We propose an efficient commutative group action suitable for non-interactive key exchange in a post-quantum setting. Our construction follows the layout of the Couveignes-Rostovtsev-Stolbunov cryptosystem, but we apply it to supersingular elliptic curves defined over a large prime field Fp, rather than to ordinary elliptic curves. The Diffie-Hellman scheme resulting from the group action allows for publickey validation at very little cost, runs reasonably fast in practice, and has public keys of only 64 bytes at a conjectured AES-128 security level, matching NIST's post-quantum security category I.

Abstract. We describe the design and implementation of efficient signature and key-exchange schemes for the AVR ATmega and ARM Cortex M0 microcontrollers, targeting the 128-bit security level. Our algorithms are based on an efficient Montgomery ladder scalar multiplication on the Kummer surface of Gaudry and Schost's genus-2 hyperelliptic curve, combined with the Jacobian point recovery technique of Costello, Chung, and Smith. Our results are the first to show the feasibility of software-only hyperelliptic cryptography on constrained platforms, and represent a significant improvement on the elliptic-curve state-of-the-art for both key exchange and signatures on these architectures. Notably, our key-exchange scalar-multiplication software runs in under 9740k cycles on the ATmega, and under 2650k cycles on the Cortex M0.

No abstract

A recent paper by Costello and Hisil at Asiacrypt'17 presents efficient formulas for computing isogenies with odd-degree cyclic kernels on Montgomery curves. We provide a constructive proof of a generalization of this theorem which shows the connection between the shape of the isogeny and the simple action of the point (0, 0). This generalization removes the restriction of a cyclic kernel and allows for any separable isogeny whose kernel does not contain (0, 0). As a particular case, we provide efficient formulas for 2-isogenies between Montgomery curves and show that these formulas can be used in isogeny-based cryptosystems without expensive square root computations and without knowledge of a special point of order 8. We also consider elliptic curves in triangular form containing an explicit point of order 3. Keywords: Vélu's formulas • Montgomery form • 2-isogenies • SIDH Post-quantum cryptography J. Renes-This work has been supported by the Technology Foundation STW (project 13499 -TYPHOON & ASPASIA), from the Dutch government.

No abstract