On division polynomials and supersingular elliptic curve

Abstract

The multiplication-by-m map on an elliptic curve E can be expressed by the division polynomials $\psi_n$, $\omega_n$ and $\phi_n$. The polynomials satisfy the relation $\psi_{nm}(M) =\psi_n(M)^{m^2}\psi_m([n])M)$. Based on this fact, we can show that if E is supersungular over $F_p$, then $\psi_p\equiv=-1$ mod p. Furthermore, p $\equiv$ 3 mod 4 or $\Bigg(\frac{\triangle}{p})\Bigg)=-1$$. And we apply this fact to test whether a prime p is supersingular or not over E.