This paper presents new algorithms for computing the Discrete Fourier Transform (DFT) of n points, which use significantly fewer multiplications than the previously known algorithm, especially for n in the range of a few tens to a few thousands. The DFT is defined as $ A_{j}=\sum_{i=0}^{n-1}w^{i j}a_{i},j=0,\ldots,n-1,w=e^{\frac{2\pi i}{n}} $. In 1965, Cooley and Tukey proposed an algorithm for computing DFT in $ n/2 \log_{2} n/2 $ complex multiplications and $ n \log_{2} n $ complex additions when n is a power of 2. The new algorithm described here uses fewer multiplications and about the same number of additions. Theoretical background shows that the minimum number of multiplications needed to compute the coefficients of $ T_{p}=R_{n}\cdot S_{n} $ mod $ P_{n} $ is $ 2n - k $, where k is the number of irreducible factors of $ P_{n} $. For prime p, computing the DFT of p elements involves computing $ \bar{A}_{k} = \sum_{j=1}^{p-1} w^{kj} a_{j} $, which can be done in $ 2(p-1)-k $ multiplications. Similar results are obtained when n is a power of a prime. The paper summarizes results showing that known algorithms for computing DFT in the minimum number of multiplications require many additions when P has large irreducible factors. Therefore, these algorithms are not practical unless n is small. Table 1 summarizes the number of multiplications and additions for small n. The paper also discusses how to compute DFT for larger n by decomposing it into smaller DFTs. For example, computing the DFT of 21 points involves computing the DFT of 3 points and substituting for each addition the addition of 7 dimensions. Table 2 provides examples of the complexity of the new algorithm. The new algorithm uses fewer multiplications and additions than the Fast Fourier Transform for n in the range of a few tens to a few thousands.This paper presents new algorithms for computing the Discrete Fourier Transform (DFT) of n points, which use significantly fewer multiplications than the previously known algorithm, especially for n in the range of a few tens to a few thousands. The DFT is defined as $ A_{j}=\sum_{i=0}^{n-1}w^{i j}a_{i},j=0,\ldots,n-1,w=e^{\frac{2\pi i}{n}} $. In 1965, Cooley and Tukey proposed an algorithm for computing DFT in $ n/2 \log_{2} n/2 $ complex multiplications and $ n \log_{2} n $ complex additions when n is a power of 2. The new algorithm described here uses fewer multiplications and about the same number of additions. Theoretical background shows that the minimum number of multiplications needed to compute the coefficients of $ T_{p}=R_{n}\cdot S_{n} $ mod $ P_{n} $ is $ 2n - k $, where k is the number of irreducible factors of $ P_{n} $. For prime p, computing the DFT of p elements involves computing $ \bar{A}_{k} = \sum_{j=1}^{p-1} w^{kj} a_{j} $, which can be done in $ 2(p-1)-k $ multiplications. Similar results are obtained when n is a power of a prime. The paper summarizes results showing that known algorithms for computing DFT in the minimum number of multiplications require many additions when P has large irreducible factors. Therefore, these algorithms are not practical unless n is small. Table 1 summarizes the number of multiplications and additions for small n. The paper also discusses how to compute DFT for larger n by decomposing it into smaller DFTs. For example, computing the DFT of 21 points involves computing the DFT of 3 points and substituting for each addition the addition of 7 dimensions. Table 2 provides examples of the complexity of the new algorithm. The new algorithm uses fewer multiplications and additions than the Fast Fourier Transform for n in the range of a few tens to a few thousands.