Vol. 40, No. 2, 1972

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Exponential sums over GF(2n)

Kenneth S. Williams

Vol. 40 (1972), No. 2, 511–519
Abstract

Let F = GF(q) denote the finite field with q = 2n elements. For f(X) F[X] we let

       ∑
S (f ) =   e(f(x)).
x∈F

A deep result of Carlitz and Uchiyama states that if f(X)g(X)2+g(X)+b,g(X) F[X],b F, then

|S(f)| ≦ (degf − 1)q1∕2

This estimate is proved in an elementary way when deg f = 3,4,5 or 6. In certain cases the estimate is improved.

Mathematical Subject Classification
Primary: 12C25
Milestones
Received: 6 August 1970
Published: 1 February 1972
Authors
Kenneth S. Williams