So, if two points are revealed, you know the lower bound of the diameter of the circle, as well as a line upon which the center of the circle must lay. So some information has leaked, but the leaked information isn’t enough to simplify determining the parameters of the circle to any significant extent.

If you think in terms of real numbers, the line still has an infinite number of points. But in practice the secret will be a finite number of bits, so all of X, Y (center coordinates) and R (radius) will be quantised.

Assume each of them encodes 40 bits, so the secret key is 120 bits, quite safe agains brute force. Given a line that must pass very close to the center reduces the search space to 40+eps bits, with very small eps. Brute forcing that is very well possible. So this is not better than splitting the key into three 40 bit shares.

]]>It will now be up to NIST to come up with the rules and standards for implementation.

“Remember what the dormouse said: feed your head.”

]]>Great description. Hopefully some kids will want to learn.

I was just thinking about a plane thru a conic at a funny angle.

]]>I had know idea what you are talking about, but if you plot lotto ticket, you can see the same disputation. Try e 2.713 with a celling at say 100 , then passed that point invert the numbers. ]]>

Please, let’s not bring up ellipse

Why not?

You only need to know pythag, and have the immagination to apply it and a rotation at right angles to the plane of the circle.

It took my son about five minutes to work out the basic equation for elipses after he asked about orbital mechanics.

For those that do not know… A circle is a special case of an ellipse where both foci are coincident. You can calculate a unity circle just using pythag it’s in most high school maths text books. What is often not is how you relate the positions of the foci on the semi-major-axis.

Think of holding the circle like a hoop at arms length one hand on either side of the hoop. For most people it is a simple matter to rotaye the hoop so the top of it as you look at it comes towards you whilst the bottom goes away. The hoop as you look at it starts as a circle, then becomes an obvious ellipse (oval) untill ay 90 degrees of rotation it looks like a single horizontal line.

Easy so far?

Now imagine that you have a friend holding the hoop and rotating it, and you are standing to one side of them and both there hands at your eye level so you can only see the hand of the side you are on. You are in effect looking down the semi-major-axis. You do not see a hoop but a vertical line that if you are standing on the persons left, rotates clockwise from the vertical untill it is horizontal.

So far so good?

Now when it’s neither vertical or horizontal it’s easy to see it forms the hypotenuse of a right angle triangle. Look at where the foci are on the semi-major-axis and you can quickly see how the verticle of the triangle relates to the distance from the edge of the circle.

Another fun thing is a graphical proof that the ellipse is not parabolic in shape even though it migh look like it.

To draw an ellipse you need two drawing pins a piece of thread and a pencil. Put two knots in the thread the same length as the semi-major-axis and pin then down at the two foci. Pull the thread with the tip of the pencil so it forms a triangle. Then keeping the thread tight with the pencil tip take the pencil round and an ellipse will be drawn out.

To draw a parabolic shape you do things slightly differently. Draw two parallel lines on your paper and draw another line that crosses them at 90 degrees. Again using a thread with two knots in it, pin one at where you want the focus point to be on the third line. Again using the pencil to keep the thread tight and draw a line you need to keep the free knot on one of the vertical paralles and move it and the pencil up and down such that a line between the free knot and the pencil will be parallel to the third line on which the fixed knot at the focus is.

Apply simple logic to the fact the ellipse has both knots at fixed points and for the parabolic curve one end is free. Then flip things over to draw a second parabolic curve on the other side it should be obvious that if you were to cut the two parabolic curves out, no matter how far appart they are a source at one parabolic dishes curve would focus back at the other parabolic dishes focus without either dish or dish focus point changing. A little further thinking will show that an ellipse can not do this as the only time the line from the curve away to the second focus is parallel to the semi-major-axis, is when it is directly coincident to the axis. Even when the ellipse is infinite in size the focus points are less than that appart therfore whilst it approches being parallel it can not ever become so.

All nice and easy to show your kids and bring math to life for them.

Now think about how you would prove it non graphically, would you even be able to work it out yourself without being first shown?

Oh and if you draw both parabolic curves close enough you can show them that they do not join smothly unlike the ellipse.

Whilst you are drawing your ellipse you can also show them what a “cusp” is caused by a “Reflective Caustic”

Basically loose half the ellipse then use the technique for drawing the parabolic curve in reverse such that the lines come in, in parallel and reflect off the eliptic curve.(drawing this takes practice).

You can see such a cusp if you have a cylindrical drinking glass with vertical sides and a flat bottom. If you project in a horizontal “slit of light” angled downwards so it reflects off of the inside of the cylinder and down onto the bottom, you get the bright line of the caustic visable on the bottom. Similar is seen in some tea cups and stainless steel saucepans on sunny winter days when the sunlight comes in a window at a shallow angle.

]]>Please, let’s not bring up ellipse. John will get confused. That 3 dimensional talk is confusing for the youngins.

]]>Which, I believe, it possibly can, via the magic of math, Just not obvious math.

Do I dare mention that a parabola and a circle are linked by a cone which is why only three points are required to describe a parabola?

As for modular fields of infinite planes, yes it’s easier with parabolas because they are open, but circles are closed thus finite.

It’s the kind of “not obvious” that often acts as an impediment to understanding.

]]>If you have two points of a circle, a line segment connecting those two points is a chord of that circle. And the perpendicular bisector of that chord passes through the center of that circle as well. It also passes through an infinite number of possible circles that include the two specified points, as well as an infinite number of possible circles that don’t include those two points.

So, if two points are revealed, you know the lower bound of the diameter of the circle, as well as a line upon which the center of the circle must lay. So some information has leaked, but the leaked information isn’t enough to simplify determining the parameters of the circle to any significant extent.

I mention circles because every semiprime is tied to a unique circle.

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