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In this work we look at cryptographic schemes that§are based on the §mathematics of quadratic number fields. These schemes§were §proposed by Johannes Buchmann and Hugh Williams in§the late 1980s. §After summarizing the necessary background, we§describe the actual §schemes and give some reasons why we think that they§are secure at §present. Finally, we investigate what happens with§the security once we §allow quantum computations into the picture, where we§mention some §recent results. §While the present work contains nothing new and is§completely §expository, we hope that it will provide a good§summary of this §exciting subject. We believe that giving the reader a§feel for the big picture will motivate him or her to go the local§library armed with the §present book go ahead and read the more technical§papers on the §subject.
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