We analyze the Commutative Supersingular Isogeny Diffie-Hellman protocol (CSIDH), a novel supersingular isogeny-based key establishment protocol. Our analysis is from three perspectives: \n \nQuantum Cryptanalysis. Building upon quantum attacks on ordinary isogeny-based cryptography, we propose a subexponential-time quantum algorithm for inverting the complex multiplication group action for supersingular elliptic curves, which uses only polynomial quantum space. This improves upon previously-known algorithms which required subexponential quantum space. \n \nOptimization. We develop more efficient algorithms for evaluating the class group action in the context of CSIDH. We consider "strategies"—formerly only considered for Supersingular Isogeny Diffie-Hellman (SIDH)—in the context of CSIDH, and develop systematic methods for optimizing "permutations" of the small primes used in CSIDH, which previously had been treated only in an ad hoc fashion. We also develop a systematic technique to optimize the CSIDH keyspace. These optimizations are complementary to prior work on optimizing CSIDH, including improved field arithmetic, Splitting Isogenies into Multiple Batches (SIMBA), and the two-point method. We apply our optimizations to the CSIDH-512 parameter set and give experimental results. \n \nFault Attacks. We consider physical attacks on static/ephemeral CSIDH in which limited information about which isogenies are "real" and which are "dummy" is revealed. We determine bounds on the number of fault injections required to recover the static secret key, and show that simply reordering the real and dummy isogenies from the ubiquitous "real-then-dummy" ordering to a dynamic random ordering dramatically increases the number of faults required, with negligible impact on the running time of the key exchange protocol (in contrast with prior fault attack countermeasures, which prevent fault attacks entirely at the cost of doubling the running time for key exchange).
We analyze the Commutative Supersingular Isogeny Diffie-Hellman protocol (CSIDH), a novel supersingular isogeny-based key establishment protocol. Our analysis is from three perspectives: \n \nQuantum Cryptanalysis. Building upon quantum attacks on ordinary isogeny-based cryptography, we propose a subexponential-time quantum algorithm for inverting the complex multiplication group action for supersingular elliptic curves, which uses only polynomial quantum space. This improves upon previously-known algorithms which required subexponential quantum space. \n \nOptimization. We develop more efficient algorithms for evaluating the class group action in the context of CSIDH. We consider "strategies"—formerly only considered for Supersingular Isogeny Diffie-Hellman (SIDH)—in the context of CSIDH, and develop systematic methods for optimizing "permutations" of the small primes used in CSIDH, which previously had been treated only in an ad hoc fashion. We also develop a systematic technique to optimize the CSIDH keyspace. These optimizations are complementary to prior work on optimizing CSIDH, including improved field arithmetic, Splitting Isogenies into Multiple Batches (SIMBA), and the two-point method. We apply our optimizations to the CSIDH-512 parameter set and give experimental results. \n \nFault Attacks. We consider physical attacks on static/ephemeral CSIDH in which limited information about which isogenies are "real" and which are "dummy" is revealed. We determine bounds on the number of fault injections required to recover the static secret key, and show that simply reordering the real and dummy isogenies from the ubiquitous "real-then-dummy" ordering to a dynamic random ordering dramatically increases the number of faults required, with negligible impact on the running time of the key exchange protocol (in contrast with prior fault attack countermeasures, which prevent fault attacks entirely at the cost of doubling the running time for key exchange).
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.
The emergence of quantum computing poses a significant threat to the security of currently deployed public-key cryptosystems, including those based on elliptic curves (ECC). Shor's algorithm can solve the underlying integer factorization and discrete logarithm problems in polynomial time, rendering established cryptographic standards obsolete. This has catalyzed the development of post-quantum cryptography (PQC), a new generation of algorithms resistant to attacks by both classical and quantum computers. Among the most promising PQC candidates are systems based on supersingular isogeny graphs. This paper provides a comprehensive analysis of the supersingular isogeny paradigm as a quantum-secure evolution of elliptic curve cryptography. It begins by introducing the mathematical foundations of elliptic curves and isogenies, which are structure-preserving maps between them. The core of isogeny-based cryptography lies in the conjectured hardness of finding an isogeny path between two given supersingular elliptic curves in the vast isogeny graph. This is fundamentally different from the discrete logarithm problem that underpins traditional ECC. We review the foundational Supersingular Isogeny Diffie-Hellman (SIDH) key exchange protocol and its subsequent realization as the Supersingular Isogeny Key Encapsulation (SIKE) mechanism, which was a candidate in the NIST PQC standardization process. We discuss its advantages, such as small key sizes and perfect forward secrecy, which made it an attractive alternative. However, the paper also presents a balanced view by thoroughly examining the devastating cryptanalytic breakthroughs of 2022 that rendered the original SIDH/SIKE schemes insecure. These attacks, which exploit auxiliary point information, underscore the dynamic nature of PQC research. Despite this setback, the isogeny-based field remains active, with ongoing research into alternative constructions like CSIDH and other variants that are not vulnerable to the known attacks. We conclude that while the initial promise of SIDH/SIKE was broken, the underlying mathematical problem of finding isogenies remains a compelling basis for quantum-secure cryptography, representing a paradigm of active research rather than a discarded concept.
