# 2 - Summary from last week We have introduced the...

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Summary from last week ! We have introduced the fundamental ideas behind cryptographic systems (ciphers). ! We have set up a methodology for attacking these systems. ! We looked at shift and affine ciphers and how to analyze them. ! We have examined a number of basic ciphers used up to the 19’th century, based on substitutions and permutations, some with additional keywords. ! We have looked at implementation on computers using binary format. ! We have discussed the issue of key generation on computer.

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Objectives This week, we consider the underlying mathematical theory on which most of them are based. This includes linear feedback shift registers ( LFSR ) and some basic number theory and modular arithmetic .
Using computers for encrypting ! Computers run on binary values rather than letters or numbers modulo 26 . ! To review binary representations, see Section 2.8. ! One of the simplest encryption ideas for a binary string is to add it mod 2 to a fixed key string . Eg. plaintext key ciphertext

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Computers for encrypting cont’d ! Decryption is easy: just add the same key to the ciphertext, then you will get the plaintext. ! The one problem with this method is that the key is the same size as the plaintext, and so can only be used for that one message. (This is known as a ‘one time pad’.) ! Key management is a big issue in information security. The key has to be easy to store and to deliver, so it needs to be quite small.
Keys ! Keys also need to be ‘random’ in a sense that they are not easily guessed from formulas. ! There is a well-known method, which is very easy to implement on a machine, for generating a long string of digits from a small string. ! The machine is called a linear feedback shift register or LFSR for short.

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Linear feedback shift registers > This picture of an LFSR is taken from Wikipedia. An animation is given on: http://en.wikipedia.org/wiki/Linear_feedback_shift_reg ister
LFSRs cont’d ! Labeling the registers from the left (not standard) as 1 through 16, the LFSR takes the bits in the 11’th , 13’th , 14’th and 16’th positions, adds these to get the output on the right and feeds it all back in on the left. ! We can represent this as an equation: > e.g. x 17 = x 11 + x 13 + x 14 + x 16

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LFSRs cont’d ! These systems are used to generate ‘random’ key stream in applications such as pay-for-TV and satellite where a REALLY long key is needed. ! The initial values loaded into the register are kept secret. ! We assume, however, that an attacker knows the length of the register and the generating formula.
LFSRs cont’d ! The output of an LFSR will repeat after a certain time, and this helps an attacker to find the initial values. ! This ‘period’ of LFSRs has been well studied. ! If we have some output values and know the recurrence, we can generate systems of equations, which may be solvable using matrices.

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LFSRs cont’d Consider for , where We can establish the table.
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2 - Summary from last week We have introduced the...

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