Evaluating Binomial Character Sums Modulo Powers of Two
  We show that for any mod $2^m$ characters, $\chi_1, \chi_2,$ the complete exponential sum, $$ \sum_{x=1}^{2^m}\chi_1(x) \chi_2(Ax^k+B), $$ has a simple explicit evaluation. 
Vincent Pigno, Chris Pinner and Joe Sheppard
math.NT