Compact Isogeny PQC can replace HD wallets, key-tweaking, silent payments

Isogenies represent a foundational concept in the realm of elliptic curve cryptography, characterized as group homomorphisms that facilitate the mapping from one elliptic curve to another while preserving the group structure. This intrinsic property, where the image of the sum of two points is equivalent to the sum of the images of these points, underscores the homomorphic nature of isogenies. Such a characteristic is not merely a theoretical requirement but serves a practical utility in cryptographic protocols, particularly in zero-knowledge proofs of knowledge (zkpok). In zkpok, the need for a one-way function that possesses a homomorphic quality is paramount, highlighting the relevance of isogenies in this domain.

Furthermore, the discussion extends into the territory of dual isogenies and their role in ensuring the soundness of cryptographic proofs. The notion that duals act as inverses, although not strictly in all contexts, plays a critical role in formulating soundness proofs. Specifically, the concept of 2-special soundness in cryptographic protocols benefits from this property. By examining specific literature, such as the document found at arXiv, it becomes evident that the utilization of dual isogenies in soundness proofs parallels the application of inverses in discrete logarithm-based cryptography. This intricate relationship between dual isogenies and the mechanism of soundness proofs underscores the nuanced understanding required to navigate the complexities of modern cryptographic practices.