Montgomery Arithmetic over Gaussian Integers

Abstract

Up to now, we have demonstrated that Gaussian integers are suitable for RSA and ECC systems. Moreover, we have illustrated that performing complex point multiplications over Gaussian integers is advantageous regarding robustness side channel attacks, computational complexity, and memory requirements. However, calculating the modular arithmetic over Gaussian integers is not trivial. Determining the modulo reduction for Gaussian integers is very computational intensive according to (3.1). In [37, 38], a new arithmetic approach was proposed based on Montgomery modular arithmetic over Gaussian integers. Due to the independent computation of real and imaginary parts of Gaussian integers, this principal reduces the hardware complexity of arithmetic operations, especially of the Montgomery reduction. It was shown in [37, 38] that a significant complexity reduction can be achieved for the RSA cryptographic system using Gaussian integers. Such a cryptographic system is presented in [103, 104]. Similarly, a Rabin cryptographic system was previously considered over Gaussian integers in [105, 106]. Moreover, coding applications over Gaussian integers [18, 19, 40, 107–109] could also benefit from an algorithm for the Montgomery reduction of Gaussian integers.