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With the advent of powerful computing tools and numerous advances in mathematics, computer science and cryptography, algorithmic number theory has become an important subject in its own right. Both external and interna1 pressures gave a powerful impetus to the development of more powerful algorithms. These in turn led to a large number of spectacular breakthroughs. To mention but a few, the LLL algorithm which has a wide range of applications, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc . . .