How can one construct the weakest S-box ever? Is it possible to come up with an NxN S-box which would have a difference distribution table with N entries of 100% probability?

I am studying the properties of S-boxes and I don't quite understand how a poorly designed S-box can destroy all the cipher's security.

For example, if one replaces the AES-256's S-box table with such an S-box I described above, would he/she be able to crack the cipher with some known plaintext/cipherext pairs?

EDIT:

Here is a variation of the most useless S-box:

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


But I cannot figure out by what rule it was constructed.
Any affine function will do. Let your Sbox be $$S(x)=Mx\oplus c$$ where $M$ is an $n\times n$ binary matrix and $c$ is an $n-$bit constant vector.

The output difference for this Sbox is, for any nonzero $a$
$$
S(x \oplus a)\oplus S(x)=(M(x\oplus a)\oplus c )\oplus Mx\oplus c= M a\oplus c
$$
which is a constant for fixed $a$ so all the output differences for that input difference take on the same value.

For nontriviality pick either $c$ nonzero or $M$ a non identity matrix. The matrix must be invertible (over $GF(2)$) for the Sbox to be invertible, i.e., a proper substitution.

AES with this type of Sbox can be trivially broken.

Edit: See the answer to this question for details.