arXiv subjects

Frederik Vercauteren

Publications and source records attributed to Frederik Vercauteren.

5 recordsLinked to original sources

arXiv2024

Generalized class group actions on oriented elliptic curves with level structure

We study a large family of generalized class groups of imaginary quadratic orders $O$ and prove that they act freely and (essentially) transitively on the set of primitively $O$-oriented elliptic curves over a field $k$ (assuming this set is non-empty) equipped with appropriate level structure. This extends, in several ways, a recent observation due to Galbraith, Perrin and Voloch for the ray class group. We show that this leads to a reinterpretation of the action of the class group of a suborder $O' \subseteq O$ on the set of $O'$-oriented elliptic curves, discuss several other examples, and briefly comment on the hardness of the corresponding vectorization problems.

math.NT↗

arXiv2023

BASALISC: Programmable Hardware Accelerator for BGV Fully Homomorphic Encryption

Fully Homomorphic Encryption (FHE) allows for secure computation on encrypted data. Unfortunately, huge memory size, computational cost and bandwidth requirements limit its practicality. We present BASALISC, an architecture family of hardware accelerators that aims to substantially accelerate FHE computations in the cloud. BASALISC is the first to implement the BGV scheme with fully-packed bootstrapping -- the noise removal capability necessary for arbitrary-depth computation. It supports a customized version of bootstrapping that can be instantiated with hardware multipliers optimized for area and power. BASALISC is a three-abstraction-layer RISC architecture, designed for a 1 GHz ASIC implementation and underway toward 150mm2 die tape-out in a 12nm GF process. BASALISC's four-layer memory hierarchy includes a two-dimensional conflict-free inner memory layer that enables 32 Tb/s radix-256 NTT computations without pipeline stalls. Its conflict-resolution permutation hardware is generalized and re-used to compute BGV automorphisms without throughput penalty. BASALISC also has a custom multiply-accumulate unit to accelerate BGV key switching. The BASALISC toolchain comprises a custom compiler and a joint performance and correctness simulator. To evaluate BASALISC, we study its physical realizability, emulate and formally verify its core functional units, and we study its performance on a set of benchmarks. Simulation results show a speedup of more than 5,000 times over HElib -- a popular software FHE library.

cs.CR↗

arXiv2022

On the decisional Diffie-Hellman problem for class group actions on oriented elliptic curves

We show how the Weil pairing can be used to evaluate the assigned characters of an imaginary quadratic order $\mathcal{O}$ in an unknown ideal class $[\mathfrak{a}] \in \mathrm{Cl}(\mathcal{O})$ that connects two given $\mathcal{O}$-oriented elliptic curves $(E, ι)$ and $(E', ι') = [\mathfrak{a}](E, ι)$. When specialized to ordinary elliptic curves over finite fields, our method is conceptually simpler and often somewhat faster than a recent approach due to Castryck, Sotáková and Vercauteren, who rely on the Tate pairing instead. The main implication of our work is that it breaks the decisional Diffie-Hellman problem for practically all oriented elliptic curves that are acted upon by an even-order class group. It can also be used to better handle the worst cases in Wesolowski's recent reduction from the vectorization problem for oriented elliptic curves to the endomorphism ring problem, leading to a method that always works in sub-exponential time.

math.NT↗

arXiv2021

Quantum Equivalence of the DLP and CDHP for Group Actions

In this short note we give a polynomial-time quantum reduction from the vectorization problem (DLP) to the parallelization problem (CDHP) for group actions. Combined with the trivial reduction from par-allelization to vectorization, we thus prove the quantum equivalence of both problems.

cs.CR↗

arXiv2007

Computing Zeta Functions of Nondegenerate Curves

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over F_{p^n}, the expected running time is O(n^3g^6 + n^2g^{6.5}), whereas the space complexity amounts to O(n^3g^4), assuming p is fixed.

math.NT↗