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In elliptic curve math, there is a point called the “point at infinity”, which roughly cor‐ responds to the role of 0 in addition. On computers, it’s sometimes represented by x = y = 0 (which doesn’t satisfy the elliptic curve equation — but it’s an easy separate case that can be checked). There is also an operator “+”, called “addition” which has some properties similar to the traditional addition of real numbers that grade school children learn. Given two points P1 and P2 on the elliptic curve, there is a third point P3 = P1 + P2, also on the elliptic curve. Geometrically, this third point P3 is calculated by drawing a line between P1 and P2. This line will intersect the elliptic curve in exactly one additional place. Call this point P3' = (x, y). Then reflect in the X axis to get P3 = (x, -y). There are a couple of special cases which explain the need for the “point at infinity”. If P1 and P2 are the same point, the line “between” P1 and P2 should extend to be the tangent on the curve at this point P1. This tangent will intersect the curve in exactly one new point. You can use techniques from calculus to determine the slope of the tangent line. These techniques curiously work even though we are restricting our interest to points on the curve with two integer coordinates! In some cases (i.e., if P1 and P2 have the same x values but different y values), the tangent line will be exactly vertical, in which case P3 = “point at infinity”. If P1 is the “point at infinity”, then the sum P1 + P2 = P2. Similary, if P2 is the point at infinity, then P1 + P2 = P1. This shows how the point at infinity plays the roll of 0. It turns out that + is associative, which means that (A+B)+C = A+(B+C). That means we can write A+B+C without parentheses without any ambiguity. Now that we have defined addition, we can define multiplication in the standard way that extends addition. For a point P on the elliptic curve, if k is a whole number, then kP = P + P + P + ... + P (k times). Note that k is sometimes confusingly called an “exponent” in this case. Generating a public key Starting with a private key in the form of a randomly generated number k, we multiply it by a predetermined point on the curve called the generator point G to produce another point somewhere else on the curve, which is the corresponding public key K. The gen‐ erator point is specified as part of the secp256k1 standard and is always the same for all keys in bitcoin. K=k*G 68 | Chapter 4: Keys, Addresses, WalletsPDF Image | Mastering Bitcoin
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