Multi-sources Randomness Extraction over Finite Fields and Elliptic Curve

This work is based on the proposal of a deterministic randomness extractor of a random Diffie-Hellman element defined over two prime order multiplicative subgroups of a finite fields ${\mathbb{F}}_{p^n}$, $G_1$ and $G_2$. We show that the least significant bits of a random element in $G_1\ast G_2$, are indistinguishable from a uniform bit-string of the same length. One of the main application of this extractor is to replace the use of hash functions in pairing by the use of a good deterministic randomness extractor.