New fast Walsh–Hadamard–Hartley transform algorithm

This paper presents an efficient fast Walsh–Hadamard–Hartley transform (FWHT) algorithm that incorporates the computation of the Walsh-Hadamard transform (WHT) with the discrete Hartley transform (DHT) into an orthogonal, unitary single fast transform possesses the block diagonal structure. The proposed algorithm is implemented in an integrated butterfly structure utilizing the sparse matrices factorization approach and the Kronecker (tensor) product technique, which proved a valuable and fast tool for developing and analyzing the proposed algorithm. The proposed approach was distinguished by ease of implementation and reduced computational complexity compared to previous algorithms, which were based on the concatenation of WHT and FHT by saving up to 3N-4 of real multiplication and 7.5N-10 of real addition

Radix-2² Algorithm for the Odd New Mersenne Number Transform (ONMNT)

\ua9 2023 by the authors. This paper introduces a new derivation of the radix- (Formula presented.) fast algorithm for the forward odd new Mersenne number transform (ONMNT) and the inverse odd new Mersenne number transform (IONMNT). This involves introducing new equations and functions in finite fields, bringing particular challenges unlike those in other fields. The radix- (Formula presented.) algorithm combines the benefits of the reduced number of operations of the radix-4 algorithm and the simple butterfly structure of the radix-2 algorithm, making it suitable for various applications such as lightweight ciphers, authenticated encryption, hash functions, signal processing, and convolution calculations. The multidimensional linear index mapping technique is the conventional method used to derive the radix- (Formula presented.) algorithm. However, this method does not provide clear insights into the underlying structure and flexibility of the radix- (Formula presented.) approach. This paper addresses this limitation and proposes a derivation based on bit-unscrambling techniques, which reverse the ordering of the output sequence, resulting in efficient calculations with fewer operations. Butterfly and signal flow diagrams are also presented to illustrate the structure of the fast algorithm for both ONMNT and IONMNT. The proposed method should pave the way for efficient and flexible implementation of ONMNT and IONMNT in applications such as lightweight ciphers and signal processing. The algorithm has been implemented in C and is validated with an example