On the use of Gaussian integers in public key cryptosystems

Abstract

We present a comparative analysis of the processes of factorization of Gaussian integers and rational integers, with the objective of demonstrating the advantages of using the former instead of the latter in RSA public key cryptosystems. We show that the level of security of a cryptosystem based on the use of the product of two Gaussian primes is much higher than that of one based on the use of the product of two rational primes occupying the same storage space. Consequently, to achieve a certain specific degree of security, the use of complex Gaussian primes would require much less storage space than the use of rational primes, leading to substantial saving of expenditure. We also set forth a scheme in which rings of algebraic integers of progressively higher and higher degrees and class numbers can be used to build cryptosystems that remain secure by forever staying ahead of advances in computing power.