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I am a little confused when it comes to linear boolean functions. According to this post:

What is a simple way of explaining what a linear boolean function means in boolean algebra and relating it to the standard definition of linearity?

We have that a function is linear if: $$f(x \oplus y) = f(x) \oplus f(y)$$

which (again according to the comments in the previously mentioned post) is equivalent to:

$$f(x_1, x_2, ... x_n) = c_0 \oplus c_1x_1 \oplus ... \oplus c_nx_n $$

where multiplication is defined as the logical AND (since wie operate mod 2). Now assume I define the function f as addition with 1, thus:

$$ f(x) = 1 \oplus x $$

This can easily be written in the above mentioned way, namely having $$c_0 = c_1 = 1$$, and thus:

$$ f(x) = 1 \oplus (1 \wedge x_1)$$

Quick double-check: If x = 1:

$$ f(1) = 1 \oplus (1 \wedge 1) = 1 \oplus 1 = 0$$

and if x = 0:

$$ f(0) = 1 \oplus (1 \wedge 0) = 1 \oplus 0 = 1$$

However, if we try to apply the above mentioned definition of linearity, and we set x to 1 and y to 1, we have: $$f(x \oplus y) = f(1 \oplus 1) = f(0) = 1$$ which is NOT the same as: $$f(x) \oplus f(y) = f(1) \oplus f(1) = 0 \oplus 0 = 0$$

Can someone clarify or explain where I am wrong in my thinking?

Thanks :)

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  • $\begingroup$ If a function is linear, then $c_0=0$. The more general solution is for the identity $f(x\oplus y\oplus z)=f(x)\oplus f(y)\oplus f(z)$. $\endgroup$ May 11, 2022 at 13:19
  • $\begingroup$ As is often the case, there is a confusion between linear and affine. $\endgroup$
    – user16034
    May 12, 2022 at 6:40

2 Answers 2

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A function satisfies $f(x \oplus y) = f(x) \oplus f(y)$ for all $x,y$ iff it is of the form $$ f(x) = c_1 x_1 \oplus \cdots \oplus c_n x_n. $$ Indeed, all functions of this form satisfy the identity. Conversely, suppose that $f(x \oplus y) = f(x) \oplus f(y)$. Then $f(0) = f(0 \oplus 0) = f(0) \oplus f(0)$. Now let $e_i$ be the vector $(0,\ldots,0,1,0,\ldots,0)$, where the unique $1$ is in the $i$'th position. Then $$ f(x) = f(x_1 e_1 \oplus \cdots \oplus x_n e_n) = f(x_1 e_1) \oplus \cdots \oplus f(x_n e_n) = x_1 f(e_1) \oplus \cdots \oplus x_n f(e_n), $$ where $f(x_i e_i) = x_i f(e_i)$ holds since if $x_i = 0$ then $f(x_i e_i) = f(0) = 0 = 0 f(e_i)$, and if $x_i = 1$ then $f(x_1 e_i) = f(e_i) = x_i f(e_i)$.


A function satisfies $f(x \oplus y \oplus z) = f(x) \oplus f(y) \oplus f(z)$ for all $x,y,z$ iff it is of the form $$ f(x) = c_0 \oplus c_1 x_1 \oplus \cdots \oplus c_n x_n. $$ Indeed, all functions of this form satisfy the identity. Conversely, let $g(x) = f(x) \oplus f(0)$. Then $$ g(x \oplus y) = f(x \oplus y \oplus 0) \oplus f(0) = f(x) \oplus f(y) \oplus f(0) \oplus f(0) = g(x) \oplus g(y). $$ Thus $g(x) = c_1 x_1 \oplus \cdots \oplus c_n x_n$, and so $f(x)$ is of the required form, with $c_0 = f(0)$.


More generally, if a function satisfies $f(x^{(1)} \oplus \cdots \oplus x^{(m)}) = f(x^{(1)}) \oplus \cdots \oplus f(x^{(m)})$ for even $m \geq 1$, then $f$ is linear; if $m \geq 1$ is odd then $f$ is affine. You can see this by taking $x^{(2)} = \cdots = x^{(m)}$ when $m$ is even, and $x^{(3)} = \cdots = x^{(m)}$ when $m$ is odd.

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From the definition, for any univariate linear function $$f(0)=f(0\oplus 0)=f(0)\oplus f(0)=0.$$

Hence the function $1\oplus a$ is not linear. (In fact, it is affine.)

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