## Elliptic Curve Public Key CryptosystemsElliptic curves have been intensively studied in algebraic geometry and number theory. In recent years they have been used in devising efficient algorithms for factoring integers and primality proving, and in the construction of public key cryptosystems. Elliptic Curve Public Key Cryptosystems provides an up-to-date and self-contained treatment of elliptic curve-based public key cryptology. Elliptic curve cryptosystems potentially provide equivalent security to the existing public key schemes, but with shorter key lengths. Having short key lengths means smaller bandwidth and memory requirements and can be a crucial factor in some applications, for example the design of smart card systems. The book examines various issues which arise in the secure and efficient implementation of elliptic curve systems. Elliptic Curve Public Key Cryptosystems is a valuable reference resource for researchers in academia, government and industry who are concerned with issues of data security. Because of the comprehensive treatment, the book is also suitable for use as a text for advanced courses on the subject. |

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### Contents

III | 1 |

IV | 3 |

V | 4 |

VI | 5 |

VII | 6 |

VIII | 7 |

IX | 10 |

X | 13 |

XXXVIII | 62 |

XXXIX | 63 |

XL | 66 |

XLI | 68 |

XLII | 69 |

XLIII | 72 |

XLIV | 77 |

XLVI | 79 |

XI | 14 |

XII | 15 |

XIII | 17 |

XIV | 19 |

XV | 20 |

XVI | 21 |

XVII | 23 |

XVIII | 28 |

XIX | 32 |

XX | 34 |

XXI | 35 |

XXII | 37 |

XXIII | 39 |

XXIV | 40 |

XXV | 41 |

XXVI | 46 |

XXVII | 48 |

XXVIII | 49 |

XXIX | 50 |

XXX | 51 |

XXXI | 52 |

XXXII | 54 |

XXXIV | 55 |

XXXV | 57 |

XXXVI | 59 |

XXXVII | 61 |

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### Common terms and phrases

abelian group Addition Formula admissible changes arithmetic baby-step giant-step change of variables Chapter classes of elliptic computing logarithms coordinates curve logarithm problem curve y2 curves isomorphic curves over F2m cyclic group defined over Fq denote determine discrete logarithm problem division polynomials divisor efficient eigenvalue ElGamal cryptosystem elliptic curve cryptosystems elliptic curve defined elliptic curve logarithm encryption exists F2iss field elements field Fq fields of characteristic finite field group G group of order group operation Hence index calculus method inversion ISBN isomorphism classes Lemma mod fi(x mod q modulo non-supersingular curve normal basis odd prime ord(P pair point of order private key probabilistic polynomial public key cryptography public key cryptosystems random integer rational function reduced Schoof's algorithm Section signature scheme solutions subexponential supersingular curves supersingular elliptic curves Theorem Type II curves verify Weierstrass equation Weil pairing x-coordinate