33. This problem deals with the Sigaba attack discussed in this chapter.

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a. Compute the average probability p,, for i = 1 , 2 , 3 , 4 , that precisely i cipher rotors step, where the average is taken over all possible index permutations and all possible control rotor outputs. Hint: Model the control rotor outputs as uniformly random. Then there are equally likely outputs of the control rotors and these outputs are combined as indicated in (2.7). Test each of these with each of the 10!/32 distinct index permutations (see Section 2.4.3). Compare your results to Problem 22, part c.

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b. How can you use the result of part a of this problem to improve on the Sigaba attack described in this chapter

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3

Stream Ciphers

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If we are carried along the stream we fear nothing,

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and it is only when we strive against it, that its progress and power are discernible. - John Owen

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Introduction

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Stream ciphers are a class of symmetric ciphers that operate something like a one-time pad. The crucial difference is that a stream cipher only requires a small key, whereas a one-time pad cipher requires a key that is the same length as the original message. While a one-time pad cipher is provably secure (provided it is used correctly), it is generally impractical since the key is the same length as the message. After all, if Alice and Bob can securely distribute a key that is the same length as the message, why not simply distribute the message by the same means as the key and do away with the cipher In a stream cipher, a relatively small key is stretched into a long keystream that can then be used just like a one-time pad. A stream cipher has far fewer keys than the number of possible keystreams, so we cannot prove that such a cipher is secure-at least not using a similar argument as is used to prove the one-time pad is secure. In effect, a stream cipher trades the provable security of a one-time pad for practicality. A generic stream cipher is illustrated in Figure 3.1, where the key is input to the stream cipher algorithm, which then generates the keystream ki, for i = 0, 1 , 2 , . . .. This keystream can be generated in bits, bytes, or other sized chunks. Encryption is accomplished by XOR of the keystream ki with the plaintext pi to yield the ciphertext ci. To decrypt, the same key is input to the stream cipher algorithm, so that the same keystream is generated. Then the keystream bits are XORed with the ciphertext to yield the original

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STREAM CIPHERS

plaintext. As with the one-time pad, this decryption relies on the fact that if c, = p , @ k, then c, @ k, = ( p , fBk , ) @ k, = p,. That is, regardless of the value of the bit k,, we have k , @ k , = 0.

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stream cipher --I

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stream cipher

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ciphertext

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Figure 3.1: Generic stream cipher. We must assunie that Trudy, the cryptanalyst, knows (or can guess) some of the plaintext. For a stream cipher, known plaintext and the corresponding ciphertext immediately enables Trudy to recover part of the keystream. If Trudy can recover more of the keystream from such a captured segment: then the stream cipher is insecure. Therefore, the security of a stream cipher depends on properties of the generated keystream. But what properties should a keystreani ideally satisfy The keystream needs to be random, but there are many definitions of randomness, and many of these random sequences would be poor keystreams. For example, a common method for generating pseudo-random sequences is to employ a linear congruential generator (LCG). The output from these generators satisfy many statistical properties that make them excellent random sources for a variety of applications (for exaniple, simulations). The bits generated by an LCG could be used as a keystream, with the seed value acting as the key. However, an LCG would make a very poor stream cipher, since given a sniall section of the keystreani it is not difficult to determine the entire sequence [9]. This is exactly what we must avoid with a stream cipher keystream. In other words, statistical randomness is insufficient to ensure the security of a keyst rcarn. The crucial property required of a keystream sequence is that it be unpredictable, or cr gptop-aphically strong. Intuitively, it is clear what we mean by unpredictable, but there is no entirely satisfactory technical definition. We discuss this problem briefly in the ncxt section. In this chapter we first discuss linear feedback shift registers (LFSRs), which are often used as building blocks for stream ciphers. We also consider correlation attacks against a particular class of LFSR-based stream cipher.

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