Cryptography Research Paper Example | Topics and Well Written Essays - 1000 words

Cryptography - Research Paper Example The construction of c is done in such a way that there are elements which are redundant in it. This will, therefore, enable the receiver to reconstruct c even if some bits of c are corrupted by noise; the receiver will eventually reconstruct m (Gary 93). In a formal manner, an error correcting code is composed of a set, C? {0, 1} n of codewords. This set has strings which enables messages to be mapped in it before they are transmitted. In this case, a code that will be used for k-bit messages, C will have 2k elements which are distinct. So that there is some redundancy, there will be a need to have n>k. codes that are used for correcting errors can be defined in spaces which are non-binary too and this paper has construction which is straightforward and extensible in these non-binary spaces (Denning 72). For error correcting codes to be used, there will be a need for functions that will enable us to encode and decode messages. In this paper we will let M = {0, 1}k be a representation of the space message. There is a translation function, g : M C, which represent a one-to-one mapping capability of messages to codewords. What this means is that g is the mapping that is used before the transmission takes place. On the other hand, g-1 is the function that is used upon receiving of messages to retrieve codes in the codeword. There is a function, referred to as decoding function that is used for mapping n-bits that are arbitrary to codewords. This is the function, f : {0, 1}1 C U {O}. If the f function is successful, it will manage to map a given string which has n-bits x to the nearest codeword that is found in C (that is, the proximity to nearness in Hamming distance). If this not the case, then f will fail and the output will be O3. The robustness that an error-correcting code has will depend on the distance between the codewords. To make this more definite, we will need some fundamental notation that regard strings of the binary digits. For this case, we will use + and – to represent bitwise XOR operator on the bit strings. We will use a measurement Hamming weight, which is the number of ‘1’ bits that are found in u. The Hamming weight is denoted by ||u|| (this is the weight of a string which has n strings). The Hamming weight has a precise definition of the number of ‘l’ bits that are found in u. In the same perspective, the Hamming distance that is found between two strings, u and v is defined as the number of digits that make two strings to be different (Gary 62). In an equivalent manner, the Hamming distance will be equal to ||u - v||. We normally take it that a function that is used for decoding, that is function f, will have a correction threshold with a size of t if it has the ability to correct any set of t bit errors. In a more definite manner, for any codeword c â‚ ¬ C, and any error term e â‚ ¬ {0, 1}n, that has || e ||? t, this is the case that f(c+e) = c. in this case, we will regard C to have a correction threshold which has a size of t if there is a function f for C for t, which also has a correction threshold of size t. there is a an observation that the distance that is found between two codewords in C should have a distance of at least 2t + 1. The neighborhood of a codeword c is defined to be f-1 (c). This means that the neighborhood of c has a subset of strings that are n-bit long where f maps to c. the function that is used for decoding, that is function f, is set in such a way that f-1(c) has a close proximity to c that any other code word that