arXiv subjects

Benjamin Smith

Publications and source records attributed to Benjamin Smith.

At least 55 records · Page 3Linked to original sources

arXiv2020

Anomaly Detection with SDAE

Anomaly detection is a prominent data preprocessing step in learning applications for correction and/or removal of faulty data. Automating this data type with the use of autoencoders could increase the quality of the dataset by isolating anomalies that were missed through manual or basic statistical analysis. A Simple, Deep, and Supervised Deep Autoencoder were trained and compared for anomaly detection over the ASHRAE building energy dataset. Given the restricted parameters under which the models were trained, the Deep Autoencoder perfoms the best, however, the Supervised Deep Autoencoder outperforms the other models in total anomalies detected when considerations for the test datasets are given.

cs.LG↗

arXiv2020

Faster computation of isogenies of large prime degree

Let $\mathcal{E}/\mathbb{F}_q$ be an elliptic curve, and $P$ a point in $\mathcal{E}(\mathbb{F}_q)$ of prime order $\ell$. Vélu's formulae let us compute a quotient curve $\mathcal{E}' = \mathcal{E}/\langle{P}\rangle$ and rational maps defining a quotient isogeny $ϕ: \mathcal{E} \to \mathcal{E}'$ in $\tilde{O}(\ell)$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is uniform in $q$.This article shows how to compute $\mathcal{E}'$, and $ϕ(Q)$ for $Q$ in $\mathcal{E}(\mathbb{F}_q)$, using only $\tilde{O}(\sqrt{\ell})$ $\mathbb{F}_q$-operations, where the $\tilde{O}$ is again uniform in $q$.As an application, this article speeds up some computations used in the isogeny-based cryptosystems CSIDH and CSURF.

cs.CR↗

arXiv2020

The supersingular isogeny problem in genus 2 and beyond

Let $A/\overline{\mathbb{F}}\_p$ and $A'/\overline{\mathbb{F}}\_p$ be supersingular principally polarized abelian varieties of dimension $g>1$. For any prime $\ell \ne p$, we give an algorithm that finds a path $ϕ\colon A \rightarrow A'$ in the $(\ell, \dots , \ell)$-isogeny graph in $\widetilde{O}(p^{g-1})$ group operations on a classical computer, and $\widetilde{O}(\sqrt{p^{g-1}})$ calls to the Grover oracle on a quantum computer. The idea is to find paths from $A$ and $A'$ to nodes that correspond to products of lower dimensional abelian varieties, and to recurse down in dimension until an elliptic path-finding algorithm (such as Delfs--Galbraith) can be invoked to connect the paths in dimension $g=1$. In the general case where $A$ and $A'$ are any two nodes in the graph, this algorithm presents an asymptotic improvement over all of the algorithms in the current literature. In the special case where $A$ and $A'$ are a known and relatively small number of steps away from each other (as is the case in higher dimensional analogues of SIDH), it gives an asymptotic improvement over the quantum claw finding algorithms and an asymptotic improvement over the classical van Oorschot--Wiener algorithm.

cs.CR↗

arXiv2019

Pre- and post-quantum Diffie-Hellman from groups, actions, and isogenies

Diffie-Hellman key exchange is at the foundations of public-key cryptography, but conventional group-based Diffie-Hellman is vulnerable to Shor's quantum algorithm. A range of "post-quantum Diffie-Hellman" protocols have been proposed to mitigate this threat, including the Couveignes, Rostovtsev-Stolbunov, SIDH, and CSIDH schemes, all based on the combinatorial and number-theoretic structures formed by isogenies of elliptic curves. Pre-and post-quantum Diffie-Hellman schemes resemble each other at the highest level, but the further down we dive, the more differences emerge-differences that are critical when we use Diffie-Hellman as a basic component in more complicated constructions. In this survey we compare and contrast pre-and post-quantum Diffie-Hellman algorithms, highlighting some important subtleties.

