Evaluating Binomial Character Sums Modulo Powers of Two

Vincent Pigno, Chris Pinner, Joe Sheppard

We show that for any mod $2^m$ characters, ${\chi }_1,{\chi }_2,$ the complete exponential sum, $$\sum _{x=1}^{2^m}{\chi }_1(x){\chi }_2(A{x}^k+B),$$ has a simple explicit evaluation.