Decisional Diffie–Hellman assumption

The decisional Diffie–Hellman (DDH) assumption is a computational hardness assumption about a certain problem involving discrete logarithms in cyclic groups. It is used as the basis to prove the security of many cryptographic protocols, most notably the ElGamal and Cramer–Shoup cryptosystems.

Consider a (multiplicative) cyclic group ${\displaystyle G}$ of order ${\displaystyle q}$, and with generator ${\displaystyle g}$. The DDH assumption states that, given ${\displaystyle g^{a}}$ and ${\displaystyle g^{b}}$ for uniformly and independently chosen ${\displaystyle a,b\in \mathbb {Z} _{q}}$, the value ${\displaystyle g^{ab}}$ "looks like" a random element in ${\displaystyle G}$.

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