I am trying to construct a theoretical Diffie-Hellman key exchange protocol. However, i cannot understand the difference between choosing an additive group or a multiplicative group. I believe an additive would make it much more simple, but would a multiplicative group make it more difficult to break? What would be the difference between the use of those 2 kinds of number groups?
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From an abstract algebraic point of view the two are one and the same, a group.
However, depending on how that group is defined additional structure may make additional operations possible or easy that are detrimental to your security.
For example: In a finite field of prime order calculating logarithms is hard but division is easy.
More mathematically the two problems are
- calculating $x$ given $g$ and $g^x$
- calculating $x'$ given $g$ and $x' \cdot g$
and they have totally different complexities.
Both addition and multiplication in this case define a group but integer multiplication is not hard to invert.
And, the usual disclaimer, please don't design your own crypto unless for exercises. ;)
Elias
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No difference. The operation of the group does not have to do with the security. The addition or multiplication is just a convention we make to write the operations between the elements of the group. Of course in some groups, say ${\bf Z}_p^{*}$ we use the multiplication symbol because the operation is multiplication $\mod p.$ In elliptic curves, always we use the addition symbol.
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