The paper by Shmuel Winograd presents new algorithms for computing the Discrete Fourier Transform (DFT) of \( n \) points, where \( n \) ranges from a few tens to a few thousands. These algorithms use significantly fewer multiplications compared to the best previously known algorithm, while the number of additions remains similar. The DFT is widely used in scientific and engineering calculations. Winograd's work builds on earlier research by Cooley and Tukey, who described an algorithm for DFT with \( n/2 \log n/2 \) complex multiplications and \( n \log n \) complex additions when \( n = 2^k \). The theoretical background involves a problem of minimizing the number of multiplications needed to compute the coefficients of the product of two polynomials modulo a given polynomial. Winograd proves that for a prime \( p \), the minimum number of multiplications required to compute the DFT of \( p \) elements is \( 2(p-1) - k \), where \( k \) is the number of irreducible factors of \( u^{p-1} - 1 \). The summary of results includes a table detailing the number of multiplications and additions used for small values of \( n \). For larger values of \( n \), Winograd proposes algorithms that decompose the computation using the Chinese Remainder Theorem, reducing the problem to smaller subproblems. This approach is demonstrated with examples, showing that the new algorithms can significantly reduce the computational complexity for DFT calculations.The paper by Shmuel Winograd presents new algorithms for computing the Discrete Fourier Transform (DFT) of \( n \) points, where \( n \) ranges from a few tens to a few thousands. These algorithms use significantly fewer multiplications compared to the best previously known algorithm, while the number of additions remains similar. The DFT is widely used in scientific and engineering calculations. Winograd's work builds on earlier research by Cooley and Tukey, who described an algorithm for DFT with \( n/2 \log n/2 \) complex multiplications and \( n \log n \) complex additions when \( n = 2^k \). The theoretical background involves a problem of minimizing the number of multiplications needed to compute the coefficients of the product of two polynomials modulo a given polynomial. Winograd proves that for a prime \( p \), the minimum number of multiplications required to compute the DFT of \( p \) elements is \( 2(p-1) - k \), where \( k \) is the number of irreducible factors of \( u^{p-1} - 1 \). The summary of results includes a table detailing the number of multiplications and additions used for small values of \( n \). For larger values of \( n \), Winograd proposes algorithms that decompose the computation using the Chinese Remainder Theorem, reducing the problem to smaller subproblems. This approach is demonstrated with examples, showing that the new algorithms can significantly reduce the computational complexity for DFT calculations.