To visualize multiplication of a point with an integer, we will use the simpler elliptic curve over the real numbers -- remember, the math is the same. Our goal is to find the multiple kG of the generator point G. That is the same as adding G to itself, k times in a row. In elliptic curves, adding a point to itself is the equivalent of drawing a tangent line on the point and finding where it intersects the curve again, then reflecting that point on the x-axis.
Starting with the generator point G, we take the tangent of the curve at G until it crosses the curve again at another point. This new point is -2G. Reflecting that point across the x-axis gives us 2G. If we take the tangent at 2G, it crosses the curve at -4G, which again we reflect on the x-axis to find G. Continuing this process, we can bounce around the curve finding the multiples of G, 2G, 4G, 8G, etc. As you can see, a randomly selected large number k, when multiplied against the generator point G is like bouncing around the curve k times, until we land on the point kG which is the public key. This process is irreversible, meaning that it is infeasible to find the factor k (the secret k) in any way other than trying all multiples of G (1G, 2G, 4G, etc) in a brute-force search for k. Since k can be an enormous number, that brute-force search would take an infeasible amount of computation and time.
Starting with the generator point G, we take the tangent of the curve at G until it crosses the curve again at another point. This new point is -2G. Reflecting that point across the x-axis gives us 2G. If we take the tangent at 2G, it crosses the curve at -4G, which again we reflect on the x-axis to find 4G. Continuing this process, we can bounce around the curve finding the multiples of G, 2G, 4G, 8G, etc. As you can see, a randomly selected large number k, when multiplied against the generator point G is like bouncing around the curve k times, until we land on the point kG which is the public key. This process is irreversible, meaning that it is infeasible to find the factor k (the secret k) in any way other than trying all multiples of G (1G, 2G, 4G, etc) in a brute-force search for k. Since k can be an enormous number, that brute-force search would take an infeasible amount of computation and time.
Sam Ritchie
10 years ago