## CryptoDB

### Benjamin Wesolowski

#### Publications

**Year**

**Venue**

**Title**

2022

EUROCRYPT

Orientations and the supersingular endomorphism ring problem
Abstract

We study two important families of problems in isogeny-based cryptography and how they relate to each other: computing the endomorphism ring of supersingular elliptic curves, and inverting the action of class groups on oriented supersingular curves. We prove that these two families of problems are closely related through polynomial-time reductions, assuming the generalized Riemann hypothesis.
We identify two classes of essentially equivalent problems. The first class corresponds to the problem of computing the endomorphism ring of oriented curves. The security of a large family of cryptosystems (such as CSIDH) reduces to (and sometimes from) this class, for which there are heuristic quantum algorithms running in subexponential time. The second class corresponds to computing the endomorphism ring of orientable curves. The security of essentially all isogeny-based cryptosystems reduces to (and sometimes from) this second class, for which the best known algorithms are still exponential.
Some of our reductions not only generalise, but also strengthen previously known results. For instance, it was known that in the particular case of curves defined over $\mathbb F_p$, the security of CSIDH reduces to the endomorphism ring problem in subexponential time. Our reductions imply that the security of CSIDH is actually equivalent to the endomorphism ring problem, under polynomial time reductions (circumventing arguments that proved such reductions unlikely).

2021

ASIACRYPT

Séta: Supersingular Encryption from Torsion Attacks
📺
Abstract

We present Séta, a new family of public-key encryption schemes with post-quantum security based on isogenies of supersingular elliptic curves.
It is constructed from a new family of trapdoor one-way functions, where the inversion algorithm uses Petit's so called \emph{torsion attacks} on SIDH to compute an isogeny between supersingular elliptic curves given an endomorphism of the starting curve and images of torsion points.
We prove the OW-CPA security of S\'eta and present an IND-CCA variant using the post-quantum OAEP transformation.
Several variants for key generation are explored together with their impact on the selection of parameters, such as the base prime of the scheme.
We furthermore formalise an ``uber'' isogeny assumption framework which aims to generalize computational isogeny problems encountered in schemes including SIDH, CSDIH, OSIDH and ours.
Finally, we carefully select parameters to achieve a balance between security and run-times and present experimental results from our implementation.

2020

CRYPTO

Random Self-reducibility of Ideal-SVP via Arakelov Random Walks
📺
Abstract

Fixing a number field, the space of all ideal lattices, up to isometry, is naturally an Abelian group, called the *Arakelov class group*. This fact, well known to number theorists, has so far not been explicitly used in the literature on lattice-based cryptography. Remarkably, the Arakelov class group is a combination of two groups that have already led to significant cryptanalytic advances: the class group and the unit torus.
In the present article, we show that the Arakelov class group has more to offer. We start with the development of a new versatile tool: we prove that, subject to the Riemann Hypothesis for Hecke L-functions, certain random walks on the Arakelov class group have a rapid mixing property. We then exploit this result to relate the average-case and the worst-case of the Shortest Vector Problem in ideal lattices. Our reduction appears particularly sharp: for Hermite-SVP in ideal lattices of certain cyclotomic number fields, it loses no more than a $\tilde O(\sqrt n)$ factor on the Hermite approximation factor.
Furthermore, we suggest that this rapid-mixing theorem should find other applications in cryptography and in algorithmic number theory.

2020

ASIACRYPT

SQISign: Compact Post-Quantum signatures from Quaternions and Isogenies
📺 ★
Abstract

We introduce a new signature scheme, \emph{SQISign}, (for \emph{Short Quaternion and Isogeny Signature}) from isogeny graphs of supersingular elliptic curves. The signature scheme is derived from a new one-round, high soundness, interactive identification protocol. Targeting the post-quantum NIST-1 level of security, our implementation results in signatures of $204$ bytes, secret keys of $16$ bytes and public keys of $64$ bytes. In particular, the signature and public key sizes combined are an order of magnitude smaller than all other post-quantum signature schemes. On a modern workstation, our implementation in C takes 0.6s for key generation, 2.5s for signing, and 50ms for verification.
While the soundness of the identification protocol follows from classical assumptions, the zero-knowledge property relies on the second main contribution of this paper.
We introduce a new algorithm to find an isogeny path connecting two given supersingular elliptic curves of known endomorphism rings.
A previous algorithm to solve this problem, due to Kohel, Lauter, Petit and Tignol, systematically reveals paths from the input curves to a `special' curve. This leakage would break the zero-knowledge property of the protocol. Our algorithm does not directly reveal such a path, and subject to a new computational assumption, we prove that the resulting identification protocol is zero-knowledge.

