TAILIEUCHUNG - Hacker Professional Ebook part 165

Tham khảo tài liệu 'hacker professional ebook part 165', công nghệ thông tin, kỹ thuật lập trình phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Fast Fourier Transform. A numerically advantageous way of computing a Fourier transform. Basically a way of transforming information from amplitude values sampled periodically through time into amplitude values sampled periodically through complex frequency. The FFT performs this transformation in time proportional to n log n for some n a power of 2. While exceedingly valuable the FFT tends to run into practical problems in use which can require a deep understanding of the process. For example the transform assumes that the waveform is stationary and thus repetitive and continuous which is rarely the case. As another example sampling a continuous wave can create spurious frequency values related to the sampling and not the wave itself. Also the range of possible values increases by a factor of 2 in both positive and negative directions in every sublayer mixing this is information expansion which we often try to avoid in cryptography. The FFT provides a strong mathematical basis for block cipher mixing such that all input values will have an equal chance to affect all output values. Cryptographic mixing then occurs in butterfly operations based on balanced block mixing structures which replace the simple add subtract butterfly in the FFT and confine the value ranges so information expansion does not occur. A related concept is the fast Walsh-Hadamard transform FWT which can use exactly the same mixing patterns as the FFT. Field In abstract algebra a commutative ring in which all non-zero elements have a multiplicative inverse. This means we can divide. In general a field supports the four basic operations addition subtraction multiplication and division and satisfies the normal rules of arithmetic. An operation on any two elements in a field is a result which is also an element in the field. Examples of fields include rings of integers modulo some prime. Here are multiplication tables under mod 2 mod 3 and mod 4 0 1 0 1 2 0 1 2 3 0 0 1 0 0 1 0 0 0 0 1 0 1 2 0 0 0 0

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