3,671 research outputs found
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
Learning with Errors over Group Rings Constructed by Semi-direct Product
The Learning with Errors (LWE) problem has been widely utilized as a
foundation for numerous cryptographic tools over the years. In this study, we
focus on an algebraic variant of the LWE problem called Group ring LWE
(GR-LWE). We select group rings (or their direct summands) that underlie
specific families of finite groups constructed by taking the semi-direct
product of two cyclic groups. Unlike the Ring-LWE problem described in
\cite{lyubashevsky2010ideal}, the multiplication operation in the group rings
considered here is non-commutative. As an extension of Ring-LWE, it maintains
computational hardness and can be potentially applied in many cryptographic
scenarios. In this paper, we present two polynomial-time quantum reductions.
Firstly, we provide a quantum reduction from the worst-case shortest
independent vectors problem (SIVP) in ideal lattices with polynomial
approximate factor to the search version of GR-LWE. This reduction requires
that the underlying group ring possesses certain mild properties; Secondly, we
present another quantum reduction for two types of group rings, where the
worst-case SIVP problem is directly reduced to the (average-case) decision
GR-LWE problem. The pseudorandomness of GR-LWE samples guaranteed by this
reduction can be consequently leveraged to construct semantically secure
public-key cryptosystems.Comment: 45 page
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 [Eisentr\"ager-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.Comment: 20 Page
Lattice-Based proof of a shuffle
In this paper we present the first fully post-quantum proof of a shuffle for RLWE encryption schemes. Shuffles are commonly used to construct mixing networks (mix-nets), a key element to ensure anonymity in many applications such as electronic voting systems. They should preserve anonymity even against an attack using quantum computers in order to guarantee long-term privacy. The proof presented in this paper is built over RLWE commitments which are perfectly binding and computationally hiding under the RLWE assumption, thus achieving security in a post-quantum scenario. Furthermore we provide a new definition for a secure mixing node (mix-node) and prove that our construction satisfies this definition.Peer ReviewedPostprint (author's final draft
Ring Learning With Errors: A crossroads between postquantum cryptography, machine learning and number theory
The present survey reports on the state of the art of the different
cryptographic functionalities built upon the ring learning with errors problem
and its interplay with several classical problems in algebraic number theory.
The survey is based to a certain extent on an invited course given by the
author at the Basque Center for Applied Mathematics in September 2018.Comment: arXiv admin note: text overlap with arXiv:1508.01375 by other
authors/ comment of the author: quotation has been added to Theorem 5.
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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