Vigenere ciphertext encrypted with another vigenere cipher I have been learning about Vigenere ciphers and then thought of this scenario: if a cryptographer encrypts a plaintext English message with a Vigenere cipher and then another cryptographer, who wants to ensure that the message is very secure, encrypts the ciphertext with another Vigenere cipher (with a different key length). Given only the ciphertext, how can you decrypt the message? A step by step explanation would be greatly appreciated.
Applying another round of vigenere would make the ciphtertext (in nearly every case) harder to break, yes. The problem is: This "new" algorithm is just a normal vigenere algorithm with a longer key (if the key lengths of both keys are not equal and not 1). You don't need to apply vigenere a second time, you can just calculate the new key in advance.

Example: We encrypt the plaintext "hello world hello world" with the key "ABC". (Encryption with A doesn't change the plaintext, it's like adding 0.)

hello world hello world  
+
ABCAB CABCA BCABC ABCAB  
=
HFNLP YOSND IGLMQ WPTLE


Now we encrypt the intermediate result again with the key "DDBC":

HFNLP YOSND IGLMQ WPTLE
+
DDBCD DBCDD BCDDB CDDBC
=
KIONS BPUQG JIOPR YSWMG


The length of the first key is 3, of the second key is 4. The least common multiple of both numbers is 12. If we now repeat the first key 4 times and "encrypt" it with the second key, we get a new key which eliminates the need for a intermediate result:

ABC ABC ABC ABC
+
DDB CDD BCD DBC
=
DED CEF BDF DCE


If we encrypt the plaintext with this new key we immediately get the final ciphertext:

hello world hello world
+
DEDCE FBDFD CEDED CEFBD
=
KIONS BPUQG JIOPR YSWMG


You see, double encryption (or more generally, multiple encryption) with vigenere doesn't add extra security by itself. You could just use a longer key from the beginning on. Search for generals attacks on vigenere to break this "double encryption".

What about the one-time-pad? You need a fully random, never before used key, which has to be as long as the plaintext for the one-time-pad. If you really use this scheme, you also don't need a double encryption: The one-time-pad is always fully secure if you tell nobody the key.