Let $E(\mathbb F_{q^k})$ be an elliptic curve on finite field $\mathbb F_{q^k}$, where $\mathbb F_{q^k}$ is an extension of $\mathbb F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a bilinear pairing, and $G_1$, $G_2$ and $G_t$ are some subgroup of $E(\mathbb F_{q^k})$ of order $r$, where $r| \# E(\mathbb F_q)$ and $r^2 | \# E(\mathbb F_{q^k})$.

Is it possible to modify (Weil or Tate) pairing such that the order of $G_t$ be $r^2$? In other word, is it possible to map (with keeping bilinearity property) two $r$-torsion point to a $r^2$-torsion point?

Let $E(\mathbb F_{q^k})$ be an elliptic curve on finite field $\mathbb F_{q^k}$, where $\mathbb F_{q^k}$ is an extension of $\mathbb F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a bilinear pairing, and $G_1$, $G_2$ are some subgroup of $E(\mathbb F_{q^k})$ of order $r$, where $r| \# E(\mathbb F_q)$ and $r^2 | \# E(\mathbb F_{q^k})$, and $G_t$ is subgroup of order of $\mathbb F^*_{q^k}$.

Is it possible to modify (Weil or Tate) pairing such that the order of $G_t$ be $r^2$? In other word, is it possible to map (with keeping bilinearity property) two $r$-torsion points to a $r^2$-root of unity?

Let $E(F_{q^k})$ be an elliptic curve on finite field $F_{q^k}$, where $F_{q^k}$ is an extension of $F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a bilinear pairing, and $G_1$, $G_2$ and $G_t$ are some subgroup of $E(F_{q^k})$ of order $r$, where $r| \sharp E(F_q)$ and $r^2 | \sharp E(F_{q^k})$.

Is it possible to modify (Weil or Tate) pairing such that the order of $G_t$ be $r^2$? In other word, is it possible to map (with keeping bilinearity property) two $r$-tortion point to a $r^2$-tortion point?

Let $E(\mathbb F_{q^k})$ be an elliptic curve on finite field $\mathbb F_{q^k}$, where $\mathbb F_{q^k}$ is an extension of $\mathbb F_q$ with $k>1$. Let $e: G_1 \times G_2 \rightarrow G_t$ be a bilinear pairing, and $G_1$, $G_2$ and $G_t$ are some subgroup of $E(\mathbb F_{q^k})$ of order $r$, where $r| \# E(\mathbb F_q)$ and $r^2 | \# E(\mathbb F_{q^k})$.

Is it possible to modify (Weil or Tate) pairing such that the order of $G_t$ be $r^2$? In other word, is it possible to map (with keeping bilinearity property) two $r$-torsion point to a $r^2$-torsion point?