I have given a ciphertext: vpitqjcahpzkcvdtmttmtnmjuigf
and two Keys: k1 = SECRET, k2 = KEY.
Decryption is done by applying the keys one by one, but you can also encrypt k1 with k2, which yields k3 = CIABIR. Then it is also possible to use k3 for decryption.
In both cases i get the same result for the plaintext: thisisashortandsecretmessage.
How can i describe this mathematically and therefore show that a single encryption with k3 is equivalent to double encryption with the keys k1 and k2?
[Math] Double Vigenere Cipher
cryptography
Best Answer
That is easy if you know write it down - assuming the keys have the same size.
$$C_i = P_i + K_{i \pmod{keyLenght}} \pmod{26}$$
For simplicity consider only the first char.
$$C_1 = P_1 + K_1 \pmod{26}$$
Now consider other two keys $K'$ and $K''$ such that $K = K' + K'' \pmod{26}$
Encrypt $P_1$ with $K'$ than $K''$ $C_1$
$$C_1'= P_1 + K_1' \pmod{26}$$ $$C_1'' = C_1' + K_1'' \pmod{26}$$
Substitute
$$C_1'' = P_1 + K_1' + K_1'' \pmod{26}$$ then
$$C_1'' = P_1 + K_1 \pmod{26}$$
So, you get the double encryption with single encryption.
Therefore, the sum of the two keys is actually another key that is equal to the double encryption.
You can similarly find a key for triple encryption, quad, etc.