DDH, CDH and discrete logarithm assumptions

Caio Nogueira

12/14/23, 1:46 PM

When we consider a group generation algorithm $\mathcal{G}$ (taken from Katz and Lindell's Introduction to Modern Cryptography), that takes as input a security parameter $1^n$ and outputs $(\mathbb{Z}_n, n, g)$, where $\mathbb{Z}_n$ is an additive group, can we say that the DDH, CDH, and discrete logarithm assumptions hold?

On the other hand, if they don't hold for any $g$, which conditions do we need to assert on the generator?

0 + 3

discrete-logarithm

diffie-hellman

poncho

12/14/23, 1:56 PM

None of the assumptions DDH, CDH or DLog are true in $\mathbb{Z}_n^+$; if you need those, you need to use a different group

Caio Nogueira

12/14/23, 2:58 PM

@poncho thank you for helping. I still don't fully understand how we can compute the discrete logarithm for $\mathbb{Z}_n$ in polynomial time. According to the book, it relies on the definition of a multiplicative group.

poncho

12/14/23, 3:30 PM

What my comments was based on was your text "... where $\mathbb{Z}_n$ is an additive group". It is quite unusual to represent $\mathbb{Z}_n^*$ as an additive group (that is, you use the group operation syntax $A+B$, which is commonly implemented as $A \times B \bmod n$); the other alternative is the group $\mathbb{Z}_n^+$ (where the above problems are easily solvable in polynomial time)

Elon Musk

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EN: DDH, CDH and discrete logarithm assumptions

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