cs.CR↗

arXiv2019

Stronger and Faster Side-Channel Protections for CSIDH

CSIDH is a recent quantum-resistant primitive based on the difficulty of finding isogeny paths between supersingular curves. Recently, two constant-time versions of CSIDH have been proposed: first by Meyer, Campos and Reith, and then by Onuki, Aikawa, Yamazaki and Takagi. While both offer protection against timing attacks and simple power consumption analysis, they are vulnerable to more powerful attacks such as fault injections. In this work, we identify and repair two oversights in these algorithms that compromised their constant-time character. By exploiting Edwards arithmetic and optimal addition chains, we produce the fastest constant-time version of CSIDH to date. We then consider the stronger attack scenario of fault injection, which is relevant for the security of CSIDH static keys in embedded hardware. We propose and evaluate a dummy-free CSIDH algorithm. While these CSIDH variants are slower, their performance is still within a small constant factor of less-protected variants. Finally, we discuss derandomized CSIDH algorithms.

cs.CR↗

arXiv2019

Hash functions from superspecial genus-2 curves using Richelot isogenies

Last year Takashima proposed a version of Charles, Goren and Lauter's hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field Fp2. In a very recent paper Flynn and Ti point out that Takashima's hash function is insecure due to the existence of small isogeny cycles. We revisit the construction and show that it can be repaired by imposing a simple restriction, which moreover clarifies the security analysis. The runtime of the resulting hash function is dominated by the extraction of 3 square roots for every block of 3 bits of the message, as compared to one square root per bit in the elliptic curve case; however in our setting the extractions can be parallelized and are done in a finite field whose bit size is reduced by a factor 3. Along the way we argue that the full supersingular isogeny graph is the wrong context in which to study higher-dimensional analogues of Charles, Goren and Lauter's hash function, and advocate the use of the superspecial subgraph, which is the natural framework in which to view Takashima's Fp2-friendly starting curve.

cs.CR↗

arXiv2018

Towards practical key exchange from ordinary isogeny graphs

We revisit the ordinary isogeny-graph based cryptosystems of Couveignes and Rostovtsev-Stolbunov, long dismissed as impractical. We give algorithmic improvements that accelerate key exchange in this framework, and explore the problem of generating suitable system parameters for contemporary pre-and post-quantum security that take advantage of these new algorithms. We also prove the session-key security of this key exchange in the Canetti-Krawczyk model, and the IND-CPA security of the related public-key encryption scheme, under reasonable assumptions on the hardness of computing isogeny walks. Our systems admit efficient key-validation techniques that yield CCA-secure encryp-tion, thus providing an important step towards efficient post-quantum non-interactive key exchange (NIKE).

cs.CR↗

arXiv2017

A new version of the CABLE land surface model, incorporating land-use change, woody vegetation demography and a novel optimisation-based approach to plant coordination of photosynthesis

CABLE is a land surface model (LSM) that can be applied stand-alone, as well as providing for land surface-atmosphere exchange within the Australian Community Climate and Earth System Simulator (ACCESS). We describe critical new developments that extend the applicability of CABLE for regional and global carbon-climate simulations, accounting for vegetation response to biophysical and anthropogenic forcings. A land-use and land-cover change module, driven by gross land-use transitions and wood harvest area was implemented, tailored to the needs of the Coupled Model Intercomparison Project-6 (CMIP6). Novel aspects include the treatment of secondary woody vegetation, which benefits from a tight coupling between the land-use module and the Population Orders Physiology (POP) module for woody demography and disturbance-mediated landscape heterogeneity. Land-use transitions and harvest associated with secondary forest tiles modify the annually-resolved patch age distribution within secondary-vegetated tiles, in turn affecting biomass accumulation and turnover rates and hence the magnitude of the secondary forest sink. Additionally, we implemented a novel approach to constrain modelled GPP consistent with the Co-ordination Hypothesis, predicted by evolutionary theory, which suggests that electron transport and Rubisco-limited rates adjust seasonally and across biomes to be co-limiting. We show that the default prior assumption - common to CABLE and other LSMs - of a fixed ratio of electron transport to carboxylation capacity is at odds with this hypothesis, and implement an alternative algorithm for dynamic optimisation of this ratio, such that co-ordination is achieved as an outcome of fitness maximisation. Results have significant implications the magnitude of the simulated CO2 fertilisation effect on photosynthesis in comparison to alternative estimates and observational proxies.

