## A Course in Computational Algebraic Number TheoryWith the advent of powerful computing tools and numerous advances in math ematics, computer science and cryptography, algorithmic number theory has become an important subject in its own right. Both external and internal pressures gave a powerful impetus to the development of more powerful al gorithms. 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 appli cations, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc ... Several books exist which treat parts of this subject. (It is essentially impossible for an author to keep up with the rapid pace of progress in all areas of this subject.) Each book emphasizes a different area, corresponding to the author's tastes and interests. The most famous, but unfortunately the oldest, is Knuth's Art of Computer Programming, especially Chapter 4. The present book has two goals. First, to give a reasonably comprehensive introductory course in computational number theory. In particular, although we study some subjects in great detail, others are only mentioned, but with suitable pointers to the literature. Hence, we hope that this book can serve as a first course on the subject. A natural sequel would be to study more specialized subjects in the existing literature. |

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### Contents

1 | |

Algorithms for Linear Algebra and Lattices | 45 |

Algorithms on Polynomials | 108 |

Algorithms for Algebraic Number Theory I | 151 |

Number Theory | 214 |

Algorithms for Quadratic Fields | 218 |

Algorithms for Algebraic Number Theory II | 297 |

Introduction to Elliptic Curves | 360 |

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### Common terms and phrases

algebraic integer algebraic number algorithm computes assume Chapter class group class number coefficients congruence coprime corresponding cubic fields defined definition degree denote discriminant dividing division divisor element elliptic curve equal equation equivalent Euclidean step example Exercise exists factor base finite following algorithm formula fractional ideal function fundamental discriminant Galois group give given go to step hence Hermite normal form Initialize integral basis inverse irreducible polynomial isomorphic kernel lattice Legendre symbol Lemma linear LLL algorithm matrix method minimal polynomial modulo monic multi-precision multiplication non-trivial non-zero norm notations Note number field number theory obtain Otherwise output prime ideals prime number primitive problem Proof Proposition prove quadratic fields quadratic form real numbers reduced form resp result roots of unity Section sieve square squarefree Sub-algorithm subfield terminate the algorithm Theorem trivial unique Z-basis Z-module Z/NZ