Spectral modular arithmetic

Graduation date: 2006In many areas of engineering and applied mathematics, spectral methods provide\ud very powerful tools for solving and analyzing problems. For instance, large to\ud extremely large sizes of numbers can efficiently be multiplied by using discrete Fourier\ud transform and convolution property. Such computations are needed when computing\ud π to millions of digits of precision, factoring and also big prime search projects.\ud When it comes to the utilization of spectral techniques for modular operations\ud in public key cryptosystems two difficulties arise; the first one is the reduction needed\ud after the multiplication step and the second is the cryptographic sizes which are much\ud shorter than the optimal asymptotic crossovers of spectral methods.\ud In this dissertation, a new modular reduction technique is proposed. Moreover,\ud modular multiplication is given based on this reduction. These methods work fully\ud in the frequency domain with some exceptions such as the initial, final and partial\ud transformations steps. Fortunately, the new technique addresses the reduction problem\ud however, because of the extra complexity coming from the overhead of the forward and\ud backward transformation computations, the second goal is not easily achieved when\ud single operations such as modular multiplication or reduction are considered. On the\ud contrary, if operations that need several modular multiplications with respect to the\ud same modulus are considered, this goal is more tractable.\ud An obvious example of such an operation is the modular exponentiation i.e., the\ud computation of c=m[superscript e] mod n where c, m, e, n are large integers. Therefore following\ud the spectral modular multiplication operation a new modular exponentiation method is\ud presented. Since forward and backward transformation calculations do not need to be\ud performed for every multiplication carried during the exponentiation, the asymptotic\ud crossover for modular exponentiation is decreased to cryptographic sizes. The method\ud yields an efficient and highly parallel architecture for hardware implementations of\ud public-key cryptosystems