On public keys compression, why an even or odd y coordinate corresponds to the positive / negative sign respectively?
There is no such thing as a negative or a positive value when you're talking in a finite field.

For example, in Z7, the field of integers modulo 7. There holds:


0 = 7 = 14 = -7
1 = 8 = 15 = -6
2 = 9 = 16 = -5
...


So you can't say that the number 2 is positive, because it's equal to -5.

Despite that, the square root still has two solutions. For example, 3^2 = 9 = 2, 4^2 = 16 = 2. Thus both 3 and 4 are square roots of 2.

So we need a way to say which solution we want. Turns out, that when reduced to a range of 0-6, the two solutions of the square, one is odd and the other is even.