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Isogenies and cryptography

open access: yes
Cryptography is an essential tool for enabling privacy of electronic communication. However, many of the cryptographic protocols we use nowadays, would be insecure after the development of large-scale quantum computers. Therefore, the cryptographic community is preparing new protocols to replace the would-be broken ones. Protocols based on lattices are
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Horizontal isogeny theorems [PDF]

open access: possibleForum Mathematicum, 2002
Summary: Let \(K\) be a field which is finitely generated over its prime field. Consider elliptic curves \(E\) and \(E'\) defined over \(K\). Suppose there exists \(c\geq 1\) and a set \(\Lambda\) of prime numbers such that \([K(E_l,E_l'): K(E_l)\cap K(E_l')]\leq c\) for all \(l\in\Lambda\).
Gerhard Frey, Moshe Jarden
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