3
$\begingroup$

I am currently reading a research paper that uses the term 2-flat. However, I don't understand what this term means.

For example, the paper mentions in its definition of APN that for all distinct $a$, $b$, $c$ and $d$ in $V$,

$$a + b + c + d = 0 \implies f(a) + f(b) + f(c) + f(d) \neq 0$$

i.e. $f$ does not sum to 0 on any 2-flat.

It would be great if anyone could explain what 2-flat means here.

An APN permutation in dimension 6

$\endgroup$
2
$\begingroup$

In the context of Boolean functions, "flat" is usually used as a synonym for "affine subspace of $\mathbb{F}_2^n$". More generally, an $n$-flat in a vector space $V$ (which may be considered as an affine space) is an $n$-dimensional affine subspace of $V$.

Note that a 2-flat of a vector space $V$ over $\mathbb{F}_2$ is then any set of the form

$$A = \{a, a + b, a + c, a + b + c\}$$

with $a, b, c \in V$ distinct. Remark that $\sum_{x \in A} x = 0$ and conversely, any four distinct elements in $V$ that sum to zero form a 2-flat.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.