A Novel Approach Towards Degree and Walsh-Transform of Boolean Functions

Boolean functions are fundamental bricks in the development of various applications in Cryptography and Coding theory by making benefit from the weights of related Boolean functions (Walsh spectrum). Towards this, the discrete Fourier transform (Walsh–Hadamard) plays a pivotal tool. The work in this paper is dedicated towards the algebraic and numerical degrees, together with the relationship between weights of Boolean function and their Walsh transforms. We introduce Walsh matrices and generalize them to any arbitrary Boolean function. This improves the complexity in computation of Walsh–Hadamard and Fourier transform in certain cases. We also discuss some useful results related to the degree of the algebraic normal form using Walsh–Hadamard transform.