Multi-sources Randomness Extraction over Finite Fields and Elliptic Curve

TL;DR

This paper proposes a deterministic randomness extractor for Diffie-Hellman elements over finite fields and elliptic curves, demonstrating its effectiveness in producing indistinguishable uniform bits, with applications in cryptographic pairings.

Contribution

It introduces a new deterministic extractor for Diffie-Hellman elements over finite fields and elliptic curves, replacing hash functions in pairing-based cryptography.

Findings

Least significant bits are indistinguishable from uniform

Extractor works over finite fields and elliptic curves

Potential to replace hash functions in cryptographic pairings

Abstract

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 $G_1*G_2$, 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.