How to calculate Cardinality of an Elliptic curve over $\mathbb Z_{23}$.
$E: y^2 = x^3 + x + 1$ defined over $\mathbb Z_{23}$.
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Use sagemath's order() ... here the code – kelalaka Feb 23 '24 at 08:07
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Welcome to crypto-SE. Hint: by definition, the cardinality (or order) of an Elliptic curve is the number of elements in the Elliptic Curve group. The Elliptic Curve group is a set, which consists of the neutral element noted $\infty$ or $\mathcal O$, and the solutions $(x,y)$ to the equation $E$, with $x,y\in\mathbb Z_{23}$. Because $23$ is so small, the simplest (and likely what you are expected to do) is to systematically find and count these solutions. It helps to note that if $(x,y)$ is a solution, then so is $(x,-y)$, or equivalently $(x,23-y)$. – fgrieu Feb 23 '24 at 08:54
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2@fgrieu though 23 is small, finding 28 points ( well 14 since reflection) is not a good exercise due to lots of repeating calculations. for x in range(0,24): check that $f(x)$ is QR in $Z_{23}$ and special case $f(x) = 0$... – kelalaka Feb 23 '24 at 10:29
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When I gave such an assignment (usually with a somewhat smaller field) the idea was: 1. List the quadratic residues. 2. Quickly generate a table of $x^3+x+1$. 3. Find overlap and take $y,-y$ corresponding to the residue. This can be done by hand – kodlu Feb 23 '24 at 19:24