Elliptic periods for finite fields

Abstract

We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind

Sub-quadratic decoding of Gabidulin codes

“…The authors would like to thank Johan S. R. Nielsen for the valuable discussions and Luca De Feo for pointing us at [18].…”

Elliptic periods and primality proving

“…To this end, we extend the methods introduced by Couveignes and Lercier [9] in two different directions. Firstly, we provide efficient explicit expressions for the constants that appear in the multiplication tensor of the ring of elliptic periods.…”

“…One may look for an auxiliary extension R ⊃ R that contains such a primitive root, but this may result in many complications and a great loss of efficiency. Another approach, already experimented in the context of normal bases [9] for finite fields extensions, consists in replacing the multiplicative group G m by some well chosen elliptic curve E over R. We then look for a section T ∈ E(R) of exact order d. Because elliptic curves are many, we increase our chances to find such a section. We call the resulting algebra S a ring of elliptic periods because of the strong analogy with classical Gauss periods.…”

On modular inverses of cyclotomic polynomials and the magnitude of their coefficients

“…In [12] we present a practical implementation of this map, whose efficiency relies on the use of a certain class of normal bases (see [10]) in the representation of field extensions.…”

“…The values of their coefficients and the bounds of Theorem 1 proved in Section 3 ensure the low cost of this computation. We make use of a certain class of normal bases [10], which allows efficient arithmetic in F q n . See [12] for more details.…”