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 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.
We study cryptosystems based on supersingular isogenies. This is an active area of research in post-quantum cryptography. Our first contribution is to give a very powerful active attack on the supersingular isogeny encryption scheme. This attack can only be prevented by using a (relatively expensive) countermeasure. Our second contribution is to show that the security of all schemes of this type depends on the difficulty of computing the endomorphism ring of a supersingular elliptic curve. This result gives significant insight into the difficulty of the isogeny problem that underlies the security of these schemes. Our third contribution is to give a reduction that uses partial knowledge of shared keys to determine an entire shared key. This can be used to retrieve the secret key, given information leaked from a side-channel attack on the key exchange protocol. A corollary of this work is the first bit security result for the supersingular isogeny key exchange: Computing any component of the j-invariant is as hard as computing the whole j-invariant.
The rise of quantum computing poses a significant threat to contemporary public-key cryptography, which relies on problems like integer factorization and the discrete logarithm, both of which are solvable in polynomial time by Shor's algorithm. This has prompted a global effort to develop and standardize post-quantum cryptographic primitives that are resistant to attacks by both classical and quantum computers. Among the various families of post-quantum candidates, isogeny-based cryptography has emerged as a particularly compelling, albeit challenging, alternative. This paper explores the mathematical foundations of elliptic curve isogenies and their application as a basis for quantum-resistant cryptographic protocols. We delve into the structure of supersingular elliptic curves and the associated isogeny graphs, which form the basis for the conjectured hardness of the underlying computational problems. The Supersingular Isogeny Diffie-Hellman (SIDH) protocol and its derivative, Supersingular Isogeny Key Encapsulation (SIKE), are examined in detail as principal examples of this cryptographic paradigm. We analyze their key advantages, most notably their exceptionally small key sizes, which offer significant benefits for constrained environments. However, the security landscape of isogeny-based cryptography has been dramatically reshaped by recent cryptanalytic breakthroughs that completely broke the SIKE protocol. This paper analyzes the nature of these attacks, discussing the vulnerabilities they exploited and the profound implications for the field. We conclude by surveying the current state of isogeny-based cryptography in the aftermath of these developments, outlining the proposed countermeasures, alternative designs, and the open research questions that define the new frontier for this promising area of post-quantum cryptography.
In this article, we discuss how algebraic geometry, especially the isogenies and elliptic curves, have been used to construct secure post-quantum cryptographic systems. Since quantum computing is a very big threat to conventional cryptographic techniques, it is essential to develop new strategies that would enhance protection of data in quantum age. Algebraic geometry provides such solutions as the elliptic curve cryptography and the protocols based on isogenic, which is also resistant to the attacks of quantum algorithms. The article explores the mathematical basis of these cryptographic techniques, their effectiveness, scalability and security. It is noteworthy that isogeny-based cryptography schemes such as the Supersingular Isogeny Diffie-Hellman (SIDH) and Supersingular Isogeny Key Encapsulation (SIKE) prove that isogeny-based cryptography is capable of secure key exchange and encapsulation. The results indicate that algebraic geometry is not only enhancing the cryptography systems, but also opening up a reasonable channel in which strong post-quantum systems can be created. The future of secure communication in a quantum-driven world has profound implications to the methods.
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.
We propose and develop new schemes for post-quantum cryptography based on isogenies over elliptic curves. First we show that ordinary elliptic curves are have less than exponential security against quantum computers. These results were used as the motivation for De Feo, Jao and Pl\\^ut's construction of public key cryptosystems using supersingular elliptic curve isogenies. We extend their construction and show that isogenies between supersingular elliptic curves can be used as the underlying hard mathematical problem for other quantum-resistant schemes. For our second contribution, we propose is an undeniable signature scheme based on elliptic curve isogenies. We prove its security under certain reasonable number-theoretic computational assumptions for which no efficient quantum algorithms are known. This proposal represents only the second known quantum-resistant undeniable signature scheme, and the first such scheme secure under a number-theoretic complexity assumption. Finally, we also propose a security model for evaluating the security of authenticated encryption schemes in the post-quantum setting. Our model is based on a combination of the classical Bellare-Namprempre security model for authenticated encryption together with modifications from Boneh and Zhandry to handle message authentication against quantum adversaries. We give a generic construction based on Bellare-Namprempre for producing an authenticated encryption protocol from any quantum-resistant symmetric-key encryption scheme together with any digital signature scheme or MAC admitting any classical security reduction to a quantum-computationally hard problem. We apply the results and show how we can explicitly construct authenticated encryption schemes based on isogenies.
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.