The contents of this blog post today will detail about a cipher in classical cryptography. It is not used nowadays because it is easily vulnerable. In fact, the way it works can also be understandable to any high schooler or first year college student who has an elementary-level knowledge of linear algebra.

Say you have a message you want to decrypt, e.g. "I like pie". We convert each character into a numerical representation (e.g. perhaps ASCII). Then we form a matrix out of these characters. Let's call it matrix B.

Let's use the concept of matrix multiplication and invertible matrices in order to decrypt and encrypt this matrix B message.

We can easily encrypt it by matrix-multiplying it with an arbitrary matrix A. The resulting matrix, AB, will be sent as the "encrypted message". Note that of course, the number of columns in matrix A must be equal to the number of rows in matrix B for matrix multiplication of A and B to work.

Now, you can send this encrypted message AB to anyone. But how can they decrypt it? Simple. We can use the property of invertible matrices.

(A^-1)(AB) = B

So, we just need to multiply the inverse of matrix A to the encrypted message AB, to get back the decrypted message B. Note that A must be an invertible matrix then.

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