DDH_hybrid.ec
require import Real.
require import Int IntDiv.
require import Prime_field.
require import Cyclic_group_prime.
require Hybrid.
op n : { int | 0 < n } as n_pos.
clone import Hybrid as H with
type input <- unit,
type output <- group * group * group,
type inleaks <- unit,
type outleaks <- unit,
type outputA <- bool,
op q <- n
proof* by smt(n_pos).
module DDHl = {
proc orcl () : group * group * group = {
var x, y: gf_q;
x <$ Dgf_q.dgf_q;
y <$ Dgf_q.dgf_q;
return (g^x, g^y, g^(x * y));
}
}.
module DDHr = {
proc orcl () : group * group * group = {
var x, y, z: gf_q;
x <$ Dgf_q.dgf_q;
y <$ Dgf_q.dgf_q;
z <$ Dgf_q.dgf_q;
return (g^x, g^y, g^z);
}
}.
module DDHb : H.Orclb = {
proc leaks () : unit = { }
proc orclL = DDHl.orcl
proc orclR = DDHr.orcl
}.
lemma islossless_leaks : islossless DDHb.leaks.
proof. proc;auto. qed.
lemma islossless_orcl1 : islossless DDHb.orclL.
proof. proc;auto;progress;smt. qed.
lemma islossless_orcl2 : islossless DDHb.orclR.
proof. proc;auto;progress;smt. qed.
section.
declare module A <: H.AdvOrclb{-Count,-HybOrcl,-DDHb}.
declare axiom losslessA : forall (Ob0 <: Orclb{-A}) (LR <: Orcl{-A}),
islossless LR.orcl =>
islossless Ob0.leaks =>
islossless Ob0.orclL =>
islossless Ob0.orclR => islossless A(Ob0, LR).main.
lemma Hybrid:
forall &m,
Pr[Ln(DDHb, HybGame(A)).main() @ &m : (res /\ HybOrcl.l <= n) /\ Count.c <= 1 ] -
Pr[Rn(DDHb, HybGame(A)).main() @ &m : (res /\ HybOrcl.l <= n) /\ Count.c <= 1 ] =
1%r / n%r *
(Pr[Ln(DDHb, A).main() @ &m : (res /\ Count.c <= n) ] -
Pr[Rn(DDHb, A).main() @ &m : (res /\ Count.c <= n) ]).
proof.
move=> &m.
apply (H.Hybrid_div (<:DDHb) (<:A) _ _ _ _ &m
(fun (ga:glob A) (gb:glob DDHb) (c:int) (r:bool), r)).
apply islossless_leaks. apply islossless_orcl1. apply islossless_orcl2. apply losslessA.
smt(n_pos).
qed.
end section.