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

These methods utilize the principles of mapping points between elliptic curves through isogenies and endomorphisms, offering smaller key and signature sizes which are highly beneficial in space-constrained applications. The focus of recent research has been on making isogeny-based cryptography accessible to bitcoin engineers, leveraging their familiarity with classical Elliptic Curve Cryptography (ECC). Isogenies play a crucial role in this context, enabling mappings between different elliptic curves or within the same curve, a process central to the development of secure cryptographic keys and processes resilient against quantum computing threats.

The challenges in isogeny crypto, such as the supersingular isogeny path problem (SIPP) and determining the endomorphism ring of a curve without explicit knowledge, underscore the difficulty and complexity inherent in these cryptographic methods. However, the ability to generate secret isogenies and compute corresponding endomorphism rings offers a promising avenue for secure communication in the quantum era. The practical applications of isogeny-based cryptography are varied, including signature schemes and key derivation protocols, yet the development of compact multisignature schemes remains a significant challenge.

In addressing zero-knowledge proofs of knowledge (zkpok) within cryptographic protocols, the transition from ECC to isogeny-based systems introduces a nuanced understanding of homomorphic properties and their application in cryptographic proofs. The discussion extends to the practical implications of adopting post-quantum cryptographic primitives, emphasizing attributes such as rerandomization, aggregatability, and batch verifiability. These considerations highlight the need for a meticulous approach in evaluating and integrating advanced cryptographic techniques, especially in the context of blockchain technologies and ensuring security against quantum-enabled adversaries.

Further correspondence delves into the specifics of isogeny-based cryptography, detailing the importance of selecting specific isogenies as secret keys and the distinction between isogeny composition and traditional point addition in ECC. Various implementations and adaptations of isogeny-based schemes are discussed, alongside the theoretical underpinnings and practical applications of dual isogenies. The dialogue also explores the comparison between BIP340 Schnorr signatures and isogeny-based signatures, highlighting differences in commitment mechanisms and security paradigms. Despite the exploration of desirable cryptographic properties and the potential for signature aggregation or batch verification, the feasibility of incorporating these features into isogeny-based protocols remains an area of active research.

Moreover, the correspondence touches upon the foundational aspects of isogenies in cryptographic protocols, particularly in the construction of zero-knowledge proofs of knowledge. The discussion highlights the critical role of dual isogenies in formulating soundness proofs and the conceptual shift required to apply these principles effectively in isogeny-based systems. Despite the complexities involved, the ongoing dialogue and research efforts aim to advance the field of post-quantum cryptography by exploring innovative approaches and overcoming the challenges associated with isogeny-based cryptographic methods.