Multilinear-Polynomial-Extension-MLE

Compute the Multilinear Polynomial Extension (MLE) of your function based on your inputs

What is Multilinear Polynomial Extension (MLE)

The multilinear polynomial extension (MLE) is a low-degree polynomial extension in which each variable has a degree of at most one. Given a function Z : {0, 1}^m → F , the multilinear extension of Z(·) is the unique multilinear polynomial $\tilde{Z}$ : F^m → F . It can be computed as follows:

How to Run

python3 multilinearPolynomial.py

Example

Consider function f:

• f:{0,1}² -> F₅ is a function mapping {0,1}² to the field F₅ and f₀ = 1, f₁ = 2, f₂ = 1, f₃ = 4:

	0	1
0	1	2
1	1	4

For computing the Multilinear Polynomial Extension ( $\tilde{f}$ ) of function f based on x₁ = 2 and x₂ = 3:

Please specify the group order:

1. 5
   Please enter your inputs, then type "end" when you're done:
2. 1
3. 2
4. 1
5. 4
6. end
   Your multilinear polynomial extension has 2 inputs.
   Please enter your inputs for your multilinear polynomial extension:
7. 2
8. 3
   Final result: 16
   16 mod 5 = 1

$\tilde{f}$ is the Multilinear Polynomial Extension of function f based on different inputs over F₅:

	0	1	2	3	4
0	1	2	3	4	0
1	1	4	2	0	3
2	1	1	1	1	1
3	1	3	0	2	4
4	1	0	4	3	2

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