Elliptic Curve and Consecutive X Coordinates

Question

My question is brief, is there an algorithm or method to calculate the consecutive X coordinates on an elliptic curve to find out if the distance between is constant?

The reason for the question is as follows: 1. Base Generator Point is given. 2. Embedding degree is given. 3. The total number of points is given (N). 4. For the curve, 1 - (N-1) are the multipliers for the points on the curve. 5. The curve is half the full curve for X-Coordinates meaning 1 and N-1 are the same X coordinate but the inverse on the Y coordinate.

Thanks.

score 1 · Answer 1 · answered May 15 '20 at 23:59

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Not sure if this is helpful, but there are a number of papers that discuss $x$-coordinates on elliptic curves that lie in arithmetic progression. Here's one paper:

A. Bremner, N. Tzanakis, J.H. Silverman, Integral points in arithmetic progression on $y^2=x(x^2-n^2)$, Journal of Number Theory 80 (2000), 187-208.

answered May 15 '20 at 23:59

Joe Silverman

2,174

Does this work has any effect on Cryptography? – kelalaka Feb 20 '21 at 19:29
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@kelalaka As far as I'm aware, the answer is no, there are no applications of the sorts of results tin our paper to cryptography. – Joe Silverman Feb 20 '21 at 21:08
I see, any possible applications? – kelalaka Feb 20 '21 at 21:10
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Our paper has been cited 30 times in other papers (according to MathSciNet). You might take a look at those to see if there are any applications that you find interesting. – Joe Silverman Feb 20 '21 at 23:29

Elliptic Curve and Consecutive X Coordinates

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