On a class of quadratic polynomials with no zeros and its application to APN functions

In [6], Lilya Budaghyan and Claude Carlet introduced a family of APN functions on F22k of the form F(x)=x(x2i+x2k+cx2k+i)+x2i(c2kx2k+δx2k+i)+x2k+i+2k. They showed that this infinite family exists provided the existence of the quadratic polynomial G(y)=y2i+1+cy2i+c2ky+1, which has no zeros such that y2k+1=1, or in particular has no zeros in F22k. However, up to now, no construction of such polynomials is known. In this paper, we show that, when k is an odd integer, the APN function F is CCZ-equivalent to the one in [2, Theorem 1]; and when k is even with 3∤k, we explicitly construct the polynomial G, and hence demonstrate the existence of F. More generally, it is well known that G relates to the polynomial Pa(x)=x2i+1+x+a∈F2n[x] and Pa has applications in many other contexts. We determine all coefficients a such that Pa has no zeros on F2n when gcd(i,n)=1 and n is even.