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

Cryptography - Research Paper ExampleThe expression of c is done in such a way that there are elements which are scanty in it. This pass on, therefore, enable the receiver to reconstruct c even if some bits of c are subvert by noise the receiver lead eventually reconstruct m (Gary 93). In a orb manner, an error correcting code is composed of a set, C? 0, 1 n of codewords. This set has forces which enables messages to be roleped in it before they are transmitted. In this skid, a code that will be apply 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 nk. codes that are used for correcting errors can be defined in spaces which are non-binary as well 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, 1k be a formation 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, 11 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 implant in C (that is, the proximity to neighborhood 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 than definite, we will need some fundamental notation that regard strings of th e 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 combining weight 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 door with a surface of t if it has the ability to correct any set of t bit errors. In a more definite manner, for any codeword c C, and any error term e 0, 1n, that has e ? t, this is the case that f(c+e) = c. in this case, we will regard C to have a correctio n 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