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

These methods utilize mappings between elliptic curves to create space-efficient solutions for cryptographic keys and signatures, often requiring less than 300 bytes. This efficiency is particularly significant when compared to lattice crypto schemes, which typically require several kilobytes. Isogenies, defined as functions that map points between different elliptic curves, and endomorphisms, which map points within the same curve, are pivotal in this approach. The challenges central to isogeny cryptography include the supersingular isogeny path problem (SIPP) and determining the endomorphism ring of a curve without explicit knowledge, known as the endomorphism ring problem (ERP). These problems are complex but crucial for generating secure cryptographic keys through the computation of secret isogenies and corresponding endomorphism rings.

The practical applications of isogeny-based cryptography extend into various areas including signature schemes and enhanced privacy measures in bitcoin transactions such as unhardened BIP32 key derivation, silent payments, and taproot-style key tweaking. However, the development of multisignature schemes remains a challenge. Engaging more bitcoin developers in this field could potentially accelerate advancements and lead to more robust cryptographic systems. For those interested in deeper insights into these topics, resources like DelvingBitcoin.org and Conduition.io provide detailed discussions.

Zero-knowledge proofs of knowledge (zkpok) leverage cryptographic protocols that utilize homomorphic properties between groups, facilitated by mappings like isogenies. While these mappings enable binding secrets to challenge values, achieving true zero-knowledge requires further manipulation due to the public nature of challenges. Pedersen commitments exemplify how commit-challenge-response protocols can be constructed to yield signatures. The transition from classical elliptic curve operations to isogeny-based systems highlights differences such as the non-commutative nature of isogeny composition compared to abelian group operations. The discourse also emphasizes critical attributes for post-quantum cryptographic primitives like rerandomization, aggregatability, and batch verifiability, underlining the necessity of evaluating these properties when advancing cryptographic standards.

Isogeny-based cryptography's theoretical foundation and practical implementations, such as CSIDH and SIDH, reveal unique approaches to non-interactive key exchange despite inherent structural challenges. Discussions also cover dual isogenies and their role in solving the isogeny path problem and the potential for integrating advanced cryptographic properties like signature aggregation or batch verification using high-dimensional isogenies. Notable efforts in this direction include the SQIsignHD, which explores the trade-offs involved in enhancing the applicability and efficiency of isogeny-based cryptography.

In summary, while isogeny-based cryptography offers promising alternatives to traditional methods, it faces significant theoretical and practical challenges. The ongoing research and development in this field are crucial for overcoming these obstacles and ensuring the robustness of cryptographic systems against emerging threats, including those posed by quantum computing technologies.