2010/05/26 by Reza Rezaeian Farashahi, Reza R. Farashahi, Farashahi, Reza R. +2
Computer Science · Mathematics · #11G05 #11T23 #14G50 #94A60 #Analytic Number Theory Research #Coding theory and cryptography #Cryptography and Residue Arithmetic #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.CR #math.NT #msc:11G05 #msc:11T23 #msc:14G50 #msc:94A60
paper · pdf · doi:10.48550/arxiv.1005.4771
arxiv created 2010/05/26 · openalex publication_date 2010/05/26 · arxiv updated 2010/05/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \E be an elliptic curve over a finite field \Fq of q elements, with gcd(q,6)=1, given by an affine Weierstraß equation. We also use x(P) to denote the x-component of a point P = (x(P),y(P))∈ \E. We estimate character sums of the form ∑n=1N χ\(x(nP)x(nQ)\) and ∑n1, …, nk=1N ψ\(∑j=1k cj x\(\(∏i =1j ni\) R\)\) on average over all \Fq rational points P, Q and R on \E, where χ is a quadratic character, ψ is a nontrivial additive character in \Fq and (c1, …, ck)∈ \Fqk is a non-zero vector. These bounds confirm several recent conjectures of D. Jao, D. Jetchev and R. Venkatesan, related to extracting random bits from various sequences of points on elliptic curves.