  Budaghyan and Carlet constructed a family of almost perfect nonlinear (APN)
hexanomials over a field with r^2 elements, and with terms of degrees r+1, s+1,
rs+1, rs+r, rs+s, and r+s, where r = 2^m and s = 2^n with GCD(m,n)=1. The
construction requires a technical condition, which was verified empirically in
a finite number of examples. Bracken, Tan, and Tan (arXiv:1110.3177 [cs.it])
proved the condition holds when m = 2 or 4 (mod 6). In this article, we prove
that the construction of Budaghyan and Carlet produces APN polynomials for all
m and n.
  In the case where GCD(m,n) = k >= 1, Budaghyan and Carlet showed that the
nonzero derivatives of the hexanomials are 2^k-to-one maps from F_{r^2} to
F_{r^2}, provided the same technical condition holds. We prove their
construction produces hexanomials with this differential property for all m and
n.
