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Unformatted text preview: Lecture 13: Elliptic Curve Cryptography In general, an elliptic curve can be defined by the equation y 2 +axy+by = x 3 +cx 2 +dx+e. For our purposes, we'll only consider those of the form: y 2 = x 3 + bx + c. Consider the following example: Let a=1, b=1. We define the set of points that satisfy the equation y 2 = x 3 + x + 1, as E(1,1). If we include "the point at infinity" with these points, we have a set of points that define an abelian group, under certain conditions. A couple notes before we define the group. 1) Note that the graph is symmetric about the xaxis. If (x,y) is a solution to the equation above, so is (x, y). For our particular graph some sample points are (0, 1), (0, 1), (1, 3) and (1,  3). 2) To ensure that the function in x on the righthand side has no repeated factors, we must make sure to pick a and b such that 4a 3 27b 2 . If we pick a and b based on the rules in #2, then we define addition amongst these points as follows: If 3 points on an elliptic curve lie on a straight line, their sum is 0, the point at infinity. If we consistently apply this definition above, we can derive the following: 1) 0 = 0, and 0 is the additive identity, thus P + 0 = P. For the rest of these rules, we will assume that P 0 and Q 0....
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 Summer '09

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