We present and analyze attacks on additive stream ciphers that rely on linear equations that hold with non-trivial probability in plaintexts that are encrypted using distinct keys. These attacks extend Biham's key collision attack and Hellman's time memory tradeoff attack, and can be applied to any additive stream cipher. We define linear redundancy to characterize the vulnerability of a plaintext source to these attacks. We show that an additive stream cipher with an n-bit key has an effective key size of n
David A. McGrew, Scott R. Fluhrer