Extracting bits from coordinates of a point of an elliptic curve

In the classic Diffie-Hellman protocol based on a generic group G, Alice and Bob agree on a common secret KAB (master secret) which is indistinguishable from another element of G but not from a random bits-string of the same length. In this paper, we present a new deterministic method to extract bits from KAB when G is an elliptic curve defined over a quadratic extension of a finite field. In the last section, we show that it is also possible to extract a few bits when G is an elliptic curve defined over a prime field.