Our main result gives a way to instantiate therandom oracle with a concrete hash function in

\"full domain hash\" applications.

The term full domain hash was first proposed by Bellare and

Rogaway and referred to a signature scheme from any

trapdoor permutation that was part of their seminal work introducing

the random oracle heuristic. Over time the term full domain hash has

informally encompassed a broader range of notable cryptographic

schemes including the Boneh-Franklin IBE scheme and

Boneh-Lynn-Shacham (BLS) signatures.

All of the above described schemes required a hash function that had

to be modeled as a random oracle to prove security. Our work utilizes

recent advances in indistinguishability obfuscation to construct

specific hash functions for use in these schemes. We then prove

security of the original cryptosystems when instantiated with

our specific hash function.

Of particular interest, our work evades the impossibility result of

Dodis, Oliveira, and Pietrzak, who showed that there can

be no black-box construction of hash functions that allow Full-Domain

Hash Signatures to be based on trapdoor permutations. This indicates

that our techniques applying indistinguishability obfuscation may be

useful in the future for circumventing other such black-box

impossibility proofs.