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Can anyone say me the steps involved in calculating Walsh Hadamard Transform $(W_f(a))$ for a Boolean function $f(x)=x^3$ in a finite field $GF=F_4$?

Jean Marie
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user378779
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1 Answers1

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The Walsh transform of the Boolean function $f \colon \mathbb{F}_2^n \to \mathbb{F}_2$ is defined as $W_f(a) = \sum_{x \in \mathbb{F}_2^n}(-1)^{f(x) + x \cdot a}$. If we identify $\mathbb{F}_2^n$ with $\mathbb{F}_{2^n}$, we instead use $tr(ax)$ instead of $x \cdot a$, where $tr(x)=\sum_{i=0}^{n-1} x^{2^i}$ is the absolute trace function.

One should also be careful by noting that $x \mapsto x^3$ is typically considered as a function over $\mathbb{F}_{2^n}$ to itself, but it so happens that the image of this map is $\mathbb{F}_2\subsetneq \mathbb{F}_4$ when $n=2$. So, we can consider $x^3$ over $\mathbb{F}_4$ as the "same" as the Boolean function whose truth table is $[0,1,1,1]$. With this fact, one can quickly compute the Walsh spectrum of $f$.

Darrion
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