notebook/02-Merklist-Field.ipynb

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Inverse?

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# Q: If you compute the inverse of one of these matrixes and multiply it to the right of the position
#    where it was originally multiplied it should "delete" that element as if the element was never added.
#    I'm uncertain of the possibility and frequency of such inverse matrices over finite fields.
#    If this scenario is possible to construct, I think we should "define" this operation as deletion.

# See: https://en.wikipedia.org/wiki/Determinant#Properties_of_the_determinant

def det2(mat, m=256):
    ((a, b),
     (c, c)) = mat
    return (a*c - b*c) % m

def det3(mat, m=256):
    ((a, b, c),
     (d, e, f),
     (g, h, i)) = mat
    return (a*e*i + b*f*g + c*d*h - c*e*g - b*d*i - a*f*h) % m

def det(mat, m=256):
    # I dislike having to round here, but it returns a float
    # I wish it returned a native "B" type
    return int(round(np.linalg.det(mat),0)) % m

a = ((-2,2,-3),(-1,1,3),(2,0,-1))
assert_equal(det(a), 18)
assert_equal(det(a), det3(a))

# ?
# assert that they didn't change since the last time I ran it
assert_equal(det(f1), 128)
assert_equal(det(f2), 160)
assert_equal(det(f3), 128)

This suggests to find the inverse matrix by first finding the adjugate then applying

(1/det(A))adj(A)=inv(A)

Under mod p

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The adjugate is the inverse of the cofactor matrix.

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The cofactor matrix is made by multiplying the matrix minor of each (i,j) of the original matrix times a sign factor.

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The (i,j)-minor of matrix A is the determinant of the matrix A with row i & column j removed.