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As you said, we can get the $M\in S$ whose first column is $y-b$, but i don't know how to generate the set $S$ by Sage code.

There exist other cases such as y = vector(GF(2), [1, 0, 1, 0, 1, 0, 0, 0]) and x = vector(GF(2), [0, 1, 1, 1, 1, 1, 0, 0]). The values of x and y are variable.

Yes, we need all possible solutions of $A$, since we have many equations like $Ax+b=y$, we will solve each equation and take the intersection to get the final $A$.

A = matrix(GF(2), 8, 8, [])
b = vector(GF(2), [0, 1, 1, 0, 1, 0, 1, 1])
y = vector(GF(2), [0, 0, 0, 0, 1, 0, 1, 1])
x = vector(GF(2), [1, 0, 0, 0, 0, 0, 0, 0])

If the matrix $A$ is unkown, we have $Ax+b = y$.

How can we solve the matrix $A$?

X = GF(2).polynomial_ring().gen()

F = GF(2^8, name="a", modulus=X^8 + X^6 + X^5 + X + 1)

As example above, the polynomial X^8 + X^6 + X^5 + X + 1 is one of irreducible polynomial defining the finite field GF(2^8).

How can I get all the irreducible polynomial which can define the finite field GF(2^8)?

A cyclic matrix means a circulant matrix. We need the first row the property that the sum of entries is not zero so that this cyclic matrix could be a invertible matrix.

How to generate all 8×8 cyclic matrix $A$ with the first row $(a_0, a_1, \ldots, a_7)$,

the matrix $A\in GL(8, 2)$ and the first row $a_0 + a_1 + \ldots + a_7 \neq 0$

Suppose that there is a permutation from (1, 2, 3, 4, 5, 6) to (2, 3, 5, 4, 6, 1), how could I check whether it is linear or not?

Now I have many permutations from 256 integers to 256 integers, how could i check?

For example, this is a permutation, from (0, 1, ..., 256) to (0, 29, 142, 147, 199, 218, 73, 84, 227, 254, 109, 112, 36, 57, 170, 183, 113, 108, 255, 226, 182, 171, 56, 37, 146, 143, 28, 1, 85, 72, 219, 198, 184, 165, 54, 43, 127, 98, 241, 236, 91, 70, 213, 200, 156, 129, 18, 15, 201, 212, 71, 90, 14, 19, 128, 157, 42, 55, 164, 185, 237, 240, 99, 126, 220, 193, 82, 79, 27, 6, 149, 136, 63, 34, 177, 172, 248, 229, 118, 107, 173, 176, 35, 62, 106, 119, 228, 249, 78, 83, 192, 221, 137, 148, 7, 26, 100, 121, 234, 247, 163, 190, 45, 48, 135, 154, 9, 20, 64, 93, 206, 211, 21, 8, 155, 134, 210, 207, 92, 65, 246, 235, 120, 101, 49, 44, 191, 162, 238, 243, 96, 125, 41, 52, 167, 186, 13, 16, 131, 158, 202, 215, 68, 89, 159, 130, 17, 12, 88, 69, 214, 203, 124, 97, 242, 239, 187, 166, 53, 40, 86, 75, 216, 197, 145, 140, 31, 2, 181, 168, 59, 38, 114, 111, 252, 225, 39, 58, 169, 180, 224, 253, 110, 115, 196, 217, 74, 87, 3, 30, 141, 144, 50, 47, 188, 161, 245, 232, 123, 102, 209, 204, 95, 66, 22, 11, 152, 133, 67, 94, 205, 208, 132, 153, 10, 23, 160, 189, 46, 51, 103, 122, 233, 244, 138, 151, 4, 25, 77, 80, 195, 222, 105, 116, 231, 250, 174, 179, 32, 61, 251, 230, 117, 104, 60, 33, 178, 175, 24, 5, 150, 139, 223, 194, 81, 76)