Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

Abstract

Let F q (q = p r ) be a finite field. In this paper the number of irreducible polynomials of degree m in F q [x] with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if m is small compared to q. As a corollary, sharp bounds are obtained for the number of elements in F q 3 with prescribed trace and norm over F q improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over F 2 r divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case r odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field F 3 r , first proved by Katz and Livne, is given.

On Kloosterman sums over finite fields of characteristic 3

“…In this section we state the known results about Kloosterman sums K(a) [20,23] and elliptic curves E(a) [9,22] over finite fields F of characteristic 3. Our interest is the divisibility of such sums by the maximal possible number of type 3 k (i.e.…”

“…Hence more simple methods of computations of Kloosterman sums are quite interesting. The values of characteristic p ∈ {2, 3} are especially interesting in connection with the number of q-rational points of some elliptic curves [17,18,20,23]. Divisibility of binary Kloosterman sums by numbers 8, 16, .…”

Slices of the unitary spread

“…Nevertheless, regarding point N , in [13] it is proven that the stabilizer in G of the corresponding slice S is isomorphic to Z q+1 × Z q+1 , indeed straightforward calculation show that N ⊥ ∩ Ω is isomorphic to an intersection set E I . These slices are partitioned into  with a ∈ F q 2 such that the polynomial x 3 + T r(a)x + 1 is irreducible over F q ; we observe that it is always possible to chose an element in F q 2 with this property, in fact this is equivalent to the existence of an element u ∈ F q 3 \ F q whose trace and norm over F q are 0 and 1, respectively and, indeed, such an element exists for any prime power q (for instance, see [17]). The hyperplane (N ) ⊥ has equation T r(az) + T r(y) + b = 0.…”

Codes Associated with and Power Moments of Kloosterman Sums

“…I a/ D 1. From (20), in [11] an explicit expression of the Kloosterman sum for GL.t; q/ was derived. Theorem 2 ( [11]).…”