q-bio.PE↗

arXiv2017

qDSA: Small and Secure Digital Signatures with Curve-based Diffie--Hellman Key Pairs

qDSA is a high-speed, high-security signature scheme that facilitates implementations with a very small memory footprint, a crucial requirement for embedded systems and IoT devices, and that uses the same public keys as modern Diffie--Hellman schemes based on Montgomery curves (such as Curve25519) or Kummer surfaces. qDSA resembles an adaptation of EdDSA to the world of Kummer varieties, which are quotients of algebraic groups by $\pm$1. Interestingly, qDSA does not require any full group operations or point recovery: all computations, including signature verification, occur on the quotient where there is no group law. We include details on four implementations of qDSA, using Montgomery and fast Kummer surface arithmetic on the 8-bit AVR ATmega and 32-bit ARM Cortex M0 platforms. We find that qDSA significantly outperforms state-of-the-art signature implementations in terms of stack usage and code size. We also include an efficient compression algorithm for points on fast Kummer surfaces, reducing them to the same size as compressed elliptic curve points for the same security level.

cs.CR↗

arXiv2017

Isogenies for point counting on genus two hyperelliptic curves with maximal real multiplication

Schoof's classic algorithm allows point-counting for elliptic curves over finite fields in polynomial time. This algorithm was subsequently improved by Atkin, using factorizations of modular polynomials, and by Elkies, using a theory of explicit isogenies. Moving to Jacobians of genus-2 curves, the current state of the art for point counting is a generalization of Schoof's algorithm. While we are currently missing the tools we need to generalize Elkies' methods to genus 2, recently Martindale and Milio have computed analogues of modular polynomials for genus-2 curves whose Jacobians have real multiplication by maximal orders of small discriminant. In this article, we prove Atkin-style results for genus-2 Jacobians with real multiplication by maximal orders, with a view to using these new modular polynomials to improve the practicality of point-counting algorithms for these curves.

math.NT↗

arXiv2017

Montgomery curves and their arithmetic

Three decades ago, Montgomery introduced a new elliptic curve model for use in Lenstra's ECM factorization algorithm. Since then, his curves and the algorithms associated with them have become foundational in the implementation of elliptic curve cryptosystems. This article surveys the theory and cryptographic applications of Montgomery curves over non-binary finite fields, including Montgomery's x-only arithmetic and Ladder algorithm, x-only Diffie--Hellman, y-coordinate recovery, and 2-dimensional and Euclidean differential addition chains such as Montgomery's PRAC algorithm.

cs.CR↗

arXiv2017

Factoring Safe Semiprimes with a Single Quantum Query

Shor's factoring algorithm (SFA), by its ability to efficiently factor large numbers, has the potential to undermine contemporary encryption. At its heart is a process called order finding, which quantum mechanics lets us perform efficiently. SFA thus consists of a \emph{quantum order finding algorithm} (QOFA), bookended by classical routines which, given the order, return the factors. But, with probability up to $1/2$, these classical routines fail, and QOFA must be rerun. We modify these routines using elementary results in number theory, improving the likelihood that they return the factors. The resulting quantum factoring algorithm is better than SFA at factoring safe semiprimes, an important class of numbers used in cryptography. With just one call to QOFA, our algorithm almost always factors safe semiprimes. As well as a speed-up, improving efficiency gives our algorithm other, practical advantages: unlike SFA, it does not need a randomly picked input, making it simpler to construct in the lab; and in the (unlikely) case of failure, the same circuit can be rerun, without modification. We consider generalizing this result to other cases, although we do not find a simple extension, and conclude that SFA is still the best algorithm for general numbers (non safe semiprimes, in other words). Even so, we present some simple number theoretic tricks for improving SFA in this case.

quant-ph↗

arXiv2016

Computing cardinalities of Q-curve reductions over finite fields

We present a specialized point-counting algorithm for a class of elliptic curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo inert primes and, more generally, any elliptic curve over F\_{p^2} with a low-degree isogeny to its Galois conjugate curve. These curves have interesting cryptographic applications. Our algorithm is a variant of the Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree endomorphism in place of Frobenius. While it has the same asymptotic asymptotic complexity as SEA, our algorithm is much faster in practice.

