The fastest algorithm for multiplying two n-digit numbers is the Fast Fourier Transform (FFT) algorithm. This algorithm is based on the principles of the Fourier transform, which is a mathematical technique for representing signals and other functions as a sum of sinusoidal waves.

The FFT algorithm takes advantage of the properties of the Fourier transform to perform fast multiplications of two n-digit numbers. It works by decomposing the two numbers into their constituent sinusoidal waves, multiplying the waves together, and then reconstructing the product.

The FFT algorithm has a time complexity of O(n log n), which means that it takes approximately n log n steps to multiply two n-digit numbers. This is significantly faster than the traditional long multiplication algorithm, which has a time complexity of O(n^2).

There are several variations of the FFT algorithm, including the Cooley-Tukey FFT and the Bluestein's FFT, which are used for different types of multiplication problems. Overall, the FFT algorithm is widely used in computer science and engineering for fast multiplications of large numbers.

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