A non-scary introduction to Elliptic Curves
Cryptography 101
Self Test
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Answer:
An elliptic curve is a relatively simple mathematical graph of the form y^2 = x^3 + ax + b.
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Answer:
The elliptic curves used in cryptography are different to continuous curves in two key respects. 1. They are defined over a finite field, meaning that the curve is only defined on discrete (integer) values on the x-axis. 2. They are defined modulo a prime number, meaning that the y-values “wrap around” some maximum (prime) value.
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Answer:
The security of elliptic curve cryptography is based on the fact that it is very difficult to find the discrete logarithm of a point on the curve. This is a fancy way of saying that it is very difficult to find the value of x if you know the value of y. This is known as the “elliptic curve discrete logarithm problem” (ECDLP).
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Answer:
Because the values “wrap around” modulo some large prime, and because the fields (ranges of values) are very, very large, the distribution of values starts to look very random. So although it is very easy to calculate a y-value given an x-value, by using the curve’s formula, it is difficult to do the reverse. Furthermore, the derivation of a secret key from a public key involves repeating this process (typically) many many trillions of times, it becomes infeasible to brute-force the solution in a reasonable amount of time.
