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2

You're essentially correct. Index calculus is impractical on elliptic curves because there is not a straightforward notion of smoothness in these groups. In prime fields, there is the easy mapping from the multiplicative group to the integers, where smoothness is well-defined. Similarly, in extension fields there is the mapping to polynomials over the ...

4

Here's the deal. The discrete log problem is feasible in the special case where the exponent is known to come from a small range of possibilities (e.g., the exponent is not too large). Suppose we are given $y=g^m$, and we want to find $m$. Suppose moreover we know that $m$ is small: $0 \le m < 2^{30}$, say. Then it turns out it is easy to recover $m$, ...

0

The size of the modulus for ssh implementations of Diffie-Hellman (according to RFC4419 from 2006) should be between 1024 and 8192 bits. This seems to be a bit off from what @poncho recommended (128 bits) in 2012.

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