I was wondering, what is to be the shortest possible key using Vigenere encryption, if a text is ciphered one time with a key of length $i$ using Vigenere and second time with a key of length $j$ using Vigenere?
From what I understand, each time it is encrypted, each character shifts using the first key and then using the second one, so I believe that for each pair there's an equivalent shifting. but if the two keys have a different length, how can I find the shortest possible key?
I think it has to be something with lowest common denominator, but I am not sure. how can this be shown mathematically?
my previous edit was a mistake and i deeply sorry for it. i read the book wrong and didn't interpret what's written well. very sorry for it. yuval explained to me well and i continue to research it using his given information