Post-Quantum Cryptography Using Supersingular Isogenies

Supersingular isogeny-based cryptography exploits the mathematical structure of certain elliptic curves to establish secure communication resistant to quantum attacks. Unlike classical schemes grounded in integer factorisation or discrete logarithms, isogeny protocols hinge on the difficulty of finding morphisms—known as isogenies—between supersingular elliptic curves. A public parameter is an elliptic curve defined over a prime field; a secret isogeny between this curve and a partner curve acts as the cryptographic key. The resulting protocols, notably Diffie–Hellman key exchange adaptations and key encapsulation mechanisms, offer remarkably small public keys and ciphertexts, making them attractive for bandwidth-constrained and embedded environments. Recent theoretical advances have improved isogeny evaluation algorithms, refined security estimates against quantum-algorithmic collisions and extended the commutative group actions to broader prime structures. In parallel, implementation research has addressed performance overhead via software optimisation on CPUs and GPUs, hardware accelerators on field-programmable gate arrays and application-specific integrated circuits, and constant-time algorithms to thwart side-channel attacks. Together, these developments chart a path towards practical deployment of supersingular isogeny cryptosystems as key components of post-quantum security standards.

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Post-Quantum Cryptography Using Supersingular Isogenies publication trend

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Technical terms

Supersingular elliptic curve: An elliptic curve over a finite field with maximal endomorphism ring complexity, used for isogeny-based protocols.

Isogeny: A structure-preserving map between elliptic curves, whose computation is hard in the supersingular case.

SIDH: Supersingular Isogeny Diffie–Hellman, a key-exchange protocol using paired isogenies on supersingular curves.

SIKE: Supersingular Isogeny Key Encapsulation, an encapsulation mechanism derived from SIDH for secure key transport.

Montgomery multiplication: An efficient algorithm for modular multiplication used in high-performance implementations.

References

1. A Compact and Scalable Hardware/Software Co-design of SIKE. IACR Transactions on Cryptographic Hardware and Embedded Systems (2020).
2. SIKE on GPU: Accelerating Supersingular Isogeny-Based Key Encapsulation Mechanism on Graphic Processing Units. IEEE Access (2021).
3. Towards High-Performance Supersingular Isogeny Cryptographic Hardware Accelerator Design. Electronics (2023).