On the uniformity of distribution of the elliptic curve ElGamal signature

We show that, under some natural conditions, the pairs (ρ, σ) produced by the elliptic curve ElGamal signature scheme are uniformly distributed. In particular, this implies that values of ρ and σ are not correlated. The result is based on some new estimates of exponential sums. For the ElGamal signature over a finite field, a similar result has been obtained by the second author.