Aichi Kudo

Improving the berlekamp algorithm for binomials x n -a

In this paper, we describe an improvement of the Berlekamp algorithm, a method for factoring univariate polynomials over finite fields, for binomials xn−a over finite fields

Tate and Ate pairings for y2 =x5−αx in characteristic five

\mathbb{F}_{q}

A remark on the computation of cube roots in finite fields.

. More precisely, we give a deterministic algorithm for solving the equation

ON P-ADIC DEDEKIND SUMS (II)

h(x)^{q} \equiv h(x) \ ({\rm mod}\ x^{n} -a)

ON A GENERALIZATION OF A THEOREM OF KUMMER

directly without applying the sweeping-out method to the corresponding coefficient matrix. We show that the factorization of binomials using the proposed method is performed in

Root Computation in Finite Fields

O \, \tilde{}\, (n \log q)

ON IWASAWA'S EXPLICIT FORMULA FOR THE NORM RESIDUE SYMBOL

operations in

A remark on the computation of cube roots in

\mathbb{F}_{q}

ON THE BREAK INTERVAL SEQUENCES OF EQUITABLE ROUND-ROBIN TOURNAMENTS

if we apply a probabilistic version of the Berlekamp algorithm after the first step in which we propose an improvement. Our method is asymptotically faster than known methods in certain areas of q, n and as fast as them in other areas.

Tate pairing for y 2 =x 5 -αx in Characteristic Five.

In this paper, we consider the Tate and Ate pairings for the genus-2 supersingular hyperelliptic curvesy2 =x5 − αx (α = ±2) defined over finite fields of characteristic five. More precisely, we construct a distortion map explicitly, and show that the map indeed gives an input for which the value of the Tate pairing is not trivial. We next describe a computation of the Tate pairing by using the proposed distortion map. We also see that this type of curve is equipped with a simple quintuple operation on the Jacobian group, which leads to an improvement for computing the Tate pairing. We further show the Ate pairing, a variant of the Tate pairing for elliptic curves, can be applied to this curve. The Ate pairing yields an algorithm which is about 50% more efficient than the Tate pairing in this case.