On a problem of H. Cohn for character sums

Cohn's problem on character sums ( see [4], p. 202 ) asks whether a multipl icative character on a finite field can be characterized by a kind of two l evel autocorrelation property. Let f be a map from a finite field F to the complex plane such that f(0) = 0, f(1) = 1, and \f(alpha)\ = 1 for all alph a not equal 0. In this paper we show that if for all a, b is an element of F*, we have [GRAPHICS] then f is a multiplicative character of F. We also prove that if F is a pri me field and f is a real valued function on F with f(0) = 0, f(1) = 1, and \f(alpha)\ = 1 for all alpha not equal 0, then Sigma(alpha is an element of F)f(alpha) f(alpha + a) = -1 for all a not equal 0 if and only if f is the Legendre symbol. These results partially answer Cohn's problem. (C) 2000 A cademic Press.