Optimization of the scalar complexity of Chudnovsky$^2$ multiplication algorithms in finite fields

On The Effective Construction of Asymmetric Chudnovsky Multiplication Algorithms in Finite Fields Without Derivated Evaluation

The Chudnovsky and Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear whith respect to the degree of the extension. Recently, Randriambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. We describe the effective algorithm of this asymmetric method, without derivated evaluation. Finally, we give examples with the finite field $\F_{16^{13}}$ using only rational places, $\F_{4^{13}}$ using also places of degree two and $\F_…

On the construction of elliptic Chudnovsky-type algorithms for multiplication in large extensions of finite fields

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, by using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in $O(n(2q)^{\log_q^*(n)})$ where $n$ corresponds to the extension degree, and $\log…

Multiplication in finite fields with Chudnovsky-type algorithms on the projective line

We propose a Recursive Polynomial Generic Construction (RPGC) of multiplication algorithms in any finite field $\mathbb{F}_{q^n}$ based on the method of D.V. and G.V. Chudnovsky specialized on the projective line. They are usual polynomial interpolation algorithms in small extensions and the Karatsuba algorithm is seen as a particular case of this construction. Using an explicit family of such algorithms, we show that their bilinear complexity is quasi-linear with respect to the extension degree n, and we give a uniform bound for this complexity. We also prove that the construction of these al…

On Chudnovsky-Based Arithmetic Algorithms in Finite Fields

Thanks to a new construction of the so-called Chudnovsky-Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation and they allow computations to be parallelized while maintaining a low number of bilinear multiplications. We give an example with the finite field ${\mathbb F}_{16^{13}}$.

有限域中乘法链的 Chudnovsky-type 算法与 k-multiplication 的张量秩

Chaining Multiplications in Finite Fields with Chudnovsky-type Algorithms and Tensor Rank of the k-multiplication

本文设计了一类 Chudnovsky-type 算法，通过代数曲线插值实现有限域中 k 个元素的乘法，证明了 k-multiplication 的张量秩关于 n 是线性的，并给出了其一致上界。

Shimura modular curves and asymptotic symmetric tensor rank of multiplication in any finite field

We obtain new asymptotical bounds for the symmetric tensor rank of multiplication in any finite extension of any finite field $\F_q$. In this aim, we use the symmetric Chudnovsky-type generalized algorithm applied on a family of Shimura modular curves defined over $\F_{q^2}$ attaining the Drinfeld-Vladut bound and on the descent of this family over the definition field $\F_q$.