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 F-22k of the form F(x) = x(x(2i) + x(2k) + cx2(k+1)) x(2)i (c(2k) x(2R) + delta x(2k-1)) x(2k+1+2k). They showed that this infinite family exists provided the existence of the quadratic polynomial G(y) = y(2i+1) + cy(2i) + C-2k y + 1, which has no zeros such that y(2k+1) = 1, or in particular has no zeros in F-22k. 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 (sic) 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 P-a(x) = x(2i+1) + x + a is an element of F-2n [x] and P-a has applications in many other contexts. We determine all coefficients a such that P-a has no zeros on F-2n when gcd(i, n) = 1 and n is even. (C) 2013 Elsevier Inc. All rights reserved.