Isogeny-based post-quantum key exchange protocols

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

Optimizations of Isogeny-based Key Exchange

Supersingular Isogeny Diffie-Hellman (SIDH) is a key exchange scheme that is believed to be quantum-resistant. It is based on the difficulty of finding a certain isogeny between given elliptic curves. Over the last nine years, optimizations have been proposed that significantly increased the performance of its implementations. Today, SIDH is a promising candidate in the US National Institute for Standards and Technology’s (NIST’s) post-quantum cryptography standardization process. This work is a self-contained introduction to the active research on SIDH from a high-level, algorithmic lens. After an introduction to elliptic curves and SIDH itself, we describe the mathematical and algorithmic building blocks of the fastest known implementations. Regarding elliptic curves, we describe which algorithms, data structures and trade-offs regard- ing elliptic curve arithmetic and isogeny computations exist and quantify their runtime cost in field operations. These findings are then tailored to the situation of SIDH. As a result, we give efficient algorithms for the performance-critical parts of the protocol

Quantum Attacks on Modern Cryptography and Post-Quantum Cryptosystems

Cryptography is a critical technology in the modern computing industry, but the security of many cryptosystems relies on the difficulty of mathematical problems such as integer factorization and discrete logarithms. Large quantum computers can solve these problems efficiently, enabling the effective cryptanalysis of many common cryptosystems using such algorithms as Shor’s and Grover’s. If data integrity and security are to be preserved in the future, the algorithms that are vulnerable to quantum cryptanalytic techniques must be phased out in favor of quantum-proof cryptosystems. While quantum computer technology is still developing and is not yet capable of breaking commercial encryption, these steps can be taken immediately to ensure that the impending development of large quantum computers does not compromise sensitive data

Towards Optimized and Constant-Time CSIDH on Embedded Devices

We present an optimized, constant-time software library for commutative supersingular isogeny Diffie-Hellman key exchange (CSIDH) proposed by Castryck et al. which targets 64-bit ARM processors. The proposed library is implemented based on highly-optimized field arithmetic operations and computes the entire key exchange in constant-time. The proposed implementation is resistant to timing attacks. We adopt optimization techniques to evaluate the highest performance CSIDH on ARM-powered embedded devices such as cellphones, analyzing the possibility of using such a scheme in the quantum era. To the best of our knowledge, the proposed implementation is the first constant-time implementation of CSIDH and the first evaluation of this scheme on embedded devices. The benchmark result on a Google Pixel 2 smartphone equipped with 64-bit high-performance ARM Cortex-A72 core shows that it takes almost 12 seconds for each party to compute a commutative action operation in constant-time over the 511-bit finite field proposed by Castryck et al. However, using uniform but variable-time Montgomery ladder with security considerations improves these results significantly

A Note on Post-Quantum Authenticated Key Exchange from Supersingular Isogenies

In this work, we study several post-quantum authenticated key exchange protocols in the setting of supersingular isogenies. Leveraging the design of the well-studied schemes by Krawczyk (2003), Boyd et al. (2008), Fujioka et al. (2013), Krawczyk and Wee (2015), and others, we show how to use the Supersingular Isogeny Diffie-Hellman (SIDH) and Supersingular Isogeny Key Encapsulation (SIKE) protocols as basic building blocks to construct efficient and flexible authenticated key exchange schemes featuring different functionalities and levels of security. This note is also intended to be a ``gentle\u27\u27 introduction to supersingular isogeny based cryptography, and its most relevant constructions, for protocol designers and cryptographers

Action de Groupe Supersingulières et Echange de Clés Post-quantique

Alice and Bob want to exchange information and make sure that an eavesdropper will not be able to listen to them, even with a quantum computer.To that aim they use cryptography and in particular a key-exchange protocol. These type of protocols rely on number theory and algebraic geometry. However current protocols are not quantum resistant, which is the reason why new cryptographic tools must be developed. One of these tools rely on isogenies, i.e. homomorphisms between elliptic curves. In this thesis the first contribution is an implementation of an isogeny-based key-exchange protocol resistant against side-channel attacks (timing and power consumption analysis, fault injection). We also generalize this protocol to a larger set of elliptic curves.Alice et Bob souhaitent échanger des informations sans qu’un attaquant, même muni d’un ordinateur quantique, puisse les entendre. Pour cela, ils ont recours à la cryptologie et en particulier à un protocole d’échange de clés. Ces protocoles reposent sur la théorie des nombres et la géométrie algébrique. Cependant les protocoles actuellement utilisés ne résistent pas aux attaques quantiques, c’est pourquoi il est nécessaire de développer de nouveaux outils cryptographiques. L’un de ces outils repose sur les isogénies, c’est-à-dire des homomorphismes entre des courbes elliptiques. Dans cette thèse nous proposons une implémentation d’un des protocoles d’échange de clés basé sur les isogénies qui résiste aux attaques par canaux auxiliaires (étude de la durée d’exécution, de la consommation de courant et injection de fautes). Nous généralisons également ce protocole à un plus grand ensemble de courbes elliptiques

Hard Homogenous Spaces and Commutative Supersingular Isogeny based Diffie-Hellman

Tema ovog rada jest proces stvaranja 3D stvarnih ili imaginarnih objekata pomoću alata SolidWorks koji je u današnje vrijeme jedan od najpoznatijih alata kod modeliranja mehaničkih i projektnih objekata. Kako bi ga što više približio svakoj osobi, ukratko sam naveo najvažnije činjenice o samom alatu, prošao kroz njegovu povijest, objasnio za što ga možemo koristiti te najvećim dijelom prikazao kako se od jednog tehničkog nacrta dođe do gotovog objekta i modela

Towards Green Computing Oriented Security: A Lightweight Postquantum Signature for IoE

[EN] Postquantum cryptography for elevating security against attacks by quantum computers in the Internet of Everything (IoE) is still in its infancy. Most postquantum based cryptosystems have longer keys and signature sizes and require more computations that span several orders of magnitude in energy consumption and computation time, hence the sizes of the keys and signature are considered as another aspect of security by green design. To address these issues, the security solutions should migrate to the advanced and potent methods for protection against quantum attacks and offer energy efficient and faster cryptocomputations. In this context, a novel security framework Lightweight Postquantum ID-based Signature (LPQS) for secure communication in the IoE environment is presented. The proposed LPQS framework incorporates a supersingular isogeny curve to present a digital signature with small key sizes which is quantum-resistant. To reduce the size of the keys, compressed curves are used and the validation of the signature depends on the commutative property of the curves. The unforgeability of LPQS under an adaptively chosen message attack is proved. Security analysis and the experimental validation of LPQS are performed under a realistic software simulation environment to assess its lightweight performance considering embedded nodes. It is evident that the size of keys and the signature of LPQS is smaller than that of existing signature-based postquantum security techniques for IoE. It is robust in the postquantum environment and efficient in terms of energy and computations.This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University. Jeddah. under grant No. (DF-457-156-1441).Rani, R.; Kumar, S.; Kaiwartya, O.; Khasawneh, AM.; Lloret, J.; Al-Khasawneh, MA.; Mahmoud, M.... (2021). Towards Green Computing Oriented Security: A Lightweight Postquantum Signature for IoE. Sensors. 21(5):1-20. https://doi.org/10.3390/s2105188312021