Cryptology ePrint Archive: Report 2019/1202

Rational isogenies from irrational endomorphisms

Wouter Castryck and Lorenz Panny and Frederik Vercauteren

Abstract: In this paper, we introduce a polynomial-time algorithm to compute a connecting $\mathcal{O}$-ideal between two supersingular elliptic curves over $\mathbb{F}_p$ with common $\mathbb{F}_p$-endomorphism ring $\mathcal{O}$, given a description of their full endomorphism rings. This algorithm provides a reduction of the security of the CSIDH cryptosystem to the problem of computing endomorphism rings of supersingular elliptic curves. A similar reduction for SIDH appeared at Asiacrypt 2016, but relies on totally different techniques. Furthermore, we also show that any supersingular elliptic curve constructed using the complex-multiplication method can be located precisely in the supersingular isogeny graph by explicitly deriving a path to a known base curve. This result prohibits the use of such curves as a building block for a hash function into the supersingular isogeny graph.

Category / Keywords: public-key cryptography / Isogeny-based cryptography, endomorphism rings, CSIDH

Original Publication (in the same form): IACR-EUROCRYPT-2020

Date: received 14 Oct 2019, last revised 9 Mar 2020

Contact author: wouter castryck at esat kuleuven be, lorenz at yx7 cc, frederik vercauteren at esat kuleuven be

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Version: 20200309:215925 (All versions of this report)

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