Elliptic curve point multiplication
Elliptic curve point multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography (ECC) as a means of producing a trapdoor function. The literature presents this operation as scalar multiplication, thus the most common name is elliptic curve scalar multiplication, as written in Hessian form of an elliptic curve.
Given a curve, E, defined along some equation in a finite field (such as E: y2 = x3 + ax + b), point multiplication is defined as the repeated addition of a point along that curve. Denote as nP = P + P + P + … + P for some scalar (integer) n and a point P = (x, y) that lies on the curve, E. This type of curve is known as a Weierstrass curve.
The security of modern ECC depends on the intractability of determining n from Q = nP given known values of Q and P if n is large (known as the "elliptic curve discrete logarithm problem"). This is because the addition of two points on an elliptic curve (or the addition of one point to itself) yields a third point on the elliptic curve whose location has no immediately obvious relationship to the locations of the first two, and repeating this many times over yields a point nP that may be essentially anywhere. Intuitively, this is not dissimilar to the fact that if you had a point P on a circle, adding 42.57 degrees to its angle may still be a point "not too far" from P, but adding 1000 or 1001 times 42.57 degrees will yield a point that may be anywhere on the circle. Reverting this process, i.e., given Q=nP and P and determining n can therefore only be done by trying out all possible n -- an effort that is computationally intractable if n is large.
With 2 distinct points, P and Q, addition is defined as the negation of the point resulting from the intersection of the curve, E, and the straight line defined by the points P and Q, giving the point, R.
Assuming the elliptic curve, E, is given by y2 = x3 + ax + b, this can be calculated as:
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