This book presents a unified approach to fast algorithms used in digital filters and discrete Fourier transform (DFT) evaluation. It consists of eight chapters, with the first two covering number theory and polynomial algebra. The rest of the book focuses on fast digital filtering and DFT algorithms, using polynomial algebra extensively to clarify relationships between algorithms and improve computation techniques. Chapter 3 reviews fast digital filtering algorithms, emphasizing algebraic methods and one-dimensional circular convolutions. Chapters 4 and 5 present the fast Fourier transform (FFT) and the Winograd Fourier transform algorithm. Chapters 6 and 7 introduce polynomial transforms, showing their importance in understanding multidimensional convolutions and DFTs, and in designing improved algorithms. Chapter 8 extends these concepts to one-dimensional convolutions using number theory. The book discusses the applications of convolutions and DFTs in physics and hopes to encourage further research in these areas. It also highlights the general nature of some methods, which may find new applications. The material is based on a graduate-level course at the University of Nice, France. The author thanks colleagues and reviewers for their contributions. The book includes detailed chapters on FFT algorithms, polynomial transforms, and number theoretic transforms, with a focus on efficient computation techniques and their implementation.This book presents a unified approach to fast algorithms used in digital filters and discrete Fourier transform (DFT) evaluation. It consists of eight chapters, with the first two covering number theory and polynomial algebra. The rest of the book focuses on fast digital filtering and DFT algorithms, using polynomial algebra extensively to clarify relationships between algorithms and improve computation techniques. Chapter 3 reviews fast digital filtering algorithms, emphasizing algebraic methods and one-dimensional circular convolutions. Chapters 4 and 5 present the fast Fourier transform (FFT) and the Winograd Fourier transform algorithm. Chapters 6 and 7 introduce polynomial transforms, showing their importance in understanding multidimensional convolutions and DFTs, and in designing improved algorithms. Chapter 8 extends these concepts to one-dimensional convolutions using number theory. The book discusses the applications of convolutions and DFTs in physics and hopes to encourage further research in these areas. It also highlights the general nature of some methods, which may find new applications. The material is based on a graduate-level course at the University of Nice, France. The author thanks colleagues and reviewers for their contributions. The book includes detailed chapters on FFT algorithms, polynomial transforms, and number theoretic transforms, with a focus on efficient computation techniques and their implementation.