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Gaps between prime numbers and tensor rank of multiplication in finite fields

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Abstract

We present effective upper bounds on the symmetric bilinear complexity of multiplication in extensions of a base finite field \(\mathbb {F}_{p^2}\) of prime square order, obtained by combining estimates on gaps between prime numbers together with an optimal construction of auxiliary divisors for multiplication algorithms by evaluation-interpolation on curves. Most of this material dates back to a 2011 unpublished work of the author, but it still provides the best results on this topic at the present time. Then a few updates are given in order to take recent developments into account, including comparison with a similar work of Ballet and Zykin, generalization to classical bilinear complexity over \(\mathbb {F}_p\), and to short multiplication of polynomials, as well as a discussion of open questions on gaps between prime numbers or more generally values of certain arithmetic functions.

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Notes

  1. Compared with (3), the (classical) bilinear complexity \(\mu _F(\mathcal {A})\) of \(\mathcal {A}\) over F is the (classical) tensor rank of \(T_\mathcal {A}\), i.e. the minimal length n of a decomposition \(T_\mathcal {A}=\sum _{1\le i\le n}\alpha _i\otimes \beta _i\otimes w_i\) of \(T_\mathcal {A}\) as a sum of n elementary tensors in \(\mathcal {A}^\vee \otimes \mathcal {A}^\vee \otimes \mathcal {A}\), where now \(\alpha _i,\beta _i\in \mathcal {A}^\vee \) are allowed to be different; equivalent definitions similiar to (1) or (2) can be given likewise.

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Acknowledgements

H. Randriam was supported by ANR-14-CE25-0015 Project Gardio and ANR-15-CE39-0013 Project Manta.

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    Hugues Randriam

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Randriam, H. Gaps between prime numbers and tensor rank of multiplication in finite fields. Des. Codes Cryptogr. 87, 627–645 (2019). https://doi.org/10.1007/s10623-018-0584-0

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