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 s +x k+s +2 k + cx k+s + c k x k +2 s + δx k , over F22k , where k is an even integer and gcd(2k, s) = 1, 3 ∤ k. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in [6] that the function is APN when there exists c such that the polynomial y s+1+ cy s + c k y+1 = 0 has no solutions in the field F22k . In [6] 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 ∤ 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. APN functions; zeros of polynomials; irreducible polynomials.