Hardness of discrete log in additive group

Question

Why is discrete log considered to be hard in multiplicative group but not in additive group?

Answer 1

It's a matter of how field elements are represented. Elements of large finite fields are usually represented as vectors over a prime field with components in ${\mathbb Z}/(p)$. So addition is easy and is just addition mod $p$. Multiplication is also not difficult and just involves working modulo a minimal polynomial.

The discrete log problem is to express a given element (represented as a vector) as a power $w^k$ of a primitive element of the field $F$, where $0 \le k < |F|-1$. This is believed to be difficult (i.e. not solvable in polynomial time), although there has been so much work done on algorithms and fast implementation, that you need to go up to very large fields before it starts to become noticeably slow.

For smaller finite fields, elements are often represented as $w^k$, in which case discrete log and multiplication are very easy, but addition is harder.