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Vulnerabilities of some semidirect products in Diffie-Hellman key exchanges

Dylan Rudy

Published May 1, 2021• 0 citations
OpenAlex

Abstract

The Diffie-Hellman key exchange protocol introduced in 1976 allows two users, Alice and Bob, to establish a shared secret key K over insecure communication channels. With the rise of quantum computers, the protocol is in danger of becoming easily breakable. In response, several modified versions of Diffie-Hellman have been proposed claiming to be resistant to attacks performed by quantum computers. \n \nTwo such protocols and use of a semidirect product creating a modified version of a Discrete Logarithm problem to provide security for Alice and Bob. \n \nRegardless of choosing a tropical semiring or a free nilpotent p-group, these protocols prove not only vulnerable to quantum attacks, but they can be attacked by a classical computer. By exploiting each of the platforms, an eavesdropper Eve needs only to figure out how to mimic Alice's contribution to the shared secret key without needing to copy Alice's exact operations. By doing so, Eve can efficiently recover the shared secret key K.

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The Diffie-Hellman key exchange protocol introduced in 1976 allows two users, Alice and Bob, to establish a shared secret key K over insecure communication channels. With the rise of quantum computers, the protocol is in danger of becoming easily breakable. In response, several modified versions of Diffie-Hellman have been proposed claiming to be resistant to attacks performed by quantum computers. \n \nTwo such protocols and use of a semidirect product creating a modified version of a Discrete Logarithm problem to provide security for Alice and Bob. \n \nRegardless of choosing a tropical semiring or a free nilpotent p-group, these protocols prove not only vulnerable to quantum attacks, but they can be attacked by a classical computer. By exploiting each of the platforms, an eavesdropper Eve needs only to figure out how to mimic Alice's contribution to the shared secret key without needing to copy Alice's exact operations. By doing so, Eve can efficiently recover the shared secret key K.
The necessity of a secure key exchange protocol arises from the critical need to establish encrypted communication over an untrusted network. Over time, a multitude of key exchange mechanisms have been developed to counteract adversarial threats. The Diffie-Hellman key exchange protocol (DHKE) is one of the most widely used protocols for symmetric key sharing. However, this protocol exhibits certain inherent limitations that attackers may exploit. It lacks authentication mechanism and is susceptible to Man-in-the-Middle (MITM) attacks and quantum attacks. To mitigate these vulnerabilities, we have designed a hybrid key exchange protocol combining DHKE with Learning With Errors (LWE), a lattice-based post-quantum primitive. This proposed protocol provides authentication via a Public Key Infrastructure (PKI) together with CRYSTALS-Dilithium digital signature, resilience against MITM attacks, and robustness against classical and quantum threats. We have done a security analysis using the Dolev-Yao threat model, extended to quantum-equipped attackers, and showed that the protocol achieves mutual authentication, session key secrecy, and forward secrecy under the hardness of LWE and DHKE. Lastly, we provided detailed parameter recommendations based on NIST standards and shed light on side-channel attacks.
The Diffie–Hellman key exchange plays a crucial role in conventional cryptography, as it allows two legitimate users to establish a common, usually ephemeral, secret key. Its security relies on the discrete-logarithm problem, which is considered to be a mathematical one-way function, while the final key is formed by random independent actions of the two users. In the present work we investigate the extension of Diffie–Hellman key exchange to the quantum setting, where the two legitimate users exchange independent random quantum states. The proposed protocol relies on the bijective mapping of integers onto a set of symmetric coherent states, and we investigate the regime of parameters for which the map behaves as a quantum one-way function. Its security is analyzed in the framework of minimum-error-discrimination and photon-number-splitting attacks, while its performance and the challenges in a possible realization are also discussed.
Authenticated key-exchange protocols allow two participants A and B , communicating over a public network and each holding an authentication means to exchange a shared secret value. Methods designed to deal with this cryptographic problem ensure A (resp. B ) that no other participants aside from B (resp. A ) can learn any information about the agreed value and often also ensure A and B that their respective partner has actually computed this value. A natural extension to this cryptographic method is to consider a pool of participants exchanging a shared secret value and to provide a formal treatment for it. Starting from the famous two-party Diffie--Hellman (DH) key-exchange protocol and from its authenticated variants, security experts have extended it to the multiparty setting for over a decade and, in the past few years, completed a formal analysis in the framework of modern cryptography. The present paper synthesizes this body of work on the provably-secure authenticated group DH key exchange.
A Virtual Private Network (VPN) ensures the confidentiality and integrity of data transferred between two endpoints, even if the means of transport are insecure. One popular protocol is Internet Protocol Security (IPSec) which operates at OSI layer 3 and protects all protocols at higher layers. Cryptographic keys in IPSec are negotiated using the Internet Key Exchange (IKE) protocol. IKE negotiates security parameters for IPSec sessions. In particular, the IKEv2 protocol uses the Elliptic Curve Diffie-Hellman (ECDH) algorithm to establish a secret key shared between two nodes on a network. Although solving such a problem remains difficult with current computing power, it is believed that generic quantum computers will be able to solve this problem, which implies that the security of IKEv2 is compromised. There are, however, several cryptographic systems that are trusted to be resistant to attacks by quantum computers. This family of cryptosystems is known as quantum-resistant cryptography (QRC). In this paper, after highlighting the requirements for a secure key exchange protocol, we briefly review the QRC solutions that have been proposed in the recent literature.