Number Theoretic Transforms
Most of the fast convolution techniques discussed so far are essentially algebraic methods which can be implemented with any type of arithmetic. In this chapter, we shall show that the computation of convolutions can be greatly simplified when special arithmetic is used. In this case, it is possible to define number theoretic transforms (NTT) which have a structure similar to the DFT, but with complex exponential roots of unity replaced by integer roots and all operations defined modulo an integer. These transforms have the circular convolution property and can, in some instances, be computed using only additions and multiplications by a power of two. Hence, significant computational savings can be realized if NTTs are executed in computer structures which efficiently implement modular arithmetic.
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