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arXiv:1805.06589

Supersingular Isogeny Oblivious Transfer (SIOT)

Paulo S. L. M. Barreto, Anderson C. A. Nascimento, Gláucio Oliveira, Waldyr Benits

Published May 17, 2018• 1 citation
arXivHTMLPDF

Abstract

We present an oblivious transfer (OT) protocol that combines the OT scheme of Chou and Orlandi together with thesupersingular isogeny Diffie-Hellman (SIDH) primitive of De Feo, Jao, and Plût. Our construction is a candidate for post-quantum secure OT and demonstrates that SIDH naturally supports OT functionality. We consider the protocol in the simplest configuration of $\binom{2}{1}$-OT and analyze the protocol to verify its security.

DOI
10.48550/arxiv.1805.06589

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Oblivious Transfer é um dos blocos de construção mais importantes em criptografia e útil para a construção de protocolos seguros. Com o advento da computação quântica houve um impulso na pesquisa e desenvolvimento de protocolos criptográficos resistentes ao processamento quântico de computadores. Assim, em 2018, o protocolo SIOT (Supersingu- lar Isogeny Oblivious Transfer) foi apresentado como o primeiro esquema OT criptográfico pós-quântico baseado em isogenias de curvas elípticas supersingulares. Inicialmente, elaborou-se um esquema de OT combinado com as primitivas criptográficas do protocolo SIDH(Supersingular Isogeny Diffie-Hellman). Além disso, o protocolo SIOT foi construído na sua configuração mais simples contra adversários semi-honestos. No entanto, o mesmo foi submetido a escrutínios que resultaram na necessidade de desenvolver novas provas de segurança. Quase em paralelo, surgiram novos e eficientes ataques criptoanalíticos ao protocolo SIDH que consequentemente comprometeram a estrutura de segurança do SIOT. Desta forma, em um primeiro momento, as novas definições de provas de segurança englobaram a compatibilização de determinados parâmetros da funcionalidade OT do próprio protocolo SIOT com hipóteses de segurança de problemas computacionais de isogenias. Após isso, foram analisadas e implementadas as contramedidas de segurança do protocolo M-SIDH (Masked-Supersingular Isogeny Diffie-Hellman). Portanto, propõem-se um protocolo OT baseado em isogenias de curvas elípticas e com resistência a ataques quânticos, em uma forma simples.
The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented.
In this paper, we have discussed isogeny-based cryptography as a rather appealing option for post-quantum security. We provide analysis of current protocols like SIDH (Supersingular Isogeny Diffie-Hellman) and SIKE (Supersingular Isogeny Key Encapsulation) setting up fundamental issues with their security and performance. In this work we introduce a novel isogeny pathfinding algorithm that employs graph theory as well as probabilistic models in order to lower its computational complexity. The results show significant improvement in both efficiency and security, strengthening the case for the use of isogeny based systems in a quantum-resilient cryptographic space.
The present thesis focus on one of the post-quantum cryptosystems, in particular, the isogenybased cryptography. Because of its certain properties like the hard problem of computing isogenies between two elliptic curves, it makes it to be one of the few candidates for post-quantum cryptography. We will study the algorithms based on this isogeny-based cryptography and why they are useful against quantum computers. One of these algorithms is the Supersingular Isogeny Diffie Hellman (SIDH), a candidate for substituing ECDH in the Signal protocol to make it quantum safe.
Isogeny-based cryptography using supersingular elliptic curves --- most prominently, the constructions of De Feo-Jao-Plut --- is one of the few practical candidates for post-quantum public key cryptography. Its formidable security claim is earned through the continual exploration of quantum algorithms for `isogeny problems' and the assessment of the threat they pose to supersingular isogeny-based cryptography. We survey the rich history of classical and quantum algorithms for isogeny problems, and close with an original result --- a quantum algorithm for the general supersingular isogeny problem, based on the discovery of Delfs and Galbraith in 2013 --- that has exponential-complexity in general and subexponential complexity in an important sub-case. As yet, this algorithm poses a limited threat to the schemes of De Feo-Jao-Plut; however, it is an important algorithm to consider, for it provides insight into the structure of supersingular curves and the isogenies between them, and may lead to newer destructive quantum algorithms.