math.NT↗

arXiv2016

$μ$Kummer: efficient hyperelliptic signatures and key exchange on microcontrollers

We describe the design and implementation of efficient signature and key-exchange schemes for the AVR ATmega and ARM Cortex M0 microcontrollers, targeting the 128-bit security level. Our algorithms are based on an efficient Montgomery ladder scalar multiplication on the Kummer surface of Gaudry and Schost's genus-2 hyperelliptic curve, combined with the Jacobian point recovery technique of Costello, Chung, and Smith. Our results are the first to show the feasibility of software-only hyperelliptic cryptography on constrained platforms, and represent a significant improvement on the elliptic-curve state-of-the-art for both key exchange and signatures on these architectures. Notably, our key-exchange scalar-multiplication software runs in under 9740k cycles on the ATmega, and under 2650k cycles on the Cortex M0.

cs.CR↗

arXiv2015

Fast, uniform, and compact scalar multiplication for elliptic curves and genus 2 Jacobians with applications to signature schemes

We give a general framework for uniform, constant-time one-and two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the x-line or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper "signed" output back on the curve or Jacobian. This extends the work of L{ó}pez and Dahab, Okeya and Sakurai, and Brier and Joye to genus 2, and also to two-dimensional scalar multiplication. Our results show that many existing fast pseudomultiplication implementations (hitherto limited to applications in Diffie--Hellman key exchange) can be wrapped with simple and efficient pre-and post-computations to yield competitive full scalar multiplication algorithms, ready for use in more general discrete logarithm-based cryptosystems, including signature schemes. This is especially interesting for genus 2, where Kummer surfaces can outperform comparable elliptic curve systems. As an example, we construct an instance of the Schnorr signature scheme driven by Kummer surface arithmetic.

math.NT↗

arXiv2015

Distributions of traces of Frobenius for smooth plane curves over finite fields

In a previous article, we obtained data on the distribution of traces of Frobenius of non-hyperelliptic genus $3$ curves over small finite fields. In the present one, we give a heuristic explanation of these data, by extrapolating from results on the distribution of traces of Frobenius for plane curves whose degree is small with respect to the cardinality of their finite base field. In particular, our methods shed some new light on the asymmetry of the distribution around its mean value, which is related to the Serre obstruction.

math.AG↗

arXiv2015

The Q-curve construction for endomorphism-accelerated elliptic curves

We give a detailed account of the use of $\mathbb{Q}$-curve reductions to construct elliptic curves over $\mathbb{F}\_{p^2}$ with efficiently computable endomorphisms, which can be used to accelerate elliptic curve-based cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case of our construction), we offer the advantage over GLV of selecting from a much wider range of curves, and thus finding secure group orders when $p$ is fixed for efficient implementation. Unlike GLS, we also offer the possibility of constructing twist-secure curves. We construct several one-parameter families of elliptic curves over $\mathbb{F}\_{p^2}$ equipped with efficient endomorphisms for every $p \textgreater{} 3$, and exhibit examples of twist-secure curves over $\mathbb{F}\_{p^2}$ for the efficient Mersenne prime $p = 2^{127}-1$.

cs.CR↗

arXiv2013

Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians

The first step in elliptic curve scalar multiplication algorithms based on scalar decompositions using efficient endomorphisms-including Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as well as higher-dimensional and higher-genus constructions-is to produce a short basis of a certain integer lattice involving the eigenvalues of the endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar coefficients, and the faster the resulting scalar multiplication. Typically, knowledge of the eigenvalues allows us to write down a long basis, which we then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more specialized algorithm. In this work, we use elementary facts about quadratic rings to immediately write down a short basis of the lattice for the GLV, GLS, GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real multiplication constructions. We do not pretend that this represents a significant optimization in scalar multiplication, since the lattice reduction step is always an offline precomputation---but it does give a better insight into the structure of scalar decompositions. In any case, it is always more convenient to use a ready-made short basis than it is to compute a new one.

math.NT↗