Diviseurs de la forme 2D-G sans sections et rang de la multiplication dans les corps finis (Divisors of the form 2D-G without sections and bilinear complexity of multiplication in finite fields)

TL;DR

This paper constructs specific divisors on algebraic curves over finite fields to improve bounds on the bilinear complexity of multiplication, fixing previous flawed proofs and extending theoretical results.

Contribution

It corrects and completes the proof of a key estimate related to the Chudnovsky-Chudnovsky method for finite field multiplication complexity.

Findings

Established a method to construct divisors with desired properties on curves.

Validated the estimate m_q ≤ 2(1+1/(A(q)-1)) for A(q) ≥ 5.

Fixed previous flawed proofs by Shparlinski-Tsfasman-Vladut and Ballet.

Abstract

Let X be an algebraic curve, defined over a perfect field, and G a divisor on X. If X has sufficiently many points, we show how to construct a divisor D on X such that l(2D-G)=0, of essentially any degree such that this is compatible the Riemann-Roch theorem. We also generalize this construction to the case of a finite number of constraints, l(k_i.D-G_i)=0, where |k_i|\leq 2. Such a result was previously claimed by Shparlinski-Tsfasman-Vladut, in relation with the Chudnovsky-Chudnovsky method for estimating the bilinear complexity of the multiplication in finite fields based on interpolation on curves; unfortunately, as noted by Cascudo et al., their proof was flawed. So our work fixes the proof of Shparlinski-Tsfasman-Vladut and shows that their estimate m_q\leq 2(1+1/(A(q)-1)) holds, at least when A(q)\geq 5. We also fix a statement of Ballet that suffers from the same problem, and then we point out a few other possible applications.