## Post-Quantum CryptographyDaniel J. Bernstein, Johannes Buchmann, Erik Dahmen Quantum computers will break today's most popular public-key cryptographic systems, including RSA, DSA, and ECDSA. This book introduces the reader to the next generation of cryptographic algorithms, the systems that resist quantum-computer attacks: in particular, post-quantum public-key encryption systems and post-quantum public-key signature systems. Leading experts have joined forces for the first time to explain the state of the art in quantum computing, hash-based cryptography, code-based cryptography, lattice-based cryptography, and multivariate cryptography. Mathematical foundations and implementation issues are included. This book is an essential resource for students and researchers who want to contribute to the field of post-quantum cryptography. |

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### Contents

1 | |

6 | |

3 Challenges in postquantum cryptography | 11 |

4 Comparison to quantum cryptography | 13 |

Quantum computing | 15 |

3 The quantum Fourier transform | 22 |

4 The hidden subgroup problem | 25 |

5 Search algorithms | 29 |

4 Codes and structures | 116 |

5 Practical aspects | 127 |

6 Annex | 137 |

References | 141 |

Latticebased Cryptography | 147 |

2 Preliminaries | 152 |

3 Finding Short Vectors in Random qary Lattices | 154 |

4 Hash Functions | 157 |

6 Outlook | 31 |

References | 32 |

Hashbased Digital Signature Schemes | 35 |

1 Hash based onetime signature schemes | 36 |

2 Merkles tree authentication scheme | 40 |

3 Onetime keypair generation using an PRNG | 44 |

4 Authentication path computation | 46 |

5 Tree chaining | 69 |

6 Distributed signature generation | 73 |

7 Security of the Merkle Signature Scheme | 81 |

References | 91 |

Codebased cryptography | 94 |

2 Cryptosystems | 96 |

3 The security of computing syndromes as oneway function | 106 |

5 Public Key Encryption Schemes | 165 |

6 Digital Signature Schemes | 180 |

7 Other Cryptographic Primitives | 185 |

8 Open Questions | 186 |

References | 187 |

Multivariate Public Key Cryptography | 192 |

2 The Basics of Multivariate PKCs | 194 |

3 Examples of Multivariate PKCs | 198 |

4 Basic Constructions and Variations | 202 |

5 Standard Attacks | 215 |

6 The Future | 229 |

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242 | |