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Title: On a class of quadratic polynomials with no zeros and its application to APN functions
Authors: Bracken, C.
Tan, C.H. 
Tan, Y. 
Keywords: APN functions
Irreducible polynomials
Zeros of polynomials
Issue Date: 2014
Source: Bracken, C., Tan, C.H., Tan, Y. (2014). On a class of quadratic polynomials with no zeros and its application to APN functions. Finite Fields and their Applications 25 : 26-36. ScholarBank@NUS Repository.
Abstract: 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 3â̂, 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 ;bsupesupbsup. © 2013 Elsevier Inc. All rights reserved.
Source Title: Finite Fields and their Applications
ISSN: 10715797
DOI: 10.1016/j.ffa.2013.08.006
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