## Mathematics of Public Key CryptographyPublic key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more. |

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### Contents

1 Introduction | 1 |

Part I Background | 11 |

Part II Algebraic groups | 59 |

Part III Exponentiation factoring and discrete logarithms | 213 |

Part IV Lattices | 335 |

Part V Cryptography related todiscrete logarithms | 403 |

Part VI Cryptography related tointeger factorisation | 483 |

VII Advanced topics in elliptic and hyperelliptic curves | 513 |

Appendix A Background mathematics | 564 |

579 | |

603 | |

608 | |

### Common terms and phrases

adversary algebraic group algebraic set attack basis bit operations char(k Chinese remainder theorem choose ciphertext collision complexity computational problem consider Corollary cryptography cryptosystem curve over 1k decryption deﬁned Deﬁnition deg(u(x degree Diffie–Hellman discrete logarithm problem distinguished point divisor class group E(Fq efficient Elgamal Elgamal encryption elliptic curve encryption equation Example Exercise expected number exponentiation factor ﬁeld finite fields ﬁrst Fixed-CDH follows Frobenius genus given gives graph group elements group operations hash function Hence heuristic homomorphism hyperelliptic curve IND-CCA input integer irreducible isogeny isomorphism kangaroo lattice Lemma Let g matrix method modulo morphism multiplication Note output pairing polynomial polynomial-time prime order probability Proof Prove public key quadratic random oracle randomised rational map reduction requires result roots Schnorr Schnorr signatures Section Show subgroup supersingular Suppose Theorem uniformiser uniformly at random values vector walk Weierstrass equation write Z/NZ