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

Hard isogeny problems over RSA moduli and groups with infeasible inversion

We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders. Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.Comment: Significant revision of the article previously titled "A Candidate Group with Infeasible Inversion" (arXiv:1810.00022v1). Cleared up the constructions by giving toy examples, added "The Parallelogram Attack" (Sec 5.3.2). 54 pages, 8 figure

SCALLOP:Scaling the CSI-FiSh

International audienceWe present SCALLOP: SCALable isogeny action based on Oriented supersingular curves with Prime conductor, a new group action based on isogenies of supersingular curves. Similarly to CSIDH and OSIDH, we use the group action of an imaginary quadratic order’s class group on the set of oriented supersingular curves. Compared to CSIDH, the main benefit of our construction is that it is easy to compute the class-group structure; this data is required to uniquely represent—and efficiently act by — arbitrary group elements, which is a requirement in, e.g., the CSI-FiSh signature scheme by Beullens, Kleinjung and Vercauteren. The index-calculus algorithm used in CSI-FiSh to compute the class-group structure has complexity L(1/2), ruling out class groups much larger than CSIDH-512, a limitation that is particularly problematic in light of the ongoing debate regarding the quantum security of cryptographic group actions.Hoping to solve this issue, we consider the class group of a quadratic order of large prime conductor inside an imaginary quadratic field of small discriminant. This family of quadratic orders lets us easily determine the size of the class group, and, by carefully choosing the conductor, even exercise significant control on it—in particular supporting highly smooth choices. Although evaluating the resulting group action still has subexponential asymptotic complexity, a careful choice of parameters leads to a practical speedup that we demonstrate in practice for a security level equivalent to CSIDH-1024, a parameter currently firmly out of reach of index-calculus-based methods. However, our implementation takes 35 seconds (resp. 12.5 minutes) for a single group-action evaluation at a CSIDH-512-equivalent (resp. CSIDH-1024-equivalent) security level, showing that, while feasible, the SCALLOP group action does not achieve realistically usable performance yet

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

A post-quantum digital signature scheme based on supersingular isogenies

We present the first general-purpose digital signature scheme based on supersingular elliptic curve isogenies secure against quantum adversaries in the quantum random oracle model with small key sizes. This scheme is an application of Unruh’s construction of non-interactive zero-knowledge proofs to an interactive zero-knowledge proof proposed by De Feo, Jao, and Plut. We implement our proposed scheme on an x86- 64 PC platform as well as an ARM-powered device. We exploit the stateof-the-art techniques to speed up the computations for general C and assembly. Finally, we provide timing results for real world applications

Computing supersingular isogenies on Kummer surfaces

We apply Scholten\u27s construction to give explicit isogenies between the Weil restriction of supersingular Montgomery curves with full rational 2-torsion over $GF(p^2)$ and corresponding abelian surfaces over $GF(p)$. Subsequently, we show that isogeny-based public key cryptography can exploit the fast Kummer surface arithmetic that arises from the theory of theta functions. In particular, we show that chains of 2-isogenies between elliptic curves can instead be computed as chains of Richelot (2,2)-isogenies between Kummer surfaces. This gives rise to new possibilities for efficient supersingular isogeny-based cryptography

Key Compression for Isogeny-Based Cryptosystems

We present a method for key compression in quantum-resistant isogeny-based cryptosystems, which reduces storage and transmission costs of per-party public information by a factor of two, with no effect on the security level of the scheme. We achieve this reduction by compressing both the representation of an elliptic curve, and torsion points on said curve. Compression of the elliptic curve is achieved by associating each j-invariant to a canonical choice of elliptic curve, and the torsion points will be represented as linear combinations with respect to a canonical choice of basis for this subgroup. This method of compressing public information can be applied to numerous isogeny-based protocols, such as key exchange, zero-knowledge identification, and public-key encryption. The details of utilizing compression for each of these cryptosystems is explained. We provide implementation results showing the computational cost of key compression and decompression at various security levels. Our results show that isogeny-based cryptosystems achieve the smallest possible key sizes among all existing families of post-quantum cryptosystems at practical security levels

Faster Isogenies for Quantum-Safe SIKE

In the third round of the NIST PQC standardization process, the only isogeny-based candidate, SIKE, suffers from slow performance when compared to other contenders. The large-degree isogeny computation performs a series of isogenous mappings between curves, to account for about 80% of SIKE’s latency. Here, we propose, implement, and evaluate a new method for computing large-degree isogenies of an odd power. Our new strategy for this computation avoids expensive recomputation of temporary isogeny results.We modified open-source libraries targeting x86, ARM64, and ARM32 platforms. Across each of these implementations, our new method achieves 10% and 5% speedups in SIKE’s key encapsulation and decapsulation operations, respectively. Additionally, these implementations use 3% less stack space at only a 48 byte increase in code size. Given the benefit and simplicity of our approach, we recommend this method for current and emerging SIKE implementations