28 research outputs found
On the Ring-LWE and Polynomial-LWE problems
The Ring Learning With Errors problem (RLWE) comes in various forms.
Vanilla RLWE is the decision dual-RLWE variant, consisting in distinguishing from uniform a distribution depending on a secret belonging
to the dual O_K^vee of the ring of integers O_K of a specified number field K.
In primal-RLWE, the secret instead belongs to O_K. Both
decision dual-RLWE and primal-RLWE enjoy search counterparts.
Also widely used is (search/decision) Polynomial Learning With Errors (PLWE),
which is not defined
using a ring of integers O_K of a number field K but
a polynomial ring ZZ[x]/f for a monic
irreducible f in ZZ[x].
We show that there exist reductions between all of these six
problems that incur limited parameter losses.
More precisely: we prove that the (decision/search) dual to
primal reduction from Lyubashevsky et al. [EUROCRYPT~2010]
and Peikert [SCN~2016]
can be implemented with a small error rate growth for all rings
(the resulting reduction is non-uniform polynomial time); we
extend it to polynomial-time reductions between (decision/search)
primal RLWE and PLWE that work for a family
of polynomials f that is exponentially large as a function
of deg f (the resulting reduction is also
non-uniform polynomial time); and we
exploit the recent technique from Peikert et al. [STOC~2017]
to obtain a search to decision reduction for RLWE for arbitrary number fields.
The reductions incur error rate increases that depend
on intrinsic quantities related to K and f
On the condition number of the Vandermonde matrix of the nth cyclotomic polynomial
Recently, Blanco-Chac\'on proved the equivalence between the Ring Learning
With Errors and Polynomial Learning With Errors problems for some families of
cyclotomic number fields by giving some upper bounds for the condition number
of the Vandermonde matrix associated to the
th cyclotomic polynomial. We prove some results on the singular values of
and, in particular, we determine for , where are integers and is an odd prime number
RLWE and PLWE over cyclotomic fields are not equivalent
We prove that the Ring Learning With Errors (RLWE) and the Polynomial
Learning With Errors (PLWE) problems over the cyclotomic field
are not equivalent. Precisely, we show that reducing one
problem to the other increases the noise by a factor that is more than
polynomial in . We do so by providing a lower bound, holding for infinitely
many positive integers , for the condition number of the Vandermonde matrix
of the th cyclotomic polynomial
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.
Trace-based cryptoanalysis of cyclotomic -PLWE for the non-split case
We describe a decisional attack against a version of the PLWE problem in
which the samples are taken from a certain proper subring of large dimension of
the cyclotomic ring with in the case
where but is not totally split over
. Our attack uses the fact that the roots of over
suitable extensions of have zero-trace and has overwhelming
success probability as a function of the number of input samples. An
implementation in Maple and some examples of our attack are also provided.Comment: 19 pages; 1 figure; Major update to previous version due to some
weaknesses detecte
Trace-based cryptanalysis of cyclotomic R_{q,0}xR_q-PLWE for the non-split case
We describe a decisional attack against a version of the PLWE problem
in which the samples are taken from a certain proper subring of large dimension
of the cyclotomic ring Fq[x]/(Φp
k (x)) with k > 1 in the case where q ≡ 1 (mod p)
but Φp
k (x) is not totally split over Fq. Our attack uses the fact that the roots of
Φp
k (x) over suitable extensions of Fq have zero-trace and has overwhelming success
probability as a function of the number of input samples. An implementation in
Maple and some examples of our attack are also provided.Agencia Estatal de InvestigaciónUniversidad de Alcal
Cryptanalysis of PLWE based on zero-trace quadratic roots
We extend two of the attacks on the PLWE problem presented in (Y. Elias, K.
E. Lauter, E. Ozman, and K. E. Stange, Ring-LWE Cryptography for the Number
Theorist, in Directions in Number Theory, E. E. Eischen, L. Long, R. Pries, and
K. E. Stange, eds., vol. 3 of Association for Women in Mathematics Series,
Cham, 2016, Springer International Publishing, pp. 271-290) to a ring
where the irreducible monic polynomial
has an irreducible quadratic factor over
of the form with of suitable multiplicative
order in . Our attack exploits the fact that the trace of the
root is zero and has overwhelming success probability as a function of the
number of samples taken as input. An implementation in Maple and some examples
of our attack are also provided.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:2209.1196
RLWE/PLWE equivalence for the maximal totally real subextension of the 2rpq-th cyclotomic field
We generalise our previous work by giving a polynomial upper
bound on the condition number of certain quasi-Vandermonde matrices to es tablish the equivalence between the RLWE and PLWE problems for the totally
real subfield of the cyclotomic fields of conductor 2r
, 2rp and 2rpq with r ≥ 1
and p, q arbitrary primes. Moreover, we give some cryptographic motivations
for the study of these subfields.Agencia Estatal de Investigació