTime = timeX pkk RO.t /\ (forall x, x < timeX pkk RO.t => TsU_1.r.[x] = hash_setX pkk RO.t x) ] = 1%r.
bypr. move => &m1 h. elim h. move => h1 h2. rewrite h2. apply (a222 &m1 pkk skk h1).
qed.
end section Adv1.
end Case4_A1_Theory.