We prove that for any pair of integers
0
≤
r
≤
g
0\leq r\leq g
such that
g
≥
3
g\geq 3
or
r
>
0
r>0
, there exists a (hyper)elliptic curve
C
C
over
F
2
\mathbb {F}_2
of genus
g
g
and
2
2
-rank
r
r
whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties
(
A
,
λ
)
(A,\lambda )
over
F
2
\mathbb {F}_2
of dimension
g
g
and
2
2
-rank
r
r
such that
Aut
(
A
,
λ
)
=
{
±
1
}
\operatorname {Aut}(A,\lambda )=\{\pm 1\}
.