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A trade-off between classical and quantum circuit size for an attack against CSIDH

Abstract : We propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryp-tosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order O). Let ∆ = Disc(O) (in CSIDH, ∆ = −4p for p the security parameter). Let 0 < α < 1/2, our algorithm requires: • A classical circuit of size 2Õ (log(|∆|) 1−α). • A quantum circuit of size 2Õ (log(|∆|) α). • Polynomial classical and quantum memory. Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity 2Õ (log(|∆|) 1/2) at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.
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Submitted on : Thursday, June 18, 2020 - 5:07:37 PM
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Jean-François Biasse, Xavier Bonnetain, Benjamin Pring, André Schrottenloher, William Youmans. A trade-off between classical and quantum circuit size for an attack against CSIDH. Journal of Mathematical Cryptology, De Gruyter, 2019, pp.1-16. ⟨hal-02423394⟩

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