Find the coordinates of three distinct points on the elliptic curve

Asked Dec 20 '19 at 23:58

Active Dec 21 '19 at 00:54

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I am studying elliptic curves. I have an operation that I want to resolve. $y^2 = x^3 + 8x + 3$ on the ﬁeld $\mathbb {F}_{23}$. And I try to find the coordinates of three distinct points on the curve that are not the inﬁnite points. Can someone help to understand how I should proceed?

edited Dec 21 '19 at 00:54

kimchi lover

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asked Dec 20 '19 at 23:58

user2388887

Welcome to MSE. Please read this text about how to ask a good question. – José Carlos Santos Dec 21 '19 at 00:03
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F := GF(23); E:= EllipticCurve([F| 8,3]); E; Points(E); http://magma.maths.usyd.edu.au/calc/ – reuns Dec 21 '19 at 00:05
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Start substituting the values of $x=0,1,2, \ldots, 22$ and find if $y$ exists for these values of $x$. For example with $x=0$, we get $y^2=3 \pmod{23}$. You should check to see if $3$ has a square root modulo $23$. Hint: $(\pm 7)^2 \equiv 3 \pmod{23}$. Thus $(0,7)$ and$(0,16)$ are two points on the curve. – Anurag A Dec 21 '19 at 00:12
Thank to Reuns and Anurag. I will try to follow your guides. – user2388887 Dec 24 '19 at 00:21

Find the coordinates of three distinct points on the elliptic curve

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