Existence of an elliptic curve and a distortion map  I would like to know if the following is true or wrong or unknown.
For any integer $m$, there is an elliptic curve over the complex numbers such that there is a (distortion) map $\phi$ so that $\{ T, \phi(T) \}$ is a basis of$E[m]$, for some point $T \in E[m]$.
I know if $m\equiv 3 (4)$ then we can take $E: y^2=x^3+x$ and $\phi$ is a complex multiplication by $i$. 
So I'd like to know if there is such an elliptic curve and a distortion map for any other $m$.
 A: I assume $n$ is $m$. Also, of course, the map $\phi$ is supposed to be an isogeny.
It's late, so this may not be quite right, but here's a thought. Take $E$ to have endomorphism ring $\mathbb{Z}[\sqrt{-D}]$. We at least want to choose $D$ so that the endomorphism $\phi=\sqrt{-D}$ does not behave like multiplication by an integer on $E[m]$. So we want $\sqrt{-D}\not\equiv a\pmod{m}$ for all integers $a$. It suffices to require that the norm of $a+\sqrt{-D}$ not be divisible by $m$. So we're looking for an integer $D$ such that $a^2+D\not\equiv0\pmod{m}$ for all integers $a$. It suffices to choose $D$ so that $-D$ is not a square modulo some prime dividing $m$.
This will at least get you a map so that if $T$ has order $m$, then the group generated by $T$ and $\phi(T)$ is strictly bigger than $m$. I'm pretty sure that with a little more work (choosing $D$ more carefully), you can get $T$ and $\phi(T)$ to generate $E[m]$, but I'll stop here.
A: The elliptic curve E would have to have complex multiplication.  If End(E) = Z then no such map will exist.
The word "distortion map" was introduced for elliptic curves over finite fields. Now the answer depends on wording.  You wrote "for some point T".  That case is easy.  The case "for all non-zero points T" is harder, and needs E to be supersingular.
The relevant literature is:


*

*E.R. Verheul, Evidence that XTR is more secure than supersingular
elliptic curve cryptosystems, Journal of Cryptology

*S. D. Galbraith and V. Rotger, Easy decision Diffie-Hellman groups, London Mathematical Society Journal of Computational Mathematics, Vol. 7 (2004) 201-218.
