Efficient Reduction Algorithms for Special Gaussian Integer Moduli

Gaussian integers are a subset of the complex numbers with integers as real and imaginary parts. When Gaussian integers are equipped with modulo operations, they form Gaussian integer rings or fields, depending on the specific choice of the modulus. Arithmetic on Gaussian integers can offer advantages in terms of operand size and improved parallelism, due to independent calculation of the real and imaginary parts. However, although Gaussian integer modulo reduction is the fundamental operation to enable computations in finite Gaussian integer rings and fields, efficient algorithms for Gaussian integer modulo reduction have not been widely investigated so far. In this work, we fill this gap and present efficient reduction algorithms for Gaussian integer moduli of special forms. Indeed, we demonstrate that there exist different classes of Gaussian integer moduli allowing for very fast reductions. Finally, we show that the computational complexity of the proposed algorithm is significantly reduced compared with generic Gaussian integer reduction methods known to date, e.g., Montgomery-based reduction for Gaussian integers.