2020

JOFC

Efficient Verifiable Delay Functions
Abstract

We construct a verifiable delay function (VDF). A VDF is a function whose evaluation requires running a given number of sequential steps, yet the result can be efficiently verified. They have applications in decentralised systems, such as the generation of trustworthy public randomness in a trustless environment, or resource-efficient blockchains. To construct our VDF, we actually build a trapdoor VDF. A trapdoor VDF is essentially a VDF which can be evaluated efficiently by parties who know a secret (the trapdoor). By setting up this scheme in a way that the trapdoor is unknown (not even by the party running the setup, so that there is no need for a trusted setup environment), we obtain a simple VDF. Our construction is based on groups of unknown order such as an RSA group or the class group of an imaginary quadratic field. The output of our construction is very short (the result and the proof of correctness are each a single element of the group), and the verification of correctness is very efficient.

2019

EUROCRYPT

Efficient Verifiable Delay Functions
📺 ★
Abstract

We construct a verifiable delay function (VDF). A VDF is a function whose evaluation requires running a given number of sequential steps, yet the result can be efficiently verified. They have applications in decentralised systems, such as the generation of trustworthy public randomness in a trustless environment, or resource-efficient blockchains. To construct our VDF, we actually build a trapdoor VDF. A trapdoor VDF is essentially a VDF which can be evaluated efficiently by parties who know a secret (the trapdoor). By setting up this scheme in a way that the trapdoor is unknown (not even by the party running the setup, so that there is no need for a trusted setup environment), we obtain a simple VDF. Our construction is based on groups of unknown order such as an RSA group, or the class group of an imaginary quadratic field. The output of our construction is very short (the result and the proof of correctness are each a single element of the group), and the verification of correctness is very efficient.

2019

CRYPTO

On the Shortness of Vectors to Be Found by the Ideal-SVP Quantum Algorithm
📺
Abstract

The hardness of finding short vectors in ideals of cyclotomic number fields (hereafter, Ideal-SVP) can serve as a worst-case assumption for numerous efficient cryptosystems, via the average-case problems Ring-SIS and Ring-LWE. For a while, it could be assumed the Ideal-SVP problem was as hard as the analog problem for general lattices (SVP), even when considering quantum algorithms.But in the last few years, a series of works has lead to a quantum algorithm for Ideal-SVP that outperforms what can be done for general SVP in certain regimes. More precisely, it was demonstrated (under certain hypotheses) that one can find in quantum polynomial time a vector longer by a factor at most
$$\alpha = \exp ({\widetilde{O}(n^{1/2})})$$
than the shortest non-zero vector in a cyclotomic ideal lattice, where n is the dimension.In this work, we explore the constants hidden behind this asymptotic claim. While these algorithms have quantum steps, the steps that impact the approximation factor
$$\alpha $$
are entirely classical, which allows us to estimate it experimentally using only classical computing. Moreover, we design heuristic improvements for those steps that significantly decrease the hidden factors in practice. Finally, we derive new provable effective lower bounds based on volumetric arguments.This study allows to predict the crossover point with classical lattice reduction algorithms, and thereby determine the relevance of this quantum algorithm in any cryptanalytic context. For example we predict that this quantum algorithm provides shorter vectors than BKZ-300 (roughly the weakest security level of NIST lattice-based candidates) for cyclotomic rings of rank larger than about 24000.

#### Program Committees

- PKC 2022

#### Coauthors

- Ronald Cramer (1)
- Koen de Boer (1)
- Luca De Feo (2)
- Léo Ducas (3)
- Tako Boris Fouotsa (1)
- Cyprien Delpech de Saint Guilhem (1)
- David Kohel (1)
- Péter Kutas (1)
- Antonin Leroux (2)
- Alice Pellet-Mary (1)
- Christophe Petit (2)
- Maxime Plançon (1)
- Javier Silva (1)