# Point Order on Elliptic Curve

i got stuck doing the exercise 4.5 d) in the Book Elliptic Curves: Number Theory and Cryptography by L. Washington and would be grateful for hints.

Let $$p \equiv 1 \text{ (mod 4)}$$ be prime and let $$E$$ be given by $$y^2 = x^3 -kx$$, where $$k \not\equiv 0 \text{ (mod p)}$$. Let $$k$$ be a square but not a fourth power mod $$p$$. Show that exactly one of the curves $$y^2 = x^3 -x$$ and $$y^2 = x^3 -kx$$ has a point of order $$4$$ defined over $$\mathbf{F_p}$$.

I tried to find a point such that $$2P = (\pm\sqrt{k},0)$$, but that did not work out and solving the division polynomial $$\psi_4$$ also didnt work.

• $$k=d^2$$ then $$y^2=x^3-kx$$ is isomorphic to $$E_d:y^2=d(x^3-x)$$.

• $$E_1:y^2=x^3-x$$.

• Since $$d$$ is not a square, for all $$a\in \Bbb{F}_p$$, either $$(a,\pm \sqrt{a^3-a})\in E_1(\Bbb{F}_p)$$ or $$(a,\pm \sqrt{d(a^3-a)})\in E_d(\Bbb{F}_p)$$

• $$\#E_1(\Bbb{F}_p)+\#E_d(\Bbb{F}_p)= 2(p-3)+8\equiv 4\bmod 8$$

• Both $$E_1(\Bbb{F}_p)$$ and $$E_d(\Bbb{F}_p)$$ contain the 2-torsion (the points coming from $$a=\infty$$ or $$a^3-a=0$$)

• Thus, exactly one of $$\# E_1(\Bbb{F}_p)$$,$$\# E_d(\Bbb{F}_p)$$ is $$\equiv 0\bmod 8$$, the other is $$\equiv 4\bmod 8$$.

• Exactly one of $$E_1(\Bbb{F}_p)$$,$$E_d(\Bbb{F}_p)$$ contains some point of order $$4$$.

• Thank you for providing hints, i have a problem getting from the second to last bullet point to the conclusion that exactly one of the elliptic curves contains some point of order 4. Can you explain it in more detail please.
– ASP
Jan 21, 2021 at 19:38
• Is $2(p-3)+8$ clear to you? Jan 22, 2021 at 0:38
• yes, since d is not a square mod p and $E_d$ is a twist of $E_1$ we have that $\#E_1(\mathbf{F}_p) + \#E_d(\mathbf{F}_p) = p +1 - a + p +1 +a = 2p+2$ and since $p\equiv 1$(mod 4) we got $2(4r +1) +2 = 8r +4 \equiv 4$ mod 8.
– ASP
Jan 22, 2021 at 1:28
• $E_1(F_p)$ is a group so the size of the 2-torsion $\# E_1(F_p)=4$ divides $\# E_1(F_p)$ and it contains some point of order 4 iff $\# E_1(F_p)> \# E_1(F_p)$ iff $8 \ | \ \# E_1(F_p)\ | \ \# E_1(F_p)$ Jan 22, 2021 at 1:59
• $2(p-3)+8$ follows from that $p-3$ is the number of $a$ such that $a^3-a\ne 0$, multiplied by 2 for the sign of $\sqrt{a^3-a}$, and $8=2.4$ is the 2-torsion $(a\in 0,-1,1,\infty$) counted two times since it is in both $E_1(F_p)$ and $E_d(F_p)$. Jan 22, 2021 at 2:03