## Abstract

CSIDH is an isogeny-based key exchange protocol proposed by Castryck, Lange, Martindale, Panny, and Renes in 2018. CSIDH is based on the ideal class group action on 𝔽

In this paper, we prove a number of theorems on the properties of Edwards curves. By using these theorems, we extend the CSIDH algorithm to that on Edwards curves with w-coordinates over 𝔽

_{p}-isomorphism classes of Montgomery curves. In order to calculate the class group action, we need to take points defined over 𝔽_{p}^{2}. The original CSIDH algorithm requires a calculation over 𝔽_{p}by representing points as x-coordinate over Montgomery curves. Meyer and Reith proposed a faster CSIDH algorithm in 2018 which calculates isogenies on Edwards curves by using a birational map between a Montgomery curve and an Edwards curve. There is a special coordinate on Edwards curves (the w-coordinate) to calculate group operations and isogenies. If we try to calculate the class group action on Edwards curves by using the w-coordinate in a similar way on Montgomery curves, we have to consider points defined over 𝔽_{p}^{4}. Therefore, it is not a trivial task to calculate the class group action on Edwards curves with w-coordinates over only 𝔽_{p}.In this paper, we prove a number of theorems on the properties of Edwards curves. By using these theorems, we extend the CSIDH algorithm to that on Edwards curves with w-coordinates over 𝔽

_{p}. This algorithm is as fast as (or a little bit faster than) the algorithm proposed by Meyer and Reith.Original language | English |
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Title of host publication | Topics in Cryptology – CT-RSA 2020 |

Subtitle of host publication | The Cryptographers’ Track at the RSA Conference 2020, San Francisco, CA, USA, February 24–28, 2020, Proceedings |

Editors | Stanislaw Jarecki |

Publisher | Springer |

Pages | 512–537 |

Number of pages | 26 |

Edition | 1 |

ISBN (Electronic) | 9783030401863 |

ISBN (Print) | 9783030401856 |

DOIs | |

Publication status | Published - 30 Jan 2020 |

### Publication series

Name | Lecture Notes in Computer Science |
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Publisher | Springer |

Volume | 12006 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

## Keywords

- Isogeny-based cryptography
- Montgomery curves
- Edwards curves
- CSIDH
- Post-quantum cryptography