# Algebraic normal form

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Algebraic normal form

In Boolean logic, the algebraic normal form (ANF) is a method of standardizing and normalizing logical formulas. As a normal form, it can be used in automated theorem proving (ATP), but is more commonly used in the design of cryptographic random number generators, specifically linear feedback shift registers (LFSRs). A logical formula is considered to be in ANF if and only if it is a single algebraic sum (XOR) of a constant $a_0$ and one or more conjunctions of the function arguments. ANF is also known as "Zhegalkin polynomials" ( _ru. полиномы Жегалкина) and as "Positive Polarity (or Parity) Reed-Muller" expression.

Putting a formula into ANF makes it easy to identify linear functions, as is needed for linear feedback in LFSRs: a linear function is one that is a sum of literals. Properties of nonlinear feedback shift registers can also be deduced from certain properties of the feedback function in ANF.

The general ANF can be written as:where $a_0, a_1, ldots, a_\left\{1,2,ldots,n\right\} in \left\{0,1\right\}^*$ fully describes $f$.

For each function $f$ there is a unique ANF. There are only four functions with one argument: $f\left(x\right)=0$, $f\left(x\right)=1$, $f\left(x\right)=x$, $f\left(x\right)=1+x$ (all of them are given in the ANF). To represent a function with multiple arguments one can use the following equality: $f\left(x_1,x_2,ldots,x_n\right) = g\left(x_2,ldots,x_n\right) + x_1 h\left(x_2,ldots,x_n\right)$, where $g\left(x_2,ldots,x_n\right) = f\left(0,x_2,ldots,x_n\right)$ and $h\left(x_2,ldots,x_n\right) = f\left(0,x_2,ldots,x_n\right) + f\left(1,x_2,ldots,x_n\right)$. Indeed, if $x_1=0$ then $x_1 h = 0$ and so $f\left(0,ldots\right) = f\left(0,ldots\right)$; if $x_1=1$ then $x_1 h = h$ and so $f\left(1,ldots\right) = f\left(0,ldots\right) + f\left(0,ldots\right) + f\left(1,ldots\right)$. Since both $g$ and $h$ have less arguments than $f$ it follows that using this process recursively we will finish with functions with one variable. For example, let us construct ANF of $f\left(x,y\right)= x lor y$ (logical or): $f\left(x,y\right) = f\left(0,y\right) + x\left(f\left(0,y\right)+f\left(1,y\right)\right)$; since $f\left(0,y\right)=0 lor y = y$ and $f\left(1,y\right)=1 lor y = 1$, it follows that $f\left(x,y\right) = y + x \left(y + 1\right)$; by opening the parentheses we get the final ANF: $f\left(x,y\right) = y + x y + x = x + y + x y$.

ee also

* Boolean function
* Conjunctive normal form
* Disjunctive normal form
* Logical graph

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