We propose an efficient commutative group action suitable for non-interactive key exchange in a post-quantum setting. Our construction follows the layout of the Couveignes–Rostovtsev–Stolbunov cryptosystem, but we apply it to supersingular elliptic curves defined over a large prime field Fp , rather than to ordinary elliptic curves. The Diffie–Hellman scheme resulting from the group action allows for public-key validation at very little cost, runs reasonably fast in practice, and has public keys of only 64 bytes at a conjectured AES-128 security level, matching NIST’s post-quantum security category I.
The Signal protocol can be considered state-of-the-art when it comes to secure messaging, but advances in quantum computing stress the importance of finding post-quantum resistant alternatives to its asymmetric cryptographic primitives. The aim is to determine whether existing post-quantum cryptography can be used as a drop-in replacement for the public-key cryptography currently used in the Signal protocol and what the performance trade-offs may be. An implementation of the Signal protocol using commutative supersingular isogeny Diffie-Hellman (CSIDH) key exchange operations in place of elliptic-curve Diffie-Hellman (ECDH) is proposed. The benchmark results on a Samsung Galaxy Note 8 mobile device equipped with a 64-bit Samsung Exynos 9 (8895) octa-core CPU shows that it takes roughly 8 seconds to initialize a session using CSIDH-512 and over 40 seconds using CSIDH-1024, without platform specific optimization. To the best of our knowledge, the proposed implementation is the first post-quantum resistant Signal protocol implementation and the first evaluation of using CSIDH as a drop-in replacement for ECDH in a communication protocol.
Our modern world is dependent on online communications and transactions. To ensure privacy in these digital actions, it is extremely common to use protocols such as the Diffie-Hellman key exchange scheme. Unfortunately, these systems have been shown to be vulnerable to attacks from a quantum adversary. This means that an overhaul of the field of cryptography is needed in order to anticipate and mitigate this threat.Over the past years, the community has begun to answer this call by proposing and analyzing new techniques that are conjectured to be quantum safe. One such example is the study of cryptographic group actions. These mappings mimic some of the structure from the Diffie-Hellman protocol,but abstract it enough so that new mathematical operations can be employed that avoid quantum attacks. Currently, the main examples of group actions come from isogenies, codes, and lattices.In order for a protocol to be used in the real-world we would hope for it to be secure, fast, and requiring little storage. This means that in these early stages of research and development of cryptographic group actions, it is essential to begin to answer some of the questions surrounding security and efficiency. This thesis contributes to this effort by focusing on these aspects across different group actions that have been proposed for use in the literature.It is certainly true that each individual group action needs its own study of security. When designers are trying to achieve some advanced functionalities in a protocol, they may sometimes alter the underlying security assumptions.Each time this is done, the new assumptions need their own careful study as well. In this thesis we consider four different variants of group action security assumptions spanning the isogeny, tensor, linear code, and matrix code group actions. We analyze their security by either offering concrete attacks or giving a polynomial time reduction to a more well-studied assumption. This helps to indicate whether a problem is appropriate for use in cryptography, and what parameters to use if so.In addition to security, this thesis is also inscribed in the effort to study the efficiency of group actions by including three works studying the isogeny group action and related subroutines. They span the core isogeny problem for oriented elliptic curves, evaluating isogenies from a kernel generator, and supersingularity testing. With an effort from the community to improve overall runtimes, we can hope to get closer to reasonable speeds for real-life use.By studying the security and efficiency of cryptographic group actions, this thesis shines light on group actions as a promising framework for post-quantum cryptographic solutions.
Abstract The Supersingular Isogeny Diffie-Hellman (SIDH) protocol is a leading contender for post-quantum cryptography. However, recent advances, specifically the Castryck-Decru attack, have exposed significant security flaws, primarily related to the predictability of torsion points and vulnerabilities in curve parameters. This study proposes and evaluates enhancements to the SIDH protocol, including increasing the size of torsion subgroups, introducing randomized isogeny generation , and employing advanced obfuscation techniques. Experimental results demonstrate that these modifications substantially improve SIDH’s security, with a manageable increase in computational overhead, positioning SIDH as a more resilient candidate for post-quantum cryptographic applications.