arXiv subjects

Katherine E. Stange

Publications and source records attributed to Katherine E. Stange.

At least 37 records · Page 2Linked to original sources

arXiv2020

Algebraic aspects of solving Ring-LWE, including ring-based improvements in the Blum-Kalai-Wasserman algorithm

We provide a reduction of the Ring-LWE problem to Ring-LWE problems in subrings, in the presence of samples of a restricted form (i.e. $(a,b)$ such that $a$ is restricted to a multiplicative coset of the subring). To create and exploit such restricted samples, we propose Ring-BKW, a version of the Blum-Kalai-Wasserman algorithm which respects the ring structure. Off-the-shelf BKW dimension reduction (including coded-BKW and sieving) can be used for the reduction phase. Its primary advantage is that there is no need for back-substitution, and the solving/hypothesis-testing phase can be parallelized. We also present a method to exploit symmetry to reduce table sizes, samples needed, and runtime during the reduction phase. The results apply to two-power cyclotomic Ring-LWE with parameters proposed for practical use (including all splitting types).

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arXiv2017

Attacks on the Search-RLWE problem with small errors

The Ring Learning-With-Errors (RLWE) problem shows great promise for post-quantum cryptography and homomorphic encryption. We describe a new attack on the non-dual search RLWE problem with small error widths, using ring homomorphisms to finite fields and the chi-squared statistical test. In particular, we identify a "subfield vulnerability" (Section 5.2) and give a new attack which finds this vulnerability by mapping to a finite field extension and detecting non-uniformity with respect to the number of elements in the subfield. We use this attack to give examples of vulnerable RLWE instances in Galois number fields. We also extend the well-known search-to-decision reduction result to Galois fields with any unramified prime modulus q, regardless of the residue degree f of q, and we use this in our attacks. The time complexity of our attack is O(nq2f), where n is the degree of K and f is the residue degree of q in K. We also show an attack on the non-dual (resp. dual) RLWE problem with narrow error distributions in prime cyclotomic rings when the modulus is a ramified prime (resp. any integer). We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks.

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arXiv2017

Security considerations for Galois non-dual RLWE families

We explore further the hardness of the non-dual discrete variant of the Ring-LWE problem for various number rings, give improved attacks for certain rings satisfying some additional assumptions, construct a new family of vulnerable Galois number fields, and apply some number theoretic results on Gauss sums to deduce the likely failure of these attacks for 2-power cyclotomic rings and unramified moduli.

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arXiv2017

A family of monogenic $S_4$ quartic fields arising from elliptic curves

We consider partial torsion fields (fields generated by a root of a division polynomial) for elliptic curves. By analysing the reduction properties of elliptic curves, and applying the Montes Algorithm, we obtain information about the ring of integers. In particular, for the partial $3$-torsion fields for a certain one-parameter family of non-CM elliptic curves, we describe a power basis. As a result, we show that the one-parameter family of quartic $S_4$ fields given by $T^4 - 6T^2 - αT - 3$ for $α\in \mathbb{Z}$ such that $α\pm 8$ are squarefree, are monogenic.

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arXiv2017

Local-global principles in circle packings

We generalize work of Bourgain-Kontorovich and Zhang, proving an almost local-to-global property for the curvatures of certain circle packings, to a large class of Kleinian groups. Specifically, we associate in a natural way an infinite family of integral packings of circles to any Kleinian group $\mathcal A\leq\textrm{PSL}_2(K)$ satisfying certain conditions, where $K$ is an imaginary quadratic field, and show that the curvatures of the circles in any such packing satisfy an almost local-to-global principle. A key ingredient in the proof of this is that $\mathcal A$ possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in $\textrm{PSL}_2(\mathcal{O}_K)$ containing a Zariski dense subgroup of $\textrm{PSL}_2(\mathbb{Z})$.

