isogenybased

Isogenybased refers to cryptographic constructions that derive security from isogenies between elliptic curves. In these schemes, the main hard problems involve computing isogenies or walking between curves in an isogeny graph. This approach is studied as a candidate for post-quantum cryptography because many standard quantum attacks on elliptic-curve discrete logarithms do not translate directly to isogeny-based problems. Isogeny-based cryptography is used for key exchange, key encapsulation, and, in some schemes, digital signatures.

An isogeny is a non-constant morphism between elliptic curves that preserves the group structure. Cryptographers typically work with supersingular elliptic curves over finite fields, where the set of all isogenies forms a rich, connectivity-rich graph. The difficulty of constructing a path between two curves, or of determining an isogeny with given endpoints, underlies the security of many schemes.

Two well-known families are SIDH (and its encapsulation form SIKE) and CSIDH. SIDH uses the structure of supersingular isogenies to allow two parties to derive a common secret with commutative isogeny steps. CSIDH instead relies on the action of a class group on the set of elliptic curves, performing exponentiations in a group to arrive at a shared secret.

Security and status: Isogeny-based schemes are believed to be quantum-resistant; however, several proposed parameter sets have been undermined by practical attacks, and implementations face performance and side-channel considerations. As a result, isogenybased schemes have not achieved broad deployment comparable with lattice-based or code-based post-quantum schemes.

Outlook: Ongoing research explores stronger security proofs, more efficient implementations, and potential standardization pathways. The field remains active, with new constructions and specialty optimizations being published as quantum-era cryptography evolves.