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On the Connection between Kloosterman Sums
and Elliptic Curves
Petr Lisonek
Department of Mathematics, Simon Fraser University
Burnaby, BC, Canada V5A 1S6
plisonek@sfu.ca
Abstract.
We explore the known connection of Kloosterman sums on
fields of characteristic 2 and 3 with the number of points on certain
elliptic curves over these fields. We use this connection to prove results on
the divisibility of Kloosterman sums, and to compute numerical examples
of zeros of Kloosterman sums on binary and ternary fields of large orders.
We also show that this connection easily yields some formulas due to
Carlitz that were recently used to prove certain non-existence results on
Kloosterman zeros in subfields.
Keywords:
Kloosterman sum, elliptic curve, finite field.
1
Introduction
Kloosterman sums have recently enjoyed much attention. Some of this interest
is due to their applications in cryptography and coding theory; see for example
[4] and [14].
Inthepresentpaperweexploittheconnection between Kloosterman sums
on
F
p
m
and the number of
F
p
m
-rational points on certain elliptic curves, where
p
. In the binary case this association was given in the well known paper
by Lachaud and Wolfmann [9]; in the ternary case this is a recent result due to
Moisio [13].
We use this connection to reprove the known characterization of binary Kloost-
erman sums divisible by 8, and we give a characterization of binary Kloosterman
sums divisible by 16. We also quote our recent result on ternary Kloosterman
sums modulo 4, which was proved elsewhere using similar methods.
There is considerable interest in non-zero elements of
F
p
m
at which the Kloost-
erman sum attains the value 0. The second objective of the paper is to provide a
practical computational method for finding numerical examples of such elements.
We also show that some older results due to Carlitz, which recently appered in a
proof that such elements can not belong to certain subfields of
∈{
2
,
3
}
F
p
m
, follow very
easily from the connection with elliptic curves.