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arXiv2017

The Dynamics of Super-Apollonian Continued Fractions

We examine a pair of dynamical systems on the plane induced by a pair of spanning trees in the Cayley graph of the Super-Apollonian group of Graham, Lagarias, Mallows, Wilks and Yan. The dynamical systems compute Gaussian rational approximations to complex numbers and are "reflective" versions of the complex continued fractions of A. L. Schmidt. They also describe a reduction algorithm for Lorentz quadruples, in analogy to work of Romik on Pythagorean triples. For these dynamical systems, we produce an invertible extension and an invariant measure, which we conjecture is ergodic. We consider some statistics of the related continued fraction expansions, and we also examine the restriction of these systems to the real line, which gives a reflective version of the usual continued fraction algorithm. Finally, we briefly consider an alternate setup corresponding to a tree of Lorentz quadruples ordered by arithmetic complexity.

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arXiv2017

Visualising the arithmetic of imaginary quadratic fields

We study the orbit of $\mathbb{R}$ under the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$, where $K$ is an imaginary quadratic field. The orbit, called a Schmidt arrangement $\mathcal{S}_K$, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. This paper presents several examples of this phenomenon. First, we show that the curvatures of the circles are integer multiples of $\sqrt{-Δ}$ and describe the curvatures of tangent circles in terms of the norm form of $\mathcal{O}_K$. Second, we show that the circles themselves are in bijection with certain ideal classes in orders of $\mathcal{O}_K$, the conductor being a certain multiple of the curvature. This allows us to count circles with class numbers. Third, we show that the arrangement of circles is connected if and only if $\mathcal{O}_K$ is Euclidean. These results are meant as foundational for a study of a new class of thin groups generalising Apollonian groups, in a companion paper.

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arXiv2016

Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$

Let $ϕ(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(ϕ^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $ϕ$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \mid ϕ^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes is finite when $d$ is even, and conjecture that it is infinite when $d$ is odd.

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arXiv2016

Arithmetic properties of the Frobenius traces defined by a rational abelian variety (with two appendices by J-P. Serre)

Let $A$ be an abelian variety over $\mathbb{Q}$ of dimension $g$ such that the image of its associated absolute Galois representation $ρ_A$ is open in $\operatorname{GSp}_{2g}(\hat{\mathbb{Z}})$. We investigate the arithmetic of the traces $a_{1, p}$ of the Frobenius at $p$ in $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ under $ρ_A$, modulo varying primes $p$. In particular, we obtain upper bounds for the counting function $\#\{p \leq x: a_{1, p} = t\}$ and we prove an Erdös-Kac type theorem for the number of prime factors of $a_{1, p}$. We also formulate a conjecture about the asymptotic behaviour of $\#\{p \leq x: a_{1, p} = t\}$, which generalizes a well-known conjecture of S. Lang and H. Trotter from 1976 about elliptic curves.

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arXiv2015

The Apollonian structure of Bianchi groups

We study the orbit of $\widehat{\mathbb{R}}$ under the Möbius action of the Bianchi group $\operatorname{PSL}_2(\mathcal{O}_K)$ on $\widehat{\mathbb{C}}$, where $\mathcal{O}_K$ is the ring of integers of an imaginary quadratic field $K$. The orbit $\mathcal{S}_K$, called a Schmidt arrangement, is a geometric realisation, as an intricate circle packing, of the arithmetic of $K$. We give a simple geometric characterisation of certain subsets of $\mathcal{S}_K$ generalizing Apollonian circle packings, and show that $\mathcal{S}_K$, considered with orientations, is a disjoint union of all primitive integral such $K$-Apollonian packings. These packings are described by a new class of thin groups of arithmetic interest called $K$-Apollonian groups. We make a conjecture on the curvatures of these packings, generalizing the local-to-global conjecture for Apollonian circle packings.

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arXiv2015

Ring-LWE Cryptography for the Number Theorist

In this paper, we survey the status of attacks on the ring and polynomial learning with errors problems (RLWE and PLWE). Recent work on the security of these problems [Eisenträger-Hallgren-Lauter, Elias-Lauter-Ozman-Stange] gives rise to interesting questions about number fields. We extend these attacks and survey related open problems in number theory, including spectral distortion of an algebraic number and its relationship to Mahler measure, the monogenic property for the ring of integers of a number field, and the size of elements of small order modulo q.

