# Period of sum of two periodic sequences

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

If you sum a sequence with period $$a$$ and a sequence with period $$b$$, then you get a sequence which is $$\mathsf{LCM}(a,b)$$-periodic. But the new sequence might have a smaller period, as the following example demonstrates (addition is modulo 10):

022441133502244113350224411335
000772999611888000772999611888
022113022113022113022113022113


The first sequence has period 10 (0224411335), the second sequence as period 15 (000772999611888), and the sum sequence has period 6 (022113), although the LCM is 30.

More generally, the sum of a sequence of period $$i$$ and a sequence of period $$j$$ could have period exactly $$k$$ iff $$\mathsf{LCM}(i,j) = \mathsf{LCM}(i,k) = \mathsf{LCM}(j,k)$$. See Restrepo and Chacon, On the period of sums of discrete periodic signals.

• so is there exists some kind of shortest key for what's described? for instence, if i cipher the message once with a key of size i(using vigenere) and second time with a key of length j(using vigenere) - is there a shortest key to decipher the message? – alberto123 Nov 21 '19 at 12:26
• Yes, but the length of the shortest key could depend on the actual keys. – Yuval Filmus Nov 21 '19 at 12:27
• thank you so much for explaining that to me and for the reference. i am going over the link and similar presentation i found at online colleges in addition to my book and it helps a lot. thanks again – alberto123 Nov 21 '19 at 13:44