On a Class of Quadratic Polynomials with no Zeros and its Application to   APN Functions
  We show that the there exists an infinite family of APN functions of the form $F(x)=x^{2^{s}+1} + x^{2^{k+s}+2^k} + cx^{2^{k+s}+1} + c^{2^k}x^{2^k + 2^s} + \delta x^{2^{k}+1}$, over $\gf_{2^{2k}}$, where $k$ is an even integer and $\gcd(2k,s)=1, 3\nmid k$. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in \cite{carlet-1} that the function is APN when there exists $c$ such that the polynomial $y^{2^s+1}+cy^{2^s}+c^{2^k}y+1=0$ has no solutions in the field $\gf_{2^{2k}}$. In \cite{carlet-1} they demonstrate by computer that such elements $c$ can be found over many fields, particularly when the degree of the field is not divisible by 3. We show that such $c$ exists when $k$ is even and $3\nmid k$ (and demonstrate why the $k$ odd case only re-describes an existing family of APN functions). The form of these coefficients is given so that we may write the infinite family of APN functions. 
Carl Bracken, Chik How Tan and Tan Yin
cs.IT math.IT