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arXiv2015

Provably weak instances of Ring-LWE

The ring and polynomial learning with errors problems (Ring-LWE and Poly-LWE) have been proposed as hard problems to form the basis for cryptosystems, and various security reductions to hard lattice problems have been presented. So far these problems have been stated for general (number) rings but have only been closely examined for cyclotomic number rings. In this paper, we state and examine the Ring-LWE problem for general number rings and demonstrate provably weak instances of Ring-LWE. We construct an explicit family of number fields for which we have an efficient attack. We demonstrate the attack in both theory and practice, providing code and running times for the attack. The attack runs in time linear in q, where q is the modulus. Our attack is based on the attack on Poly-LWE which was presented in [Eisenträger-Hallgren-Lauter]. We extend the EHL-attack to apply to a larger class of number fields, and show how it applies to attack Ring-LWE for a heuristically large class of fields. Certain Ring-LWE instances can be transformed into Poly-LWE instances without distorting the error too much, and thus provide the first weak instances of the Ring-LWE problem. We also provide additional examples of fields which are vulnerable to our attacks on Poly-LWE, including power-of-$2$ cyclotomic fields, presented using the minimal polynomial of $ζ_{2^n} \pm 1$.

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arXiv2014

Integral points on elliptic curves and explicit valuations of division polynomials

Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant C such that for any elliptic curve E/Q and non-torsion point P in E(Q), there is at most one integral multiple [n]P such that n > C. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence of valuations of the division polynomials. For P of non-singular reduction, such sequences are already well described in most cases, but for P of singular reduction, we are led to define a new class of sequences called elliptic troublemaker sequences, which measure the failure of the Neron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on h(P)/h(E) for integer points having two large integral multiples.

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arXiv2014

An arborist's guide to the rationals

There are two well-known ways to enumerate the positive rational numbers in an infinite binary tree: the Farey/Stern-Brocot tree and the Calkin-Wilf tree. In this brief note, we describe these two trees as `transpose shadows' of a tree of matrices (a result due to Backhouse and Ferreira) via a new proof using yet another famous tree of rationals: the topograph of Conway and Fung.

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arXiv2014

The sensual Apollonian circle packing

The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway.

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arXiv2012

A duality principle for selection games

A dinner table seats k guests and holds n discrete morsels of food. Guests select morsels in turn until all are consumed. Each guest has a ranking of the morsels according to how much he would enjoy eating them; these rankings are commonly known. A gallant knight always prefers one food division over another if it provides strictly more enjoyable collections of food to one or more other players (without giving a less enjoyable collection to any other player) even if it makes his own collection less enjoyable. A boorish lout always selects the morsel that gives him the most enjoyment on the current turn, regardless of future consumption by himself and others. We show the way the food is divided when all guests are gallant knights is the same as when all guests are boorish louts but turn order is reversed. This implies and generalizes a classical result of Kohler and Chandrasekaran (1971) about two players strategically maximizing their own enjoyments. We also treat the case that the table contains a mixture of boorish louts and gallant knights. Our main result can also be formulated in terms of games in which selections are made by groups. In this formulation, the surprising fact is that a group can always find a selection that is simultaneously optimal for each member of the group.

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arXiv2011

Algebraic divisibility sequences over function fields

We study the existence of primes and of primitive divisors in classical divisibility sequences defined over function fields. Under various hypotheses, we prove that Lucas sequences and elliptic divisibility sequences over function fields defined over number fields contain infinitely many irreducible elements. We also prove that an elliptic divisibility sequence over a function field has only finitely many terms lacking a primitive divisor.

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arXiv2011

How to make the most of a shared meal: plan the last bite first

If you are sharing a meal with a companion, how best to make sure you get your favourite mouthfuls? Ethiopian Dinner is a game in which two players take turns eating morsels from a common plate. Each morsel comes with a pair of utility values measuring its tastiness to the two players. Kohler and Chandrasekaharan discovered a good strategy -- a subgame perfect equilibrium, to be exact -- for this game. We give a new visual proof of their result. The players arrive at the equilibrium by figuring out their last move first and working backward. We conclude that it's never too early to start thinking about dessert.

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