Abstract:
In this paper, we consider the maximum absolute value Δ f in the autocorrelation spectrum (not considering the zero point) of a function f. In an even number of variable...View moreMetadata
Abstract:
In this paper, we consider the maximum absolute value Δ
f
in the autocorrelation spectrum (not considering the zero point) of a function f. In an even number of variables n, bent functions possess the highest nonlinearity with Δ
f
= 0. The long standing open question (for two decades) in this area is to obtain a theoretical construction of balanced functions with Δ
f
<; 2
n/2
. So far, there are only a few examples of such functions for n = 10, 14, but no general construction technique is known. In this paper, we mathematically construct an infinite class of balanced Boolean functions on n variables having absolute indicator strictly lesser than δ
n
= 2
n/2
- 2(
(n+6)/4)
, nonlinearity strictly greater than ρ
n
= 2
n-1
-2
n/2
+2
n/2-3
-5·2
((n-2)/4)
and algebraic degree n - 1, where n ≡ 2 (mod 4) and n ≥ 46. While the bound n ≥ 46 is required for proving the generic result, our construction starts from n = 18, and we could obtain balanced functions with Δ
f
<; 2
n/2
and nonlinearity > 2
n-1
- 2
n/2
for n = 18, 22, and 26.
Published in: IEEE Transactions on Information Theory ( Volume: 64, Issue: 1, January 2018)