31-03-2012, 09:43 AM
APPLICATION OF FOURIER ANALYSIS IN CRYPTOGRAPHY
APPLICATION OF FOURIER ANALYSIS IN CRYPTOGRAPHY.docx (Size: 16.11 KB / Downloads: 33)
Cryptography-
Cryptography can be defined as the conversion of data into a scrambled code that can be deciphered and sent across a public or private network. Cryptography uses two main styles or forms of encrypting data; symmetrical and asymmetrical. Symmetric encryptions, or algorithms, use the same key for encryption as they do for decryption. Other names for this type of encryption are secret-key, shared-key, and private-key. The encryption key can be loosely related to the decryption key; it does not necessarily need to be an exact copy.
Fourier analysis
It is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions. The attempt to understand functions (or other objects) by breaking them into basic pieces that are easier to understand is one of the central themes in Fourier analysis. Fourier analysis is named after Joseph Fourier who showed that representing a function by a trigonometric series greatly simplified the study of heat propagation.
Today the subject of Fourier analysis encompasses a vast spectrum of mathematics with parts that, at first glance, may appear quite different. In the sciences and engineering the process of decomposing a function into simpler pieces is often called an analysis. The corresponding operation of rebuilding the function from these pieces is known as synthesis. In this context the term Fourier synthesis describes the act of rebuilding and the term Fourier analysis describes the process of breaking the function into a sum of simpler pieces. In mathematics, the term Fourier analysis often refers to the study of both operations.
In Fourier analysis, the term Fourier transform often refers to the process that decomposes a given function into the basic pieces. This process results in another function that describes how much of each basic piece are in the original function. However, the transform is often given a more specific name depending upon the domain and other properties of the function being transformed, as elaborated below. Moreover, the original concept of Fourier analysis has been extended over time to apply to more and more abstract and general situations, and the general field is often known as harmonic analysis.
When processing signals,
such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection and/or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.
{The Fourier transform can separate low- and high- frequency information of a signal.
• Low frequencies background, overall shape
• High frequencies details, edges,}
Some examples include:
• Telephone dialing; the touch-tone signals for each telephone key, when pressed, are each a sum of two separate tones (frequencies). Fourier analysis can be used to separate (or analyze) the telephone signal, to reveal the two component tones and therefore which button was pressed.
• Removal of unwanted frequencies from an audio recording (used to eliminate hum from leakage of AC power into the signal, to eliminate the stereo subcarrier from FM radio recordings);
• Noise gating of audio recordings to remove quiet background noise by eliminating Fourier components that do not exceed a preset amplitude.