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| | When it comes to hashing, sometimes 64 bit is not enough, for example, because of birthday paradox -- the hacker can iterate through random $latex 2^{32}$ entities and it can be proven that with some constant probability they will find a collision, i.e. two different objects will have the same hash. $latex 2^{32}$ is around...
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| So far in this series we've seen elliptic curves from many perspectives, including the elementary, algebraic, and programmatic ones. We implemented finite field arithmetic and connected it to our elliptic curve code. So we're in a perfect position to feast on the main course: how do we use elliptic curves to actually do cryptography? History As the reader has heard countless times in this series, an elliptic curve is a geometric object whose points have a surprising and well-defined notion of addition.