TY - JOUR
T1 - Explicit criterions for p-ary functions being non-bent
AU - Hyun, Jong Yoon
AU - Lee, Yoonjin
N1 - Publisher Copyright:
© 2015 Elsevier Inc.
PY - 2016/1/15
Y1 - 2016/1/15
N2 - There has been only limited information on the existence of p-ary bent functions. Recently there has been a result by the authors on finding necessary conditions for the existence of regular p-ary bent functions (from Zpn to Zp), where p is a prime. The general case of p-ary bent functions is, however, an open question for finding necessary conditions for their existence. In this paper we complete this open case. We state our main result in more detail. We find an explicit family of non-bent functions. We also show that there is no p-ary bent function f in n variables with w(Mf)>n2 if n is even (w(Mf)>n+32 if n is odd, respectively), and for a given nonnegative integer k there is no p-ary bent function f in n variables with w(Mf)=n2-k (w(Mf)=n+32-k, respectively) for an even n≥Np, k (for an odd n≥Np, k, respectively), where Np, k is some positive integer which is explicitly determined and w(Mf) is some explicit value related to the power of each monomial of f. We point out that if f is not a p-ary bent function in n variables and g is any p-ary function in n variables such that L(Gf)=Gg for some CCZ-transformation L of f, then g cannot be bent either. This shows that our result produces a larger family of non-bent functions.
AB - There has been only limited information on the existence of p-ary bent functions. Recently there has been a result by the authors on finding necessary conditions for the existence of regular p-ary bent functions (from Zpn to Zp), where p is a prime. The general case of p-ary bent functions is, however, an open question for finding necessary conditions for their existence. In this paper we complete this open case. We state our main result in more detail. We find an explicit family of non-bent functions. We also show that there is no p-ary bent function f in n variables with w(Mf)>n2 if n is even (w(Mf)>n+32 if n is odd, respectively), and for a given nonnegative integer k there is no p-ary bent function f in n variables with w(Mf)=n2-k (w(Mf)=n+32-k, respectively) for an even n≥Np, k (for an odd n≥Np, k, respectively), where Np, k is some positive integer which is explicitly determined and w(Mf) is some explicit value related to the power of each monomial of f. We point out that if f is not a p-ary bent function in n variables and g is any p-ary function in n variables such that L(Gf)=Gg for some CCZ-transformation L of f, then g cannot be bent either. This shows that our result produces a larger family of non-bent functions.
KW - Boolean function
KW - P-ary bent function
KW - P-ary function
KW - Weakly regular p-ary bent function
UR - http://www.scopus.com/inward/record.url?scp=84942198544&partnerID=8YFLogxK
U2 - 10.1016/j.jmaa.2015.08.045
DO - 10.1016/j.jmaa.2015.08.045
M3 - Article
AN - SCOPUS:84942198544
SN - 0022-247X
VL - 433
SP - 1177
EP - 1189
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
M1 - 19742
ER -