If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in E(\mathbb{F}_p)$ with $mT= \mathcal{O}$. My question is in finite field $\mathbb{F}_q, 1=q+1$. It follows that $T=(q+1)T$. Does it imply $q$ have to be divided by $m$? Is there any other possibility? Thank you~

If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in E(\mathbb{F}_p)$ with $mT= \mathcal{O}$. My question is in finite field $\mathbb{F}_q, 1=q+1$. It follows that $T=(q+1)T$. Does it imply $q$ have to be divided by $m$? Is it possible that $q$ isn't divided by $m$ with other condition while $1=q+1, T=(q+1)T$? Thank you~

If \mathbb{F}_q is a finite field and the elliptic curve E is defined over finite field \mathbb{F}_p such that ECDLP is hard in E(\mathbb{F}_p), where q, p are prime and q \ll p. Let T \in E(\mathbb{F}_p) with mT= \mathcal{O}. My question is in finite field \mathbb{F}_q, $1=q+1$. It follows that T=(q+1)T. Does it imply m have to be divided by q? Is there any other possibility? Thank you~

If $\mathbb{F}_q$ is a finite field and the elliptic curve E is defined over finite field $\mathbb{F}_p$ such that ECDLP is hard in E($\mathbb{F}_p$), where $q, p$ are prime and $q \ll p$. Let T $\in E(\mathbb{F}_p)$ with $mT= \mathcal{O}$. My question is in finite field $\mathbb{F}_q, 1=q+1$. It follows that $T=(q+1)T$. Does it imply $q$ have to be divided by $m$? Is there any other possibility? Thank you~