592 research outputs found
Efficiently processing complex-valued data in homomorphic encryption
We introduce a new homomorphic encryption scheme that is natively capable of computing with complex numbers. This is done by generalizing recent work of Chen, Laine, Player and Xia, who modified the Fan–Vercauteren scheme by replacing the integral plaintext modulus t by a linear polynomial X − b. Our generalization studies plaintext moduli of the form Xm + b. Our construction significantly reduces the noise growth in comparison to the original FV scheme, so much deeper arithmetic circuits can be homomorphically executed
An Octanomial Model for Cubic Surfaces
We present a new normal form for cubic surfaces that is well suited for
p-adic geometry, as it reveals the intrinsic del Pezzo combinatorics of the 27
trees in the tropicalization. The new normal form is a polynomial with eight
terms, written in moduli from the E6 hyperplane arrangement. If such a surface
is tropically smooth then its 27 tropical lines are distinct. We focus on
explicit computations, both symbolic and p-adic numerical.Comment: 20 pages; clarified exposition at key points; final versio
Series expansions of the percolation probability for directed square and honeycomb lattices
We have derived long series expansions of the percolation probability for
site and bond percolation on directed square and honeycomb lattices. For the
square bond problem we have extended the series from 41 terms to 54, for the
square site problem from 16 terms to 37, and for the honeycomb bond problem
from 13 terms to 36. Analysis of the series clearly shows that the critical
exponent is the same for all the problems confirming expectations of
universality. For the critical probability and exponent we find in the square
bond case, , , in the
square site case , ,
and in the honeycomb bond case , . In addition we have obtained accurate estimates for the critical
amplitudes. In all cases we find that the leading correction to scaling term is
analytic, i.e., the confluent exponent .Comment: LaTex with epsf, 26 pages, 2 figures and 2 tables in Postscript
format included (uufiled). LaTeX version of tables also included for the
benefit of those without access to PS printers (note that the tables should
be printed in landscape mode). Accepted by J. Phys.
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