Is it just me, or is there something wrong with the points given? For example (6, 3) is not a point on that curve since 6^3 +1 mod 11 = 8, and 3^2 mod 11 = 9. On the other hand, (2, 3) and (2, 8) should be points on the curve.
In point addition (using chord method), is it always the case that the line passing through the points P and Q crosses the elliptic curve at some point? How about in the case of point doubling (using tangent method), what do we say if the tangent at point P doesn't cross elliptic curve at another point?
Sorry, I have not made any video to explain that. I will make it in future. You may simply choose an x value from the set {0, 1, 2, .... m-1} where m is the mod on which elliptic curve is defined. Then find corresponding y value(s) for that specific x using simple modular arithmetic.
@@AdvancedMath So I simply choose values of x from {0,1, 2, ..., m-1} one by one and substitute in the curve equation to get the corresponding y value. At the end I will get all points on the curve. Got it. Your effort, sincerity and prompt response is much appreciated and very inspiring. Thank you very much and lots of respect.
Thanks a lot sir for the video. Only one question at [02:38] in the video. Why is (6,3) an element in your set? I added x=6 and y=3 in your equation y^2≡(x^3+1)mod(11) to check if (6,3) is an element. I did the following: 3^2≡(6^3+1)mod(11) 9≡(216+1)mod(11) 9≡(216+1)mod(11) 9≡(217)mod(11) 9≡(217)mod(11) 9≡8. As you can see 9 and and 217 are not in the same class of reminders. So, why is then (6,3) an element in your set? THX and BR
I have tried to find numbers on the curve but for (3,6) and (6,8) I find that when using x=3: y^2=6 and x=6:y^2=8. What am I missing? Why is it y^2 and not y?
I can't believe it took me only a few minutes to understand these concepts, thanks
Is it just me, or is there something wrong with the points given? For example (6, 3) is not a point on that curve since 6^3 +1 mod 11 = 8, and 3^2 mod 11 = 9. On the other hand, (2, 3) and (2, 8) should be points on the curve.
I have not properly calculated the points on elliptic curve for the example. You are right (6, 3) does not lie on the curve.
In point addition (using chord method), is it always the case that the line passing through the points P and Q crosses the elliptic curve at some point?
How about in the case of point doubling (using tangent method), what do we say if the tangent at point P doesn't cross elliptic curve at another point?
Sir, great explanation! How did you obtain the points of the elliptic curve at 2:20 ?
Sorry, I have not made any video to explain that. I will make it in future. You may simply choose an x value from the set {0, 1, 2, .... m-1} where m is the mod on which elliptic curve is defined. Then find corresponding y value(s) for that specific x using simple modular arithmetic.
@@AdvancedMath So I simply choose values of x from {0,1, 2, ..., m-1} one by one and substitute in the curve equation to get the corresponding y value. At the end I will get all points on the curve. Got it. Your effort, sincerity and prompt response is much appreciated and very inspiring. Thank you very much and lots of respect.
@@anusha5788 True.
@@anusha5788 True. But do not forget to use modular arithmetic while simplifying equation.
@@AdvancedMath Thank you I have learnt calculation of points without given initial point.
Can I ask about the case of singularity? Why does the curve have no well-defined tangents when 4a^3+27b^2=0?
What is simplified equation to find value of y when x is known? Or without using modular square foot?
Thanks a lot sir for the video. Only one question at [02:38] in the video. Why is (6,3) an element in your set? I added x=6 and y=3 in your equation y^2≡(x^3+1)mod(11) to check if (6,3) is an element. I did the following: 3^2≡(6^3+1)mod(11) 9≡(216+1)mod(11) 9≡(216+1)mod(11) 9≡(217)mod(11) 9≡(217)mod(11) 9≡8. As you can see 9 and and 217 are not in the same class of reminders. So, why is then (6,3) an element in your set? THX and BR
That is a mistake. Sorry about that.
I have tried to find numbers on the curve but for (3,6) and (6,8) I find that when using x=3: y^2=6 and x=6:y^2=8. What am I missing? Why is it y^2 and not y?
y^2=6 then y= sqrt(6), this means u have to find a number mod 11 whose square is 6, but u have only {1,3,4,5,9} as QR and NQR= {2,6,7,8,10)
Thanks for the video but the given points were big disaster.