Design and Acceleration of CSIDH-Like Post-Quantum Key Exchange Protocols

The rapid development of quantum computing poses a threat to existing public-key cryptographic systems, and isogeny-based cryptography has become an important component in advancing post-quantum cryptographic research. The CSIDH protocol, owing to its balance of security and efficiency, is one of the few practical isogeny-based key exchange protocols available today. This paper investigates and accelerates CSIDH-like protocols by utilizing Montgomery supersingular elliptic curves defined over a prime field with p≡7 (mod 8), whose endomorphism ring is isomorphic to $\mathcal{O}=\mathbb{Z}[\sqrt{p}]$. And a new isogeny-based key exchange protocol called CSIDH-G is constructed. The paper establishes a one-to-one correspondence between the coefficients of Montgomery supersingular elliptic curves and their isomorphism classes, hereby preserving the advantage of low public key size. Furthermore, it is proved that the proposed protocol avoids the key collisions of order 3 present in CSIDH and those of order 1 in CSURF, and a compact interval for user private key selection is proposed. Finally, experiments on protocol efficiency are conducted. By optimizing the key selection interval, specifically, by expanding the exponent selection ranges for low-degree isogenies such as 3-isogenies and 5-isogenies, the proposed protocol achieves about a 5.95% improvement in operational efficiency over the original CSIDH protocol, validating its accuracy and effectiveness. This algorithm offers multiple advantages, including improved efficiency of isogeny-based key exchange protocols, simplicity in the public key verification process, avoidance of private key collisions existing in current CSIDH-like protocols, and increased diversity in Diffie-Hellman-style post-quantum cryptographic systems.