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

The correspondence delves into the complexities and nuances of isogeny-based cryptography (IBC), emphasizing the foundational role of isogenies in constructing one-way functions necessary for secure cryptographic protocols. It explains that the base curve, $E_0$, plays a critical role because its endomorphism ring, $\text{End}(E_0)$, is a known constant which is vital for computing and concealing the endomorphism ring of another curve, $\text{End}(E)$. This highlights the necessity of selecting specific isogenies as secret keys to ensure computational feasibility and security.

Further discussion reveals that isogenies, while being group homomorphisms facilitating the addition of mapped points, diverge from classical elliptic curve cryptography (ECC) in their inability to be composed in a straightforward manner akin to point addition in ECC. This distinction underpins the innovation within IBC, particularly through the concept of composable maps or isogeny composition, albeit with limitations due to the complex nature of isogeny degrees and the absence of a direct analogy to abelian group operations.

The correspondence also touches upon different implementations and adaptations of isogeny-based schemes such as CSIDH and SIDH, noting their unique approaches to achieving non-interactive key exchange and the inherent challenges posed by their structure, like the security vulnerabilities in SIDH exposed by Kani's lemma. The discussion extends to the innovative aspects of dual isogenies and their role in navigating the isogeny path problem, offering insights into the practical applications and theoretical underpinnings of IBC.

In terms of cryptographic proofs and security assurances, similarities between BIP340 Schnorr signatures and SQIsign are drawn, highlighting the application of fiat-shamir transforms and sigma protocols to establish correctness, special soundness, and honest-verifier zero-knowledge properties. Yet, PRISM is distinguished by its less conventional approach, lacking commitment mechanisms and thus presenting a different security paradigm.

A pivotal focus of the dialogue is the exploration of desirable cryptographic properties such as rerandomization, aggregation, and batch verification, especially in the context of blockchain technologies. Despite the recognition of these features' importance, the feasibility of integrating them into IBC protocols remains uncertain. The potential for signature aggregation or batch verification within IBC, possibly through advanced techniques involving high-dimensional isogenies, points to an ongoing area of research. A referenced endeavor in this direction is showcased by SQIsignHD, which demonstrates a method for compressing signatures at the expense of verifier performance, illustrating the trade-offs and exploratory nature of current research in enhancing IBC's applicability and efficiency.