Ekin Ozman

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 the Decision Ring-LWE problem. 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.

Unramified Brauer Classes on Cyclic Covers of the Projective Plane

Let ( {X} rightarrow mathbb{P}^{2}) be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut( ({X}/mathbb{P}^{2})) arise as pull-backs of certain Brauer classes on ( {rm{k}}(mathbb{P}^{2})) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on ( {rm{k}}(mathbb{P}^{2})) and relate the kernel of the pullback map to the Picard group of X.

Bad Reduction of Genus Three Curves with Complex Multiplication

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes (mathfrak{p}) of M such that the stable reduction of C at (mathfrak{p}) contains three irreducible components of genus 1.

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 [Eisentrager-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.

Non-ordinary curves with a Prym variety of low

If

p

pi: Y to X

-rank

is an unramified double cover of a smooth curve of genus

The Distribution of q-Points on Cyclic ℓ-Covers of Genus g

g

Points on quadratic twists of X₀(N)

, then the Prym variety

A bound on the primes of bad reduction for CM curves of genus 3

P_pi