## Abstract

Let be an elliptic curve over a finite field of elements, with gcd(q, 6)= 1, given by an affine Weierstraß equation. We use x(P) to denote the x-component of a point P = (x(P), y(P)) ε E. We estimate character sums of the form ^{N}Σ _{n=1}X(x(nP)x(nQ)) and ^{N}Σ _{n1,...,nk=1}Π ( ^{k}Σj=1cjx(( ^{j}Π _{i=1}ni)R)) on average over all F _{q}rational points P,Q, and R on E, where X is a quadratic character, φ is a nontrivial additive character in Fq, and (c _{1},...,ck) ε F ^{k} _{q}is a nonzero vector. These bounds confirm several recent conjectures of Jao, Jetchev, and Venkatesan, related to extracting random bits from various sequences of points on the elliptic curves.

Original language | English |
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Article number | 6043877 |

Pages (from-to) | 1242-1247 |

Number of pages | 6 |

Journal | IEEE Transactions on Information Theory |

Volume | 58 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2012 |