"kr t" - 1 õppematerjal

Boolean Functions and their Cryptographic Criteria

The product of any two distinct rows or columns of 𝐻 is zero. Hadamard matrix is often defined recursively by: 𝐻𝑛−1 𝐻𝑛−1 𝐻𝑛 = ( ) , 𝐻𝑜 = (1) 𝐻𝑛−1 −𝐻𝑛−1 For example: 1 1 1 1 1 1 𝐻1 = ( ), 𝐻2 = ( 1 −1 1 −1) etc. 1 −1 1 1 1 1 1 −1 1 −1 2𝑛 × 2𝑛 matrices 𝐻𝑖 are also called Sylvester-Hadamard matrices. Hadamard matrix can also be described as a Kronecker product: 𝐻𝑛 = 𝐻1 ⊗ 𝐻𝑛−1 , where the Kronecker product is defined as follows: 𝑎11 𝐵 ⋯ 𝑎1𝑛 𝐵 𝐴⊗B=( ⋮ ⋱ ⋮ ), 𝑎𝑚1 ⋯ 𝑎 𝑚𝑛 𝐵 where 𝐴 is a 𝑚 × 𝑛 matrix. It is important to note, that Kronecker product is not commutative, although it is associative and distributive. Now the Fourier transform of function 𝑓 can be described through a Sylvester-Hadamard function as follows: 𝐹𝑓 = 𝑓𝐻𝑛 . Inverse of that is similarly 𝑓 = 2 −𝑛 𝐹𝑓 𝐻𝑛 . 8 4. Correlation immunity and algebraic immunity 4.1 Cross-correlation and autocorrelation Definition 4.1. The following function is called the cross-correla...