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Computational Explorations on Semifields

Jean-Guillaume Dumas, Stefano Lia, John Sheekey

TL;DR

The paper addresses the problem of quantifying the computational cost of multiplication in finite semifields and small extension fields by jointly analyzing multiplicative complexity via tensor rank and additive complexity via base-field operations. It advances the methodology with lrp representations, folding techniques for extensions, and the construction of short straight-line programs (SSLP) tied to linear codes, enabling new upper and lower bounds and concrete algorithms. Key contributions include establishing tensor ranks for algebras of order $243$, deriving additive-complexity bounds and improved algorithms for extensions of degree $4$ and $5$, and connecting algebraic representations to efficient, constant-time implementations. These results have implications for fast arithmetic in cryptography and coding theory, clarifying when semifields can offer multiplicative advantages over fields while maintaining efficient additive costs.

Abstract

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.

Computational Explorations on Semifields

TL;DR

The paper addresses the problem of quantifying the computational cost of multiplication in finite semifields and small extension fields by jointly analyzing multiplicative complexity via tensor rank and additive complexity via base-field operations. It advances the methodology with lrp representations, folding techniques for extensions, and the construction of short straight-line programs (SSLP) tied to linear codes, enabling new upper and lower bounds and concrete algorithms. Key contributions include establishing tensor ranks for algebras of order , deriving additive-complexity bounds and improved algorithms for extensions of degree and , and connecting algebraic representations to efficient, constant-time implementations. These results have implications for fast arithmetic in cryptography and coding theory, clarifying when semifields can offer multiplicative advantages over fields while maintaining efficient additive costs.

Abstract

A finite semifield is a division algebra over a finite field where multiplication is not necessarily associative. We consider here the complexity of the multiplication in small semifields and finite field extensions. For this operation, the number of required base field multiplications is the tensor rank, or the multiplicative complexity. The other base field operations are additions and scalings by constants, which together we refer to as the additive complexity. When used recursively, the tensor rank determines the exponent while the other operations determine the constant of the associated asymptotic complexity bounds. For small extensions, both measures are of similar importance. In this paper, we establish the tensor rank of some semifields and finite fields of characteristics 2 and 3. We also propose new upper and lower bounds on their additive complexity, and give new associated algorithms improving on the state-of-the-art in terms of overall complexity. We achieve this by considering short straight line programs for encoding linear codes with given parameters.
Paper Structure (28 sections, 17 theorems, 9 equations, 7 tables, 2 algorithms)

This paper contains 28 sections, 17 theorems, 9 equations, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. Fields and Semifields
  5. Isotopy and Isotropy
  6. lrp representation and straight-line programs
  7. Subspace of matrices from tensors and decompositions
  8. Multiplicative complexity
  9. Tensor ranks
  10. Fields and semifields of order $243$
  11. Additive complexity of linear codes
  12. Shortest straight-line program
  13. Calculating $\mathrm{SSLP}_q(r,k,d)$ for small parameters
  14. Folding for extensions
  15. Additive complexity; extensions of degree $4$
  16. ...and 13 more sections

Key Result

Theorem 5

The tensor rank of a field or semifield of order $q^n$ over $\mathbb{F}\xspace_q$ is at least $A_q(n,n)$.

Theorems & Definitions (38)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Theorem 5: brockett
  • Theorem 6
  • proof
  • Theorem 7
  • Theorem 8
  • proof
  • ...and 28 more