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51 13 Trapdoors and signatures It might be thought that secure codes are all that are needed to ensure the security of communications but this is not so. It is not necessary to read a message to derive information from it17. In the same way, it may not be necessary to be able to write a message in order to tamper with it. Here is a somewhat far fetched but worrying example. Suppose that by wire tapping or by looking over peoples' shoulders I nd that a bank creates messages in the form M1 , M2 where M1 is the name of the client and M2 is the sum to be transfered to the client's account.

1. (iv) Note that the larger is the slower n ; m ; n increases. This corresponds to the very general statement that the higher the information rate of the messages the harder it is to break the code in which they are sent. 45 The ideas just introduced can be formalised by the notion of unicity distance. 11. The unicity distance of a code is the number of bits of message required to exceed the number of bits of information in the key plus the number of bits of information in the message. If the reader complains that there is a faint smell of red herring about this de nition, I would be inclined to agree.

Z (iii) (Z ) m=0 P (iv) ! j = r( u+1) v for all 0 j 2t ; 1. (vi) 0 = X u+v=j u+v=j r( u+1 ) v for all t j 2t ; 1. (vii) The conditions in (vi) determine completely. 21 completes our search for a decoding method, since determines E , E determines e and e determines c. It is worth noting that the system of equations in part (v) su ce to determine the pair and ! directly. Compact disk players use BCH codes. Of course errors are likely to occur in bursts (corresponding to scratches etc) and this is dealt with by distributing the bits (digits) in a single codeword over a much longer stretch of track.

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