On the Sum-Product Problem on Elliptic Curves
Omran Ahmadi, Igor Shparlinski

TL;DR
This paper investigates sum-product type problems on elliptic curves over finite fields, demonstrating that certain combined sets derived from elliptic curve points are necessarily large, extending sum-product phenomena to elliptic curve contexts.
Contribution
The paper establishes lower bounds on the size of sum and product sets of elliptic curve points, extending sum-product results to the setting of elliptic curves over finite fields.
Findings
Either the sum set or the product set is large for subsets of the elliptic curve points.
The results connect sum-product phenomena with elliptic curve arithmetic.
Provides new insights into algebraic structures over finite fields.
Abstract
Let be an ordinary elliptic curve over a finite field of elements and denote the -coordinate of a point on . Given an -rational point of order , we show that for any subsets of the unit group of the residue ring modulo , at least one of the sets is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures.
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Taxonomy
TopicsCoding theory and cryptography · Cryptography and Residue Arithmetic · Finite Group Theory Research
