Trace- and norm-compatible extensions of finite fields

Scheerhorn, Alfred

doi:10.1007/bf01268660

Cited by 10 publications

(4 citation statements)

References 4 publications

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“…To finish the proof of the Theorem 1 we show how the problem (17) is in fact solved by the Lemma: If for some 1 < i < n -d the element flff~l is zero then this means by This definition is an adaption to the present situation of concepts that were introduced by Scheerhorn in [10] and which prove to be useful if one wants to do calculations in any finite interval of the lattice of subfields of F (q). …”

Section: Proofmentioning

confidence: 94%

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ExplicitN-polynomials of 2-power degree over finite fields, I

Meyn

1995

Des Codes Crypt

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Abstract.For an odd prime power q the infinite field GF(q 2~ ) = U,_>0 GF(q2~ ) is explicitly presented by a sequence (f,,),,_>l of N-polynomials. This means that, for a suitably chosen initial polynomial fl, the defining polynomials f, ~ GF(q ) [x ] of degrees 2 n are constructed by iteration of the transformation of variable x ~+ x + 1/x and have linearly independent roots over GF(q). In addition, the sequences are trace-compatible in the sense that the relative traces map the corresponding roots onto each other. In this first paper the case q ---1 (mod 4) is considered and the case q ---3 (rood 4) will be dealt with in a second paper. This specific construction solves a problem raised by A. Scheerhorn in [11].

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Section: Proofmentioning

confidence: 94%

“…By the Example on p. 207 in [10] Scheerhorn could show that iteration of the Q-transformation leads to sequences of N-polynomials over GF (2). For a generalization to fields of characteristic 2 see his further work [12].…”

Section: If Fl (1) F L (-1) Is a Nonsquare In Gf(q ) Then All Membermentioning

confidence: 97%

ExplicitN-polynomials of 2-power degree over finite fields, I

Meyn

1995

Des Codes Crypt

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“…This is the norm compatible condition, which is also defined in [36]. To ensure the stability of the subgridding method, (25) should be satisfied.…”

Section: Stable Coupling For Multiple Block Meshesmentioning

confidence: 99%

A SBP-SAT FDTD Subgridding Method Using Staggered Yee's Grids Without Modifying Field Components

Wang¹,

Yu²,

Wang³

et al. 2022

Preprint

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A summation-by-parts simultaneous approximation term (SBP-SAT) finite-difference time-domain (FDTD) subgridding method is proposed to model geometrically fine structures in this paper. Compared with our previous work, the proposed SBP-SAT FDTD method uses the staggered Yee's grid without adding or modifying any field components through field extrapolation on the boundaries to make the discrete operators satisfy the SBP property. The accuracy of extrapolation keeps consistency with that of the second-order finite-difference scheme near the boundaries. In addition, the SATs are used to weakly enforce the tangential boundary conditions between multiple mesh blocks with different mesh sizes. With carefully designed interpolation matrices and selected free parameters of the SATs, no dissipation occurs in the whole computational domain. Therefore, its long-time stability is theoretically guaranteed. Three numerical examples are carried out to validate its effectiveness. Results show that the proposed SBP-SAT FDTD subgridding method is stable, accurate, efficient, and easy to implement based on existing FDTD codes with only a few modifications.

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“…Let us finally mention materials not discussed here: representation based on Zech logarithms or Conway polynomials [44], deterministic algorithms [1,47], as well as the construction of irreducible polynomials with extra properties (sparseness, primitivity, . .…”

Section: Introductionmentioning

confidence: 99%

Algorithms for Finite Field Arithmetic

Schost

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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We review several algorithms to construct finite fields and perform operations such as field embedding. Following previous work by notably Shoup, de Smit and Lenstra or Couveignes and Lercier, as well as results obtained with De Feo and Doliskani, we distinguish between algorithms that build "towers" of finite fields, with degrees of the form , 2 , 3 , . . . and algorithms for composita. We show in particular how techniques that originate from algorithms for computing with triangular sets can be useful in such a context.

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Trace- and norm-compatible extensions of finite fields

Cited by 10 publications

References 4 publications

ExplicitN-polynomials of 2-power degree over finite fields, I

ExplicitN-polynomials of 2-power degree over finite fields, I

A SBP-SAT FDTD Subgridding Method Using Staggered Yee's Grids Without Modifying Field Components

Algorithms for Finite Field Arithmetic

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