# Cascade Ciphers: The Importance of Being First

## Ueli Maurer and James L. Massey

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```The security of cascade ciphers, in which by definition the keys of the
component ciphers are independent, is considered. It is shown by a
counterexample that the intuitive result, formally stated and proved in
the literature, that a cascade is at least as strong as the strongest
component cipher, requires the uninterestingly restrictive assumption
that the enemy cannot exploit information about the plaintext
statistics. It is proved, for very general notions of breaking a
cipher and of problem difficulty, that a cascade is at least as
difficult to break as the first component cipher. A consequence of this
result is that, if the ciphers commute, then a cascade is at least as
difficult to break as the most-difficult-to-break component cipher,
i.e., the intuition that a cryptographic chain is at least as strong as
its strongest link is then provably correct. It is noted that additive
stream ciphers do commute, and this fact is used to suggest a strategy
for designing secure practical ciphers. Other applications in
cryptology are given of the arguments used to prove the cascade cipher
result.