Congruence Properties Of Binary Kloosterman Sums

Let Fpmdenote the finite field with pmelements, where p is a prime and m is a positiveinteger. For a∈Fpm, the p-ary Kloosterman sum is defined bywhere χ is the canonical additive character of Fpm.In this paper, we characterize binary Kloosterman sums using the Degree of self-inverseof a particular set from the point of number theory, find a new proof of binary Kloostermansums module8, and get some congruence properties module an odd prime of binary Klooster-man sums in a few spacial cases. Correspondingly, we obtain some results about the Degree ofself-inverse of a particular set using some known results of binary Kloosterman sums. Finally,we apply the the results to elliptic curves, irreducible polynomials over finite field and weightdistribution of some special cyclic codes.