To multiply two points on an elliptic curve, you would use the elliptic curve point multiplication algorithm. This algorithm takes two points on the curve, P and P1, and a coefficient n, and returns a third point, nP1 + P.
To perform the multiplication, you would first choose a coefficient n and calculate the value of nP1. Then, you would add P to the result to get the final result, nP1 + P.
Here is an example of how you might implement this in code:
def multiply_points(P, P1, n):
result = [0, 0]
for i in range(n):
result = add_points(result, P1)
return add_points(result, P)
above function takes two points, P and P1, and a coefficient n, and returns the result of the multiplication nP1 + P. It does this by first initializing a result point to the identity element (0, 0) and then repeatedly adding P1 to the result point n times. Finally, it adds P to the result to get the final result.
Q: how can I faster do it?
One way to make the point multiplication algorithm faster is to use the double-and-add method. This method takes advantage of the fact that multiplying a point by a number n is the same as adding the point to itself n-1 times.
Here is an example of how you might implement the double-and-add method in code:
def multiply_points_fast(P, P1, n):
result = [0, 0]
while n > 0:
if n % 2 == 1:
result = add_points(result, P1)
P1 = add_points(P1, P1)
n = n // 2
return result
Above function takes two points, P and P1, and a coefficient n, and returns the result of the multiplication nP1 + P. It does this by repeatedly doubling P1 and adding it to the result whenever the coefficient n is odd. By using this method, you can reduce the number of point additions required to calculate the result, making the overall calculation faster.
Q: what is n
In the context of elliptic curve point multiplication, n is a coefficient that determines how many times the point P1 should be added to itself. It is used to calculate the result of the multiplication nP1 + P.
For example, if n is 3 and P1 is (2, 3), then the result of the multiplication would be 3P1 + P = P1 + P1 + P1 + P. In this case, P1 would be added to itself 3 times to get the final result.
The value of n can be any integer, positive, negative, or zero. When n is positive, the result of the multiplication will be a point on the curve that is farther away from the identity element (0, 0). When n is negative, the result will be a point that is closer to the identity element. When n is zero, the result will be the identity element itself.
Another algorithm for multiplying two points on an elliptic curve is the sliding window method. This method works by breaking the coefficient n into "windows" of bits and precomputing a table of values that can be used to quickly compute the result of the multiplication.
Here is an example of how you might implement the sliding window method in code:
def multiply_points_sliding_window(P, P1, n, window_size):
# Precompute a table of values
table = [P1]
for i in range(1, 1 << window_size):
table.append(add_points(table[-1], P1))
result = [0, 0]
while n > 0:
# Compute the next window of bits
window = n & ((1 << window_size) - 1)
for i in range(window_size - 1, -1, -1):
if window & (1 << i) != 0:
result = add_points(result, table[i])
n = n >> window_size
P1 = add_points(P1, P1)
return result
This function takes two points, P and P1, and a coefficient n, as well as a window size, and returns the result of the multiplication nP1 + P. It does this by precomputing a table of values using the double-and-add method and then using the table to quickly compute the result by breaking the coefficient n into windows of bits. The size of the window can be adjusted to trade off space for time, depending on the desired performance.
