Resiliency and Nonlinearity Profiles of Some Cryptographic Functions

Abstract

Boolean functions are important in terms of their cryptographic and combinatorial properties for different kinds of cryptosystems. The nonlinearity and resiliency of cryptographic functions are crucial criteria with respect to protection of ciphers from affine approximation and correlation attacks. In this article, some constructions of disjoint spectra Boolean that function by concatenating the functions on a lesser number of variables are provided. The nonlinearity and resiliency profiles of the constructed functions are obtained. From the profiles of the constructed functions, it is observed that the nonlinearity of these functions is greater than or equal to the nonlinearity of some existing functions. Furthermore, in the security analysis of cryptosystems, 4th order nonlinearity of Boolean functions play a crucial role. It provides protection against various higher order approximation attacks. The lower bounds on 4th order nonlinearity of some classes of Boolean functions having degree 5 are provided. The lower bounds of two classes of functions have form Tr1n(λxd) for all x∈F2n,λ∈F2n*, where (i) d=2i+2j+2k+2ℓ+1, where i,j,k,ℓ are integers such that i>j>k>ℓ≥1 and n>2i, and (ii) d=24ℓ+23ℓ+22ℓ+2ℓ+1, where ℓ>0 is an integer with property gcd(ℓ,n)=1, n>8 are provided. The obtained lower bounds are compared with some existing results available in the literature.

Keywords

• affine approximation attack
• Boolean functions
• disjoint spectra functions
• higher order nonlinearities
• resiliency
• Walsh–Hadamard transform (WHT)