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Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. (English) Zbl 1372.94419

Summary: We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the non-commutativity of the endomorphism ring. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of PQCrypto 2011 [Lect. Notes Comput. Sci. 7071, 19–34 (2011; Zbl 1290.94094)]. We add a new zero-knowledge identification scheme and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.

MSC:

94A60 Cryptography
14G50 Applications to coding theory and cryptography of arithmetic geometry
11Y16 Number-theoretic algorithms; complexity
14K02 Isogeny

Citations:

Zbl 1290.94094