Constructions of Almost Resilient Functions

Abstract

The relation between almost resilient function and its component functions is investigated in this paper. We prove that if each nonzero linear combination of f ₁,f ₂,⋯,f _m is an ε-almost(n,1,k)-resilient function, then F=(f ₁,f ₂,⋯,f _m ) is a $\frac{2^{m}-1}{2^{m}-1}\epsilon$-almost(n,m,k)-resilient function. In the case ε equals 0, the theorem gives another proof of Linear Combination Lemma for resilient functions. As applications of this theorem, we introduce a method to construct a balanced $\frac{9}{2}\epsilon$-almost (3n,2,2k+1)-resilient function from a balanced ε-almost (n,1,k)-resilient function and present a method of improving the degree of the constructed functions with a small trade-off in the nonlinearity and resiliency. At the end of this paper, the relation between balanced almost CI function and its component functions are also concluded.