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

The discussion delves into the intricacies of constructing zero-knowledge proofs of knowledge (zkpok) through the lens of cryptographic protocols and their applicability in various cryptographic landscapes, particularly emphasizing the transition from elliptic curve cryptography to isogeny-based systems. The foundational concept hinges on the idea that any one-way mapping endowed with a homomorphic property between groups can facilitate the creation of a zkpok. This principle, rooted in the general meta-argument, suggests that by starting with an instance of the "language," which comprises a public key and its corresponding private key, one can leverage the map's one-way and homomorphic characteristics to bind a secret to a challenge value. However, this process, while binding, does not inherently conceal, thus not achieving true zero-knowledge—a limitation circumvented by division due to the public nature of the challenge.

Further exploration reveals how Pedersen commitments exemplify this principle through their structure ($C = xG + rH$) and properties, illustrating the feasibility of constructing commit-challenge-response protocols that yield signatures. This exploration extends to the realm of isogenies, where the traditional notions of homomorphism transform into a form of composable maps, notably through function (isogeny) composition, which diverges from the commutative operations observed in abelian groups. Despite these differences, the concept of "duals" in isogenies introduces an analogous mechanism to inverses, albeit with distinctions such as the invertibility of ideals in quaternion algebra, indicating a nuanced shift in applying these principles to isogeny-based systems.

The discourse transitions towards the practical implications and considerations for adopting post-quantum cryptographic primitives, underscoring the importance of evaluating the properties of these primitives beyond mere functionality. The emphasis on 'rerandomization,' 'aggregatability,' and 'batch verifiability' as critical attributes reflects a broader concern regarding the selection of cryptographic foundations for future-proofing security protocols. The potential challenges of industry adoption, especially when contrasted with the established deployment of elliptic curve cryptography, highlight the need for a holistic assessment of practical support and experience in integrating these advanced cryptographic techniques. This reflection points to the necessity of a meticulous and forward-thinking approach in the evolution of cryptographic standards, particularly in the context of preparing for post-quantum scenarios.