Codes and Cryptography [Lecture notes] by T. K. Carne

By T. K. Carne

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Example text

First calculate R(αj ) for j = 1, 2, . . , δ − 1. If these are all 0 then R(X) is a code word and there have been no errors (or at least δ of them). Otherwise let E = {i : ei = 0} be the set of indices at which errors occur and assume that 0 < |E| The error locator polynomial is σ(X) = (1 − αi X) . t. i∈E This is a polynomial (over K) of degree |E| with constant term 1. If we know σ(X) then we can easily find which powers α−i are roots of σ(X) and hence find the indices where errors have occurred.

These fields are unique, up to isomorphism, and will be denoted by Fq . Example: For each prime number p, the integers modulo p form a field. So Fp = Z/pZ. The non-zero elements of a finite field Fq form a commutative group denoted by F× q . This is a cyclic group of order q − 1. Any generator of F× is called a primitive element for F . Let α be a primitive q q k are the powers α for k = 1, 2, 3, . . , q − 1. These element. Then the other elements of the group F× q q − 1 are all distinct. The order of the element αk is , so the number of primitive elements is given (q − 1, k) by Euler’s totient function ϕ(q − 1) = |{k : k = 1, 2, 3, .

In this case we talk about the message being encrypted or enciphered. This is an important and very actively studied area. As computers grow more powerful we need to devise ciphers that are more difficult to break. We wish to transmit a message or plaintext which consists of a string of letters taken from a finite alphabet A. Usually there are a variety of different methods to encode or encipher the message, each depending on a key taken from a finite set K of possible keys. We will write ek : A → B for the encoding function or encrypting function corresponding to the key k.

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