Characterization of $p$ -ary Bent Functions in Terms of Strongly Regular Graphs
Jong Yoon Hyun, Yoonjin Lee · IEEE Transactions on Information Theory · 2018
A p-ary function f in n variables is an l-form if f(tu) = tlf (u) for any nonzero t in Zpand u in Zpn. Let n be a positive even integer, p an odd prime, and l an element of {1, 2, . . . , p -1} provided that l ≠ p -1 if p > 3. Let f be a p-ary bent function in n variables of l-form with f (0) = 0 and gcd(l - 1, p - 1) = 1, and let Hl= {tl: t ∈ Zp*}. We denote by Gf,lthe Cayley graph Cay(Zpn, ∪s∈Hlf-1(s)). Our main results are as follows: 1) if there is weakly regular p-ary bent f which is not regular, then l is 2; 2) if l = 2, then f is weakly regular p-ary bent if and only if the Cayley graph G f,l is strongly regular; 3) if l ≠ 2, then f is regular p-ary bent if and only if the Cayley graph Gf,lis strongly regular; 4) Gf,lcan be replaced by Cay(Zpn, f-1(0)\{0}) in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of Gf,l.