Classification of reduced weierstrass form Elliptic Curves Over Z_n

Computational results for classification of such elliptic curves are demonstrated via this HTML page. Results are obtained using C++ program and then converted into HTML tables.

Elliptic Curves over Z_n, gcd(n,6)=1

Reduced weierstrass equation for elliptic curves are derived by applying below admissible change of variables to generalized weierstrass form $\mathbf{y^2+a_1 x y+a_3 y=x^3+a_2 x^2+a_4 x+a_6}$ , $a_i\in\mathbb{Z}_n$
$(x,y) \underset{}{\rightarrow} (x,y - \frac{a_1}{2}x - \frac{a_3}{2}) \Rightarrow y^2 = x^3 + b_2x^2 + b_4x + b_6 \\(x,y) \underset{}{\rightarrow} (\frac{x - 3b_2}{36},\frac{y}{216}) \\\Rightarrow y^2 = x^3 + ax + b, a,b \in Z_n, gcd(n,6)=1\\ \Delta = -16(4a^3 + 27b^2)$

where $\mathbf{\Delta}$ is Discriminant for such elliptic curve.

Isomorphism between two elliptic curves over Z_n, gcd(n,6)=1

reference : Elliptic Curve Public Key Cryptosystems by Alfred Menezes, chapter 2

Pseudocode for generating all non-singular elliptic curves over Z_n, gcd(n,6)=1

Pseudocode for classifying all non-singular elliptic curves over Z_n, gcd(n,6)=1

Note : Point at infinity(O) is not included in #Points (one of the column in classification results Table. go to classification results to see data.)

To see the classification results click here:

Created By: Param Parekh, Paavan Parekh
Mentored By: Prof. Manish Gupta, Sourav Deb
Contact Us: ecc[at]guptalab.org

Classification of reduced weierstrass form Elliptic Curves Over Zn

Elliptic Curves over Zn, gcd(n,6)=1

Isomorphism between two elliptic curves over Zn, gcd(n,6)=1