# Full Domain Hash

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 hash

In the random oracle model, if RSA is ${\displaystyle (t',\epsilon ')}$-secure, then the full domain hash RSA signature scheme is ${\displaystyle (t,\epsilon )}$-secure where,

{\displaystyle {\begin{aligned}t&=t'-(q_{\text{hash}}+q_{\text{sig}}+1)\cdot {\mathcal {O}}\left(k^{3}\right)\\\epsilon &=\left(1+{\frac {1}{q_{\text{sig}}}}\right)^{q_{\text{sig}}+1}\cdot q_{\text{sig}}\cdot \epsilon '\end{aligned}}}.

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

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

## References

• Jean-Sébastien Coron(AF): On the Exact Security of Full Domain Hash. CRYPTO 2000: pp. 229–235 (PDF)
• Mihir Bellare, Phillip Rogaway: The Exact Security of Digital Signatures - How to Sign with RSA and Rabin. EUROCRYPT 1996: pp. 399–416 (PDF)