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Suppose I encrypted a homophonic substitution cipher at the level of syllables, trigrams, digraphs, and individual letter with a Vigenere cipher using the keyword cryptography. I would then encrypt the resulting complex ciphertext with an ADFGVX cipher in addition to using a code. How strong can a multiple-encryption be?

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If the ciphers are different, with independent keys, you can say that it is at least as strong as the first cipher. If the ciphers commute, like with stream ciphers, you can even say that it is at least as strong as the strongest. See Cascade Ciphers: The Importance of Being First.

That's really all you can say in general.

In practice, the combinations you suggest are not cryptographically secure. You would need at least one secure cipher in the mix to have reason to believe it would be. (Though that's not strictly required in theory: a particular combination of insecure ciphers could be secure.)

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In general, all you can say is it can be as weak as the weakest encryption layer, if you're lucky. Edit: It can also be even weaker, for badly chosen components that cancel out some mathematically desirable properties, as pointed out in the comment.

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    $\begingroup$ Couldn't it be even weaker? For instance mathematically, combining two different operations could cancel out terms that provide security. Also, things like one layer could provide known plain text that is passed into the next layer. $\endgroup$ – Alan Wolfe Aug 30 '15 at 1:28

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