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I've started learning about complex multiplication (CM) on elliptic curves. For clarity (and intuition), I want to make some basic example of elliptic curves endomorphism construction for a concrete elliptic curve. I decided to construct a curve over $\mathbb{F}_{101}$ according to Atkin–Morain method, described, for example, in "Prime Numbers: A Computational Perspective" by Crandall & Pomerance. Used parameters are the following:

  1. Elliptic curve is constructed over $\mathbb{F}_{101}$, $p=101$,
  2. Number field to work is $\mathbb{Q}(\sqrt{-20})$, $D=-20$,
  3. Class number $h(D)=2$ and Hilbert polynomial is $H_D(x)=x^2 - 1264000x - 681472000$,
  4. Atkin–Morain method provides coefficients for the curve (or its twist) $a=81,b=47$

That means I've constructed a curve with these parameters $a,b$. Now I want to construct an endomorphism for better understanding how endomorphism ring is structured (do it by my hand). My question is how can I do it, where I can read some accessible information about performing it, maybe there are some examples already published?

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Regarding your need for some basic examples of endomorphisms, I would suggest the book Rational Points on Elliptic curves by Joseph H. Silverman and John T. Tate on chapter 6, Section 4 (Complex Multiplication).

Recall that an elliptic curve $E$ has complex multiplication if there is an endomorphism $\phi : E \to E$ which is not a multiplication by $n$ map.

More concretely, take the elliptic curve $E$ over $\mathbb{C}$ (or if you want over $\mathbb{F}_{p^2}$ with $p \equiv 3\ mod \ 4$ so that we can define $\sqrt{-1}$) \begin{equation*}E: y^2 = x^3 +x \end{equation*} Then, you can check that $E$ has the endomorphism (simply substitute the values and see that you get the same $E$)\begin{equation*} \phi(x,y) = (-x,\sqrt{-1}y) \end{equation*} that of course is not a multiplication by $n$ map and thus $E$ has complex multiplication.

More generally, let $P$ be a point of $E$.
If $\phi_1$ and $\phi_2$ are endomorphisms of $E$, then we can define a new endomorphism $\phi_1 + \phi_2$, as "addition" by \begin{equation*} (\phi_1 + \phi_2): E \to E,\ (\phi_1 + \phi_2)(P) = \phi_1(P) + \phi_2(P) \end{equation*} and we can also get a new endomorphism, as "multiplication" by taking the composition, \begin{equation*} (\phi_1\phi_2):E \to E,\ (\phi_1\phi_2)(P) = \phi_1(\phi_2(P)) \end{equation*}

With these operations we defined above, the set of endomorphisms of $E$ becomes a ring $End(E)$. If $E$ does not have complex multiplication, then $End(E)$ is isomorphic to $\mathbb{Z}$. Else, the endomorphism ring is strictly larger than $\mathbb{Z}$. (A subring (order) of either a quadratic imaginary field or a quaternion algebra)

Edit: Now that I'm rereading your question, you probably want some basic examples of endomorphisms for a specific curve so my answer probably is redundant. Either way, I leave the answer just in case.

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    $\begingroup$ Thank your for references! Yeah, as for my question I want to understand how to construct endomorphism for a particular elliptic curve over finite field (obviously with CM), associated with multiplication by complex number, related to $\sqrt{-D}$. $\endgroup$ Commented May 21 at 17:25
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I am not sure if I understand your question. You can have trivial endomorphisms like multiplication by $n$ maps ([$n$]) or the Frobenius endomorphism, which can be computed easily (see Mappings of Elliptic Curves or Division polynomials).

The endomorphism ring can be only $\mathbb{Z}$, an order in an imaginary quadratic field or an order in a quaternion algebra (for that see Silverman's The Arithmetic of Elliptic Curves Section III.9 or https://math.mit.edu/classes/18.783/2022/LectureNotes12.pdf, in general I recommend all the Sutherland notes at https://math.mit.edu/classes/18.783/2022/lectures.html).

I haven't played with elliptic curves for a while but I think that, e.g., if the end. ring is the order in a quadratic field (you have an ordinary curve - as is your case, or a rational subset of an end. ring of a supersing. curve), then it can look like this $\mathbb{Z}[\phi]$ where $\phi$ is the Frobenius endomorphism. So, any endomorphism you can get by combining some $[n]$ with $\phi$, e.g., $[n]+\phi$ is another endomorphism and you calculate it by using the addition formulae and the isogeny descriptions as rational functions.

Also, in general for supersingular curves (yours is not), non-trivial examples seems to be a hard problem https://arxiv.org/pdf/2309.10432 since the end. ring is some order in some quaternion algebra.

Also, I think you might get better answers at math.stackexchange since this is not directly related to crypto.

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  • $\begingroup$ Thank your for answer and references! As for my question I want to understand how to construct an endomorphism for a particular elliptic curve, specified in my question, associated with multiplication by complex number, related to $\sqrt{-D}$. By the way, is there an opportunity to transfer this question to math.stackexchange? $\endgroup$ Commented May 21 at 17:26

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