Abstract
Cryptosystems based on the discrete logarithm problem in the infrastructure of a real quadratic number field [7],[19],[2] are very interesting from a theoretical point of view, because this problem is known to be at least as hard as, and when considering todays algorithms - as in [11] - much harder than, factoring integers. However it seems that the cryptosystems sketched in [2] have not been implemented yet and consequently it is hard to evaluate the practical relevance of these systems. Furthermore as [2] lacks any proofs regarding the involved approximation precisions, it was not clear whether the second communication round, as required in [7],[19], really could be avoided without substantial slowdown. In this work we will prove a bound for the necessary approximation precision of an exponentiation using quadratic numbers in power product representation and show that the precision given in [2] can be lowered considerably. As the highly space consuming power products can not be applied in environments with limited RAM, we will propose a simple (CRIAD1-) arithmetic which entirely avoids these power products. Beside the obvious savings in terms of space this method is also about 30% faster. Furthermore one may apply more sophisticated exponentiation techniques, which finally result in a ten-fold speedup compared to [2]. CRIAD is an abbreviation for Close Reduced Ideal and Approximated relative Distance
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Hühnlein, D., Paulus, S. (2001). On the Implementation of Cryptosystems Based on Real Quadratic Number Fields (Extended Abstract). In: Stinson, D.R., Tavares, S. (eds) Selected Areas in Cryptography. SAC 2000. Lecture Notes in Computer Science, vol 2012. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44983-3_21
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DOI: https://doi.org/10.1007/3-540-44983-3_21
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