# Full Domain Hash

566pages on
this wiki

In cryptography, the Full Domain Hash (FDH) is an RSA-based signature scheme that follows the hash-and-sign paradigm. It is provably secure (i.e, is existentially unforgeable under adaptive chosen-message attacks) in the random oracle model. FDH involves hashing a message using a function whose image size equals the size of the RSA modulus, and then raising the result to the secret RSA exponent.

## Exact security of full domain hashEdit

In the random oracle model, if RSA is $(t',\epsilon')$-secure, then the full domain hash RSA signature scheme is $(t,\epsilon)$-secure where, $t=t'-(q_{hash}+q_{sig}+1) \cdot \mathcal{O}(k^3)$ and $\epsilon = \left(1+\frac{1}{q_{sig}}\right)^{q_{sig}+1} \cdot q_{sig} \cdot \epsilon'$.

For large $q_{sig}$ this boils down to $\epsilon \sim exp(1)\cdot q_{sig} \cdot \epsilon'$.

This means that if there exists an algorithm that can forge a new FDH signature that runs in time t, computes at most $q_{hash}$ hashes, asks for at most $q_{sig}$ signatures and succeeds with probability $\epsilon$, then there must also exist an algorithm that breaks RSA with probability $\epsilon'$ in time $t'$.

## ReferencesEdit

• Jean-Sébastien Coron: On the Exact Security of Full Domain Hash. CRYPTO 2000: pp229–235 (PDF)