# Decision Diffie-Hellman (DDH)

The Decision Diffie-Hellman (DDH) assumption is a common assumption made to construct a wide variety of primitives. It is the weakest of assumption of the common Diffie-Hellman type assumptions. And being able to break Computational Diffie-Hellman (CDH) and Discrete Logarithm (DLOG) immediately break DDH.

## Formal assumption

Let be a group ensemble. Then, we say that the Decision Diffie-Hellman (DDH) assumption holds for if for any efficient algorithm , there is a negligible function such that,

where is a random generator of the group , are all uniform on .

The problem of distinguishing between these two inputs is sometimes also referred to as the Decision Diffie-Hellman problem.

## Known attacks

- A quantum computer can compute discrete logarithms (and therefore break DDH) in any group [Sho94]
- The Baby-Step-Giant-Step algorithm computes discrete logarithms in any group in time and space for groups of order . (Which is known to be optimal for generic groups [Sho97])

## Relationship to other assumptions

## Implied primitives

- Key Agreement (KA) due to the original work [DH76]

## Other notes

Some works define the DDH problem for a fixed generator rather than a random generator. The differences between these two definitions are discussed in [BMZ19].