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Unformatted text preview: the contents, click the chapter and section titles. Applied Cryptography, Second Edition: Protocols, Algorthms, and Source Code in C (cloth)
Brief Full Advanced Search Search Tips (Publisher: John Wiley & Sons, Inc.) Author(s): Bruce Schneier ISBN: 0471128457 Publication Date: 01/01/96 Search this book:
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----------- There are a few things to note here. First, the carry register is not a single bit; it is a number. The size of the carry register must be at least log2t, where t is the number of taps. There are only two taps in the previous example, so the carry register only has to be 1 bit wide. If there were four taps, the carry register would have to be 2 bits wide, and could be either 0, 1, 2, or 3. Second, there is an initial transient before the FCSR settles down into its repeating period. In the previous example, only one state never repeated. For larger and more complicated FCSRs, there may be more. Third, the maximum period of a FCSR is not 2n - 1, where n is the length of the shift register. The maximum period is q - 1, where q is the connection integer. This number gives the taps and is defined by: q = 2q1 + 22q2 + 24q4 +...+ 2nqn - 1 (Yes, the qis are numbered from left to right.) And even worse, q has to be a prime for which 2 is a primitive root. The rest of this discussion assumes q is of this form. In this example, q = 2*0 + 4*1 + 8*1 - 1 = 11. And 11 is a prime with 2 as a primitive root. So the maximum period is 10. Not all initial states give you the maximum period. For example, look at the FCSR when the initial value is 101 and the carry register is set to 4. Figure 17.4 3-bit FCSR. Shift Register Carry Register 101 110 111 111 4 2 1 1 At this point the register spits out a neverending stream of 1s. Any initial state will result in one of four things. First, it is part of the maximum period. Second, it will fall into the maximum period after an initial transient. Third, it will fall into a sequence of all zeros after an initial transient. Fourth, it will fall into a sequence of all ones after an initial transient. There is a mathematical formula for determining what will happen to a given initial state, but it’s much easier to just test it. Run the FCSR for a while. (If m is the initial memory, and t is the number of taps, then log2(t) + log2(m) + 1 steps are enough.) If it degenerates into a neverending stream of 0s or 1s within n bits, where n is the length of the FCSR, don